Logical Localism in the Context of Combining Logics

Logical Localism in the Context of Combining Logics Carlos Benito-Monsalvo Aquesta tesi doctoral està subjecta a la llicència Reconeixement 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution 4.0. Spain License. Tesi doctoral Logical Localism in the Context of Combining Logics Autor/a: Carlos Benito-Monsalvo Director/a: Dr. José Martínez Fernández Director/a: Dr. Elia Zardini Tutor/a: Dr. Manuel García-Carpintero Programa de Doctorat: Ciencia Cognitiva y Lenguaje Facultat de Filosofia Enero de 2023 Logical Localism in the Context of Combining Logics Carlos Benito-Monsalvo A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy Supervisors: Dr. José Martínez Fernández Dr. Elia Zardini PhD program: Cognitive Science and Language University of Barcelona Philosophy Department Carlos Benito-Monsalvo: Logical Localism in the Context of Combining Logics, 2023 SUPERVISORS: Dr. José Martínez Fernández and Dr. Elia Zardini TUTOR: Dr. Manuel García-Carpintero This dissertation has been possible thanks to the FI-DGR-EMC/2199/2017 grant of the Generalitat de Catalunya; the FPU(18/06283) grant of the Spanish Ministry of Universities; the mobility grant (EST21/00505) for beneficiaries of the Training Programme for Academic Staff (FPU); the project on Localism and Globalism in Logics and Semantics (FFI2015-70707-P) from the Spanish Ministry of Economy and Competitiveness and the project ‘Worlds and Truth Values: Challenges to Formal Semantics (MUNVAL)’(2019PIDPID-107667GB-I00) from the Spanish Ministry of Science and Innovation. Abstract Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logicsthat go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems. i Acknowledgements I am quite attracted to the idea that all intelligence is collective intelligence, in the sense that there is no such thing as an indivisible unit of intelligence that we can pinpoint. So, although the neurons of my brain have been the ones struggling to fire in the right paths in order to ultimately produce this dissertation, other neurons from other brains have certainly been of an invaluable help for the correct firings to occur. I am, therefore, thankful to everyone who, in some way or another, interacted with me. Among those interactions, the first person that showed me a true passion for logic and philosophy was Jesus Mari Larrazabal. He was my logic teacher and my bachelor’s thesis supervisor, but, most of all, he was a mentor and a friend. I cannot express how indebted I am to him. I just would like to leave, to whom it may concern, a glimpse of his mind: ‘Philosophy is an intellectual activity that will often lead you to loneliness, but this loneliness is necessary if you want to think carefully, freely, and not be affected by all the mess. However, you must know that you are not alone and that other people also deeply care about the same things as you do. Ultimately, the owl of Minerva always spreads its wings with the falling of the dusk to bring order to chaos’. Hopefully, this and other future works will be a worthy testament of his intellectual honesty and passion. A clear proof of the influence that Jesus Mari had on me and my career is that he introduced me to my supervisors, José Martínez and Elia Zardini. I met them in a workshop that Jesus Mari was organizing at Donostia. It was my last year as an undergraduate and I was in the process of deciding what to do next. Since Jesus Mari knew them both and highly appreciated and respected them, he encouraged me to go and talk to them and seek their advice. I ended up studying the master in Analytic Philosophy, where Pepe taught me the course on Philosophical Logic and supervised my Master’s Thesis, with an eye already on a PhD. He introduced me to every topic that ii this dissertation is concerned with and has guided me through every thread to get at where I am, while giving me enough freedom. I am deeply thankful for all this. Also, for the time, effort and rigour that he has dedicated to improving this dissertation and for giving me the confidence on myself and my work when I most needed it. I feel lucky to have had the opportunity of working together with someone whom I have admired since he was my teacher. Elia became my co-supervisor right on time, but I certainly wish he would have joined us before. Since the first talk that I witnessed at Donostia, I have thought that he is one of the sharpest and most brilliant philosophers that I know. Very few other times I have met someone who is able to offer you some helpful and deep insights in almost any topic, from cuisine to philosophical logic. When Pepe informed me about the possibility of having Elia as a supervisor I was really excited, but also a little bit intimidated. In the end, he has been an indispensable part of the last period of my work, when most of the coolest ideas have originated. Not only has he given me outstanding insights to improve my research, but also has had the most kind and supportive words to inspire me at the lowest points of the process. Coming from him, they meant the world to me. There are many other people from the academia that have been important in all these years. I would like to thank every member of Logos for being so close and kind. It has been an honour to be part of the best research group in analytic philosophy and be surrounded by the best minds in the field. I especially thank Andrea Rivadulla, with whom I started the process of the PhD after we finished the master. I know she has a brilliant future ahead as a researcher if she wants to. I would also like to mention and thank the people that have participated in the Philosophical Logic reading group organized by Pepe. Obviously, many of the papers that we read and discussed were related to my research, so I have greatly benefited from our discussions. I am particularly grateful to Sergi Oms, Sven Rosenkranz, Niccolò Rossi and Pilar Terrés. During my PhD I completed a research stay at the Buenos Aires Logic Group. Given the situation with COVID-19, it was quite difficult to make the stay possible and I had to postpone it almost two years. But, finally, at the end of February 2022 I could travel to Argentina. There were many reasons for choosing this particular research group. First of all, they are an outstanding team, formed by very talented and brilliant researchers who are able to systematically publish in the top level international journals like a iii well-oiled machine. Second, I was lucky enough to know some of the members of the group before my stay. I met Damian Szmuc at the ‘PhDs in Logic’ celebrated in Prague and had him afterwards in the tiny guest room of my flat at Barcelona (sorry for that). I also knew Lucas Rosenblatt, who was at Logos when I joined and left shortly after to return home, and Eduardo Barrio, whom I had met in a workshop that Pepe organized. Damian and Eduardo were very supportive and helpful with all the paperwork previous to the stay and, most importantly, keeping the hope that I could finally travel. I am deeply grateful for that. Lastly, let me be completely honest, I wanted to travel to Argentina to climb and trek at Patagonia, which I definitely did. Those months with the Buenos Aires Logic Group were a gift and completely surpassed my expectations. I want to thank all the members of the group for making me feel at home and for all the great discussions which were a true inspiration for my work. Special thanks go to Eduardo, Damian, Lucas, Natalia Buacar and Paula Teijeiro. I cannot thank Mariela Rubin (Maru) enough for everything she did for me. You are my favourite person in the American continent. I have had the chance of presenting parts of my dissertation in various venues. I would like to thank the people that have attended and helped me with questions or comments. Among all, let me mention Pablo Cobreros, who might not know that he has always given me wise advise whenever we have encountered and Antonio Yuste-Ginel who has become a friend with whom I can share philosophy, logic, climbing and beers. Of course, there is life outside the academia too and most of the people that are essential to me and to what I am are outsiders. I am deeply grateful to many of my friends: Adur, Raúl, Fynn, Álvaro, Andrea, Gonzalo, Laura and Guillermo. Also to the friends of my partner who have welcomed me and become friends of mine too: Alejandro, María, Mario, Sandra, Mauro, Laura and Sara. And to the ones who know me since I was a 5 years old kid: Iñaki Arbide, Julen Garmendia and Julen Arruti. The other day I realized that you have never asked me what my thesis was about, and I am truly grateful for that. You are in many of my dearest and happiest memories. I cannot forget to thank my family too: Julio, Carmen, Jorge and Maria, thanks for all the warmth and support. I know that my grandfather, Eugenio Monsalvo, will also be proud and shed some tears when reading his name, despite not knowing what to say when asked what his grandson does for a living. I am lucky to have a big family and there are many other members of it to whom I feel deeply grateful. If you are reading this, I hope you know who iv Introduction The first premiss and the conclusion are moral claims and, by assumption, are governed by logic L1. The second premiss, on the other hand, is a claim about the observable fact of someone (acting as an executor of the power of the U.S. government) pouring water over a cloth covering someone else’s face. So, it belongs to the domain of middle-sized objects or events formalized by L2. Nevertheless, the argument seems to be intuitively valid. But, the questions are, which is the logic that accounts for the validity of the argument? And which are the principles of reasoning that allow us to reason across domains? These are not minor issues for a philosophical thesis claiming that the correct application of logic systems is local, since reasoning across domains is something quite pervasive. Thus, the philosophical thesis that I am going to analyse is logical localism and the challenge to localism that I will try to address, the main challenge of localism, indeed, is the problem of mixed inferences. Logical localism is often considered as a form of logical pluralism. That is, as a thesis claiming that there is more than one legitimate relation of logical consequence. These types of philosophical positions regarding logic were not a serious option before the appearance and consolidation of non-classical logics at the beginning of the 20th century. These logics reject some of the classical principles, thereby allowing to question the general validity or uniqueness of classical logic. The emergence of non-classical logics -notably, relevant, intuitionistic and many-valued logicsgave rise to some of the most important questions in the philosophy of logic: are alternative logics really ‘logic’? Do they deserve the same status as classical logic? Is there just one correct logic or are there many? Some of the proponents of alternative logics continued defending monist theses, now against classical logic and in favour of one of the alternatives. Michael Dummett, for example, argued for the correctness of intuitionistic logic, claiming even that classical connectives had no meaning at all. However, the fact of having a plurality of systems and of being able to check out that some of them appeared to have applications where they excelled the most, made pluralistic proposals more and more plausible. These proposals have had various forms and I will present the most relevant ones for my purposes later on. The problem of mixed inferences is the challenge around which the philosophical and the technical aspects of the dissertation revolve. After presenting the philosophical framework I will introduce the challenge to localism and I will mostly focus on the version that Chase Wrenn offers, which is, to 3 Introduction my mind, the best and most detailed presentation of the problem of mixed inferences for logical localism. My approach in order to offer a solution to the challenge is one that has not been explored, or considered, in the literature as far as I am aware. The strategy will be to explore different methods for combining logics, concentrating specially on the method of juxtaposition, by Joshua Schechter, and trying to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. But combinations of logics bring about other potentially problematic issues regarding interactions between connectives and, in the limit case, they bring about what are known as collapse theorems. These problems will be analysed and I will develop the combination mechanisms having in mind that the collapse has to be avoided. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways for combining logic systems, specifically classical and intuitionistic logics, that go beyond the standard for combinations, which is based on minimality conditions so as to avoid collapse theorems. Besides the technical developments, the new combination mechanisms introduce further subtleties within the philosophical debate around localism and allow for a more fine-grained analysis of alternative kinds of localisms and even of domains. That is, the analysis suggests that there are some properties of the domains that are not captured only by the logic that corresponds to a given domain. Instead, those properties are revealed when the domain interacts with another, i.e. when reasoning across domains. Thus, it might be the case that reasoning across the domain of middle-sized objects and the mathematical domain requires different interactions from those required by reasoning across the domains of middle-sized objects and ethics, even if one 4 Introduction believes that the logic of the mathematical and the ethical domain is the same, say, intuitionistic logic. Thus, I think that combining logics can bring us closer to a solution to the problem of mixed inferences and, moreover, help us discern with more accuracy the intricacies of the philosophical debate. The structure in which I will unravel these ideas is the following: in chapter 2, I present the conceptual framework in which logical localism is going to be characterized. In this conceptual framework, the topics of logical and alethic pluralisms play a crucial role, so it will be relevant to clarify what they are and how they relate to localism. Then, I will conclude the chapter by presenting the main challenge for logical localism, namely, the problem of mixed inferences and I will go through some of the most notable attempts to solve it. The last part will be devoted to Chase Wrenn’s version of the problem, which is the most elaborate version of the problem in the literature. Chapter 3 is meant to establish the connection between the problem of mixed inferences and the field of combining logics. In order to justify that bridge, I start by focusing on some logical conceptions about mixed reasoning and I introduce the notions of ‘interaction principles’, ‘bridge principles’ and ‘collapse theorems’, existent in the literature, and elaborate them. Then, I present some popular methods for combining logics, paying special attention to the one upon which I am going to build my own mechanisms, namely, juxtaposition. In chapter 4 I develop a solution to the problem of mixed inferences. First, I approach the problem by applying the original method of juxtaposition and discuss its virtues and potential shortcomings. Based on the limitations of the method, I argue that the combination mechanisms should allow for more interaction between the logics being combined, in order to get a more encompassing solution to the problem of mixed inferences. Therefore, I propose some new alternative mechanisms in which the desired bridge principles can naturally emerge in the combination process. Finally, chapter 5 concludes the dissertation by looking into some promising future work. As I will try to show, the analysis suggests that there are many interesting philosophical and logical/algebraic issues to be sorted out in the vicinity. 5 Chapter 2 Logical Localism 2.1 What is Logical Localism? A conceptual framework In this section I will try to set the framework of the discussion that concerns me for the dissertation. As I already advanced in the introduction, the topic of localism is very closely related to, if not included in, the debate around logical pluralism, which is a central debate within the philosophy of logic. Given its centrality, there are many other issues in the field that affect the discussion, such as the debate around which the chief aim of logic is, the normative status of logic, the sense in which logic is formal, whether logical consequence should be spelled out model-theoretically, proof-theoretically or kept primitive, and so on and so forth. Those topics are huge and each one of them deserves more than a dissertation, as shown, for instance, by J. G. MacFarlane with his brilliant thesis on formality (MacFarlane (2000)). However, it is beyond the scope of my thesis to deepen on those topics and I reckon that it would not even be helpful for fulfilling my more modest and specific aim, namely, to work out methods for combining logics in order to find possible solutions to the problem of mixed inferences that challenges the philosophical position of localism. Nevertheless, in characterizing logical localism those crucial issues will inevitably come up, since, as I said, they affect how one thinks about the nature of logic1. Thus, I will try to make explicit, where applicable, what I 1Notice that there are, at least, three senses of ‘logic’: a consequence relation, a particular logic system or the discipline. I believe that there will be no confusion throughout 6 Logical Localism am assuming or how those issues might affect the characterization of localism that I will be developing. Let me, then, before going into the details of localism, start by laying down a general assumption I will make concerning the chief aim of logic. Following the mainstream tradition and, more concretely, Graham Priest (Priest (2006)) and J. C. Beall & G. Restall (Beall and Restall (2000,2001, 2006)), which are arguably the most important and influential defenders of monism and pluralism, respectively, I will assume that the chief subject matter of logic is logical consequence. Beall and Restall very nicely put it at the beginning of their book Logical Pluralism: Logical consequence is the heart of logic; it is also at the centre of philosophy and many theoretical and practical pursuits besides. Logical consequence is a relation among claims (sentences, statements, propositions) expressed in a language. An account of logical consequence is an account of what follows from what -of what claims follow from what claims (in a given language, whether it is formal or natural). An account of logical consequence yields a way of evaluating the connections between a series of claimsor, more specifically, of evaluating arguments. (Beall and Restall,2006, p. 3) And also in Beall and Restall (2000): The chief aim of logic is to account for consequence, to say, accurately and systematically, what consequence amounts to, which is normally done by specifying which arguments (in a given language) are valid. All of this, at least today, is common ground. (Beall and Restall,2000, p. 475) Similarly, Priest writes: What is logic? Uncontroversially, logic is the study of reasoning.[...]The study of reasoning, in the sense in which logic is interested, concerns the issue of what follows from what. Less cryptically, some things -call them premisesprovide reasons for others -call them conclusions.[...]The relationship between premise and conclusion in each case is, colloquially, an argument, implication, or inference. Logic is the investigation of that relationship. A good inference may be called a valid one. Hence, logic is, in a nutshell, the study of validity. (Priest,2006, p. 176) the text since I usually use those more specific words instead of the more general ‘logic’. In any case, the context should suffice in order to disambiguate the generic uses. 7 Logical Localism Thus, I will adhere to this widely accepted tradition. Logic is the systematic study of what follows from what; of which premises stand in the logical consequence relation to which conclusions. That is, the aim of logic is to account for the validity of arguments. This is not, however, the only existing position regarding what logic is about. J. van Benthem, for instance, has a more ‘liberal’ conception of logic and argues that the view of logic as being about consequence relations may have had some sense when it was thought to provide the foundations of mathematics. But, since the 1930s the field has changed and broadened its scope. Logic is now, van Benthem claims, about definability, computation and more (van Benthem,2008, p. 183). Indeed, van Benthem defends that the main issue of logic is “the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication” (van Benthem,2008, p. 182). I do not have any particular concern with this conception and I believe that the discussion on whether logic is X or Y is not very fruitful. However, it does affect the plausibility of pluralism and localism how broad the domain of application of logic is. To put it simply, if logic is about so many things beyond logical consequence, as van Benthem claims, it will be more probable that there is more than one ‘correct logic’ and it will be less likely that one logic does all the job. When I use ‘correct logic’ I mean, roughly, the logic that is most fruitful, most adequate to the data, overall simplest, etc. Thus, in this case, I do not aim to imply any metaphysical view on whether there is, or not, an objective reality that logic seeks to capture. Thus, it is a sense of ‘correctness’ available both to a realist and an instrumentalist (in the sense of Haack (1978)).The difference between the realist and the instrumentalist arises, though, with respect to which the True logic is. Since for the instrumentalist there is no extra-systemic validity, but just valid-in-L, there is no True logic. For the realist on the other hand the intra-systemic notion of validity is trying to capture ‘real’ external validity. But, then, it is logically possible to conceive a world in which the correct logic, after weighing the theoretical virtues, is not the True logic. Imagine, for instance, that the True logic is one in which every inference rule has some counterexamples. Still, it could be the case that the most fruitful and adequate logic to be applied, i.e., the correct logic, was one with universal inference rules. On this note, and in order to continue laying down some assumptions, it 8 Logical Localism is relevant to the discussion that we clarify a bit more the notion of ‘application’. It is largely uncontentious, nowadays, that the pure/applied distinction holds when speaking about logics. Notably, Priest (2003,2006) justifies the distinction by drawing an analogy with geometry and arithmetic (an idea due to Łukasiewicz). The analogy tries to establish that, in the same way that there are many pure geometries (Euclidean, Riemannian, Lobachevskian, etc.), there are many pure logics (classical, intuitionistic, paraconsistent, connexivist, etc.). These are “well-defined mathematical structure[s] with a proof-theory, model theory, etc.” (Priest,2006, p. 195). We define them, we study their properties, prove results about them, relations between them, and so on. With respect to pure logics, much like pure geometries, there is no doubt about pluralism. It is an uncontentious fact that there is a plurality of pure logics. There are, however, other aspects of doing logic that have to do with the application of pure logics to different domains and problems. This is a common practice within philosophical logic, for instance, where pure logics are often applied in order to deal with paradoxes, to systematically account for reasoning about knowledge, necessity, obligation, morality, etc. But, it is also the case, as R. Cook points out, that “logics have been central to the study of a number of phenomena, including many that have, at best, an indirect connection to human reasoning such as electronic circuit design, database management, and internet security”(Cook,2010, p. 494). Despite the variety of applications that logics have been used for, though, some people argue that there is a privileged application of logic, a canonical application, which is the analysis of reasoning. Quoting Priest, the most important and traditional application of a pure logic [is] the canonical application: the application of a logic in the analysis of reasoning [...]. The central purpose of an analysis of reasoning is to determine what follows from what -what premises support what conclusionsand why. An argument where this is, in fact, the case is valid. (Priest,2006, p. 196) Thus, the more interesting pluralist thesis would be with respect to its canonical application. It is not enough the mere existence of a variety of pure logics, not even the fact that there might be various rival logics competing for being the best codification of reasoning (what Priest (2006) calls theoretical pluralism). As Cook stresses, 9 Logical Localism if logical pluralism is to be a substantial and controversial thesis, something more must be intended. That something more is the notion of logical consequence -that is, a logic is ‘correct’, or ‘acceptable’, etc., if and only if it is a correct (or acceptable, etc.) codification of logical consequence. The idea that the philosophically primary (but obviously not only) goal of logical theorizing is to provide a formal codification of logical consequence in natural language traces back (at least) to the work of Alfred Tarski. (Cook,2010, p. 495) Let us, therefore, assume that the chief subject matter of logic is logical consequence and that logic’s canonical application is the analysis of reasoning. One could think that a legitimate way of arguing for pluralism would be to emphasize that there are different ways of accounting for logical consequence, e. g. model-theoretically, proof-theoretically or regarding it as a primitive notion. I have tried to argue for the model-theoretic approach in Benito-Monsalvo (2022) and I believe that the debate is philosophically substantial, but this will not be relevant to our discussion here, since it is not the source of plurality that is interesting for present purposes. We are interested in the thesis that there are different logics which capture different legitimate relations of logical consequence when canonically applied (regardless of whether the logic is presented model-theoretically or proof-theoretically). Logical localism is one of those pluralist theses, but logical pluralism, understood à la Beall and Restall, for instance, is also a position in favour of that kind of plurality. Thus, what I want to do, now, is to start singling out and delimiting the logical localist position. This is a delicate task, because logical pluralism is not an unequivocal thesis and encompasses many positions under the same naming. But I am less interested in the exegetical work than in providing a conceptual map of the available theoretical positions and addressing where localism stands in that map. This is why I will be taking some authors almost like archetypical figures of the different positions. Let me proceed, then, to clarify how I understand logical pluralism, best represented by Beall and Restall, in opposition to logical monism, which has Priest as one of its most popular defenders. This opposition is also interesting because both sides agree on the chief subject matter of logic, namely, logical consequence. 10 Logical Localism 2.1.1 Logical Pluralism It sounds like a truism, but it is always nice to remember that in order to even conceive logical pluralism we require to know of the existence of different, alternative logic systems. However, it is also always amusing to remember Kant’s thesis on Newtonian physics and Aristotelian logic, i. e. syllogistic. Some authors refer now to Hugh McColl as the first proposal of what could be considered a logical pluralist philosophy of logic. It is no coincidence that he was also a pioneer in the development of non-classical logics, including, many-valued, probability, relevant and connexive logics (see Rahman and Redmond (2008)). His pluralism, though, seems to be closer to what I call ‘localism’ than to Beall and Restall’s pluralism, and it would certainly fall under what Cook, in Cook (2010), calls ‘relativism’. This is what Rahman and Redmon comment on this respect: MacColl’s philosophy is a kind of instrumentalism in logic which led him to set the basis of what might be considered to be the first pluralism in logic. The point condensed in the epigraph amounts to the following: it could well be that in some contexts of reasoning the existing argumentation demands a type of logic which is not applicable in others. When constructing a symbolic system for a particular type of logic, the corresponding expressions in use should therefore be taken into careful consideration. (Rahman and Redmond,2008, pp. 540–541) So, we can say that McColl’s pluralism is one that claims that the application of logic is relative to ‘contexts of reasoning’ and, therefore, that there might be different correct logics for different contexts. In this sense, the application of logic is local, despite logic being a theory of reasoning, because this varies from context to context. In other words: the canonical application of logic is the analysis of reasoning, but there are, so to speak, ‘sub-canonical applications’. A second milestone in the history of logical pluralism is Rudolf Carnap and his principle of tolerance. In The Logical Syntax of Language (1937) Carnap says: In logic there are no morals. Everyone is at liberty to build his own logic, i.e. his own language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments. (Carnap,1937, §17) 11 Logical Localism A crucial point that results from this liberty for building our own logic from our own language, is that there is no external logical reality that forces a particular One True Logic. The result is a kind of conventionalism, similar to that of McColl, by which one gets different correct2logics by varying the linguistic framework. Cook summarises Carnap’s pluralism in a very illuminating way: Carnap’s view certainly amounts to a form of logical pluralism. [...] But this is a dependent pluralism, resulting from an underlying relativism – that is different logics result from varying the language in question (it is worth noting that Carnap does not advocate pluralism within a framework – different linguistic frameworks might be governed by different logics, but within a particular framework there is a single logic that correctly codifies the (internal) logical consequence relation of that framework). Thus, Carnap’s tolerance amounts to a version of logical pluralism, but not a version of SLP [Substantial Logical Pluralism]. (Cook,2010, pp. 497–498) What Cook means by ‘substantial logical pluralism’, is a pluralism such that the language is kept fix, the demarcation of the logical/nonlogical vocabulary is also fixed and, yet, there are (at least) two logical consequence relations that capture two legitimate different senses of ‘follows from’. This is, in fact, what Beall and Restall want to achieve. So, let me present the fundamentals of their proposal. 2.1.2 Beall and Restall’s Logical Pluralism As we said before, Beall and Restall adhere to the mainstream tradition of taking logic to be about the consequence relation. About systematically determining what follows from what. The account of logical consequence that Beall and Restall (2000,2001,2006) deploy is a generalization of the traditional semantic one, i.e. of Tarski’s account of logical consequence. They call it Generalised Tarski Thesis (GTT): •(GTT) An argument is validxif and only if, in every casexin which the premises are true, so is the conclusion. 2I would say that ‘correct’ for Carnap means just the logic (or the logics) that better fulfils some pragmatic goal set by the subject. 12 Logical Localism me contrast localism with globalism for the sake of making localism more clear and distinct. Localism against Globalism The opposite thesis of localism, as I will understand the notions, is not monism but globalism. Globalism is the thesis according to which the application of logic is global, i.e. independent of the subject-matter or the domain of reasoning to which it applies. Therefore, globalism stays within the orthodoxy of logic in that it retains the alleged topic-neutrality of logic and seems to support the traditional idea of the universality of reason. I agree with some of the intuitions that sustain the idea of universality of reason, but I think it is an oversimplification to infer from that desideratum that, since logic is a theory of correct reasoning and reasoning is universal, then logic has to be applied globally in order for it to be a candidate for being correct. One could push back by arguing that reason can be universal, in the sense of being a human faculty that we can employ irrespective of subject-matter, in any domain of inquiry, despite actual reasoning taking various forms and principles in different domains. To give an analogy, there is something universal to everyone getting a gold medal at the Olympics, namely, they did better and won over their competitors. But, at the same time, winning has many forms and it materializes in different ways, whether you win a gold medal in climbing or in pole vault, for instance. There is an important empirical argument on the side of the globalist though. We do seem to reason across domains, from moral and factual premises to moral conclusions, from mathematical and physical to physical, reasoning about the interplay of macro and micro-objects, and so on and so forth. This is, to my mind, the most important empirical fact that localism has to account for. It is, indeed, the fundamental fact that sustains the problem of mixed inferences. Thus, even if the application of logic is local, we must be able to give an explanation of how those domains might interact. Moreover, I reckon that there is also important empirical evidence in favour of localism and challenging globalism, namely, the amount of logical systems that are used and are constantly being developed, not just for any application, but for canonical applications, or sub-canonical applications, like formalising reasoning about vague phenomena, reasoning with inconsistent information, reasoning about truth, quantum-mechanical phenomena, etc. Even within the domain of mathematics, that one could regard as a single 19 Logical Localism homogeneous domain, there are proposals for employing different logics in different branches, like paraconsistent, constructive or classical mathematics (see, for instance, Priest (2019); Shapiro (2014a,b)). In fact, there is no reason to suppose that we have reached the ultimate stage of varieties of reasoning and that, therefore, no more new domains of reasoning will appear, which, potentially, could require even new logical systems to be systematised. Thus, I believe that globalism faces an important challenge too, similar to the scope problem that traditional theories of truth have to face, namely, that ‘the plausibility of each inflationist’s candidate for the [truth] property Fdiffers across different regions of discourse’ Pedersen and Wright (2018). Similarly, globalism has to face the empirical fact that the plausibility of any logic system, L, differs across different regions of discourse or domains of reasoning. Localism as a form of Relativism Following with the characterization of localism, I believe it is interesting to place localism within the excellent taxonomy of ‘pluralist’ theories that Cook (2010) provides. Cook’s view on relativism is such that, One is a relativist about a particular phenomenon X if and only if one thinks that the correct account of X is a function of some distinct set of facts Y. Thus, relativism about X amounts to acceptance of the following schema: The correct account of X is relative to Y. (Cook,2010, pp. 492–493) In this sense, it is clear that localism is a form of relativism, namely, one holding that the correct account of ‘follows from’ or ‘consequence’ or ‘valid’, is relative to domains (leaving open whether domains are individuated by subject-matter, by the ontological properties of the objects within that domain, or whatnot). However, localism is not a type of Substantial Logical Pluralism, under Cook’s categorization, since it is difficult to hold that, if the logic for reasoning about truth is a paraconsistent logic, the logic of evaluative discourse is intuitionistic and the logic for reasoning about middle-size phenomena is classical logic, for instance, those logics’ corresponding connectives, say, negations, are going to have the same meaning. Whether one thinks that the 20 Logical Localism meaning of the connectives is determined by the rules of inference or by their truth-conditions or satisfaction conditions, it is reasonable to assume that the meanings will change7. Therefore, since Cook’s conception of Substantial Logical Pluralism implies that the pluralism arises within a fixed language and a fixed interpretation of the logical/non-logical divide, localism cannot be a Substantial Logical Pluralism. This should not be interpreted as localism not being a substantial or interesting thesis. Cook makes this clear, but I guess that the literature has had a tendency to focus more on the type of pluralism claiming that ‘there is no genuine debate between advocates of different all-purpose logics’ (Field,2009, p. 344). The reason for this bias might be that the most popular, detailed and best developed account of logical pluralism is Beall and Restall’s and they explicitly say that their intended pluralism is not a relativism. The plurality of logics, according to them, are applicable to any domain or subject-matter. Neither should we conclude that every form of relativism is a localism. A popular relativist view is defended by Achille Varzi in Varzi (2002). According to him, which conclusions are in a logical consequence relation to which premises depends on how one conceives the logical/non-logical divide. For instance, whether or not one regards identity as a logical symbol affects the possible models that one will accept. If one takes it as a logical symbol, she will accept only the models that do justice to the intended interpretation of the predicate. Otherwise, one could accept further models. Regardless of whether we agree with Varzi or not, it seems clear to me that his relativism is not a form of localism. It is not that the plurality of accounts for the logical consequence relation depends on a domain of reasoning, a subject-matter, some ontological property, etc. His relativism does not have anything to do with domains, but with the specific set of logical constants that one adopts. To conclude this section on the characterization of logical localism, let me next refer to some localist proposals in the literature. We have already said that McColl’s and Carnap’s pluralisms can be taken as localist accounts. Now, I will move on to explain how localism has been conceptualized by other authors. 7However, I am going to discuss more this issue once my improvements on juxtaposition have been presented, since I believe that the method might allow for some interpretations compatible with localism without meaning-variance for connectives. 21 Logical Localism Some Localist proposals in the literature As I defined it above, logical localism is the thesis stating that different logic systems are required in order to systematically account for correct reasoning in different domains. The main difference between the alternative proposals resides in how one individuates the domains. In the case of McColl, the domains are contexts of reasoning and his conventionalism seems to leave quite some freedom with respect to what determines a context of reasoning. It might be a particular problem, a topic, etc. There is no specific feature that forces a new context of reasoning, simply some contexts require different tools in order to account for what follows from what in that context. Let me give some other examples of localism now. Newton da Costa’s localism8 Newton da Costa’s localism is also based on the observation that there are a plurality of logics because different domains ask for different logic systems. Moreover, in da Costa’s view, there is a concrete reason for this domain variation, namely, that reasoning about different kinds of objects requires different logics. Therefore, domains of reasoning are determined by the ontological properties of the objects belonging to the domain. An example of these kinds of objects and the variation of logics that they require is that of macro-objects versus quantum objects: [...] It is clear that for common objects, such as a book or a person, [...] [∀x(x=x)] applies apparently without a single important difficulty. Any person whatever, say A, even though they undergo multiple modifications in the course of their life, remains in a certain sense identical with themself: A = A . That appears even more clearly as concerns abstract objects: for example, the equality 1 = 1 seems evident and indisputable [...]. However, things are not as simple as naive realism would lead one to believe. In quantum physics, elementary particles, according to all appearances, transgress the principle of identity. Thus Schrödinger affirmed that the relation of identity between particles was devoid of sense: “it is not a problem that depends on our capacity for proving the identity in certain cases and our incapacity for proving it in other cases. It is certain that the issue 8I have not been able to find the relevant material in English and, since my Portuguese and French are not good enough, I will rely on Priest’s interpretation of da Costa and in some translations to English made by himself. 22 Logical Localism of ‘identity’ is, really and truly, devoid of sense”. It could be that the position of Schrödinger is acceptable only temporarily and that the future will show us that he is mistaken. Nonetheless the fact is that quantum physics shows the possibility of dialectising the idea of identity, and consequently, the very law that corresponds to it. (da Costa,1997, p. 120) as cited from (Priest,2000, p. 440) On da Costa’s view, then, the ontological differences between objects enforce different logical properties too. But the account goes even further, as Priest observes: Da Costa’s pluralism is more radical than I have so far indicated, though. He envisages not only that objects of different kinds may have different logical properties, but that different logical operators may also need to be used in reasoning about different kinds of objects. Thus, for example, classical negation is appropriate for dealing with platonic objects, and intuitionist negation is appropriate for dealing with mental constructions. (Priest,2006, p. 198) Then, da Costa’s localism could imply9that there is meaning-variance both at connective level and also with respect to logical consequence and validity. That is, different kinds of objects constitute different domains and the reasoning within those domains varies because the diverse ontological properties of the objects carry over different logical properties. Therefore, different logical connectives and consequence relations are required in each domain in order to provide theories of correct reasoning within those domains. We will soon see that a counterargument that Priest gives against da Costa’s localism constitutes one of the major challenges that I want to address with the aid of the methods for combining logics. But, there is an issue that Priest does not mention and that to me seems problematic or, at least, dubious. This is the domain individuation just in virtue of the ontological properties of objects. I think there are good reasons, given by D. Edwards in Edwards (2018), for instance, to argue against that criterion. To illustrate, take the statements ‘the number πis irrational’ and ‘the number πis beautiful’. Both statements are about the same object, namely, the mathematical object π. However, while we would certainly assign the first proposition to 9I take the precaution of using ‘could’ because I have not found out whether da Costa defends that there is an extra-systemic notion of validity. If there is one, then it could be argued that the meaning of ‘valid’ does not really change. 23 Logical Localism the mathematical domain, we will most likely not assign the second one to the domain of mathematics, but to the aesthetical domain. As Edwards suggests, it is not what a sentence is about that we should be considering for domain membership, it is rather how the thing the sentence is about is represented, by the use of a predicate to attribute a property. (Edwards,2018, p. 96) That is, domain membership of a proposition seems to have to do with how the object is represented, i.e. with the predicate, rather than with the object. Another example might come from vague phenomena. It is well known that vagueness has been a fertile field for proposing alternative non-classical logics, but it seems that the objects of which we might predicate vague properties will appear in classical domains too. We might say, for instance, ‘Putin is bald’, but also ‘Putin is the president of Russia’. However, if da Costa’s example regarding quantum objects and identity is accepted, that would constitute an example in which the kind of object would be enough to individuate a domain. So, maybe the right criterion for domain individuation has to go beyond da Costa’s and Edwards’ accounts and make room for both. Domain individuation and the enforcement of a different logic system within a domain could be a matter of kinds of objects and the way objects are represented. Diderik Batens’ localism In Batens (1990), D. Batens presents a series of arguments against the idea of global paraconsistency, i.e. the idea that the correct logic is paraconsistent and its application is global because inconsistencies are inherent to human reasoning. His view, against what he takes to be a dogmatic attitude towards paraconsistency, is that ‘each logic [has] a particular set of domains in which it is adequate’ (Batens,1990, p. 209). The way in which we should conceive of these domains, following Priest’s terminology in Priest (2006), is as ‘problem-solving situations’. One of those situations that Batens uses as example is a meta-theoretical situation. According to Batens, despite there being domains in which a paraconsistent logic is required, e. g. inconsistent domains, classical descriptions of many logical systems, including paraconsistent ones, are possible, and [...] whenever this is so, paraconsistent descriptions are too poor to be adequate. The same applies to other 24 Logical Localism domains: whenever a domain is consistent, a paraconsistent description is incomplete. (Batens,1990, p. 227) The reason why paraconsistent logics are too poor for certain situations or domains, is because they lack the expressive strength that classical logic has. For instance, according to Batens, there is a sense of ‘rejection’ that the classical negations express that cannot be captured by a paraconsistent negation that allows both Aand ¬pAto be true. In this sense, domain individuation seems to be something more pragmatic, having to do with the utility of a logic for a given problem, similarly to McColl’s position, than the more substantial criterion of what constitutes a domain that da Costa defends. Stewart Shapiro’s relativism as localism We have briefly mentioned above S. Shapiro’s localism. I think it is worth commenting on it here because it represents an interesting case. So, prima facie, one would think that, for most of the possible sound accounts regarding domain individuation, mathematics would be a domain. Whether you characterize domains by the kinds of objects, by the predicates used for representing the objects, by the type of reasoning or problem-solving situations, it seems that mathematics is a good candidate for being a domain. In spite of all that, Shapiro argues for localism within mathematics. As with the previous proposals, there are many details that I am not interested in addressing here. My purpose is to illustrate different existing approaches to localism depending on how one understands domains. So, with Shapiro, the way in which he separates the domains of application of logics is by taking domains as structures, where a structure is a legitimate branch of mathematics. Hence, according to Shapiro, ‘logical consequence and validity are relative to structure. That is, one cannot say what the proper logic is until one says which structure is being discussed’ (Shapiro,2014a, p. 321). And, the point is that Shapiro argues for a variety of interesting and applicable mathematical branches, i.e. structures, that require different logics. For instance, intuitionistic analysis is inconsistent if one has classical logic as its background logical theory. But, there is no reason to dismiss such theory. It is a legitimate branch of mathematics with potential applications and, yet, it requires us to drop some classical principles (most famously, excluded middle) on pain of inconsistency. 25 Logical Localism That is not the only example that Shapiro points out. Similar situations are found with respect to other intuitionistic theories and also paraconsistent ones. Thus, Shapiro’s localism is a localism already within mathematics that, potentially, could be extended if those mathematical structures are applied, say, in some physical theory. Just within mathematics, then, we have legitimate structures some of which require classical logic, others intuitionistic logic and some others paraconsistent logic. Pedersen and Lynch: from alethic pluralism to localism Nikolaj J. L. L. Pedersen and Michael P. Lynch argue for a form of localism that Lynch names ‘domain-specific logical pluralism’ (DLP), in Pedersen (2014) and Lynch (2008), for instance. I group them together because of how similarly motivated they are, namely, they try to connect alethic pluralism with domain-specific logical pluralism. That is, the idea that truth varies across domains (the property of being true or the way propositions are true, for instance) with the idea that this variation of truth forces a variation on logic. Here is how Pedersen frames it: features of the truth property of a domain play a crucial role in determining the logic of the domain. In particular, the truth properties of some domains have the feature of being epistemically constrained and go hand in hand with cases that deliver intuitionistic logic, while the truth properties of other domains have the feature of being epistemically unconstrained and go hand in hand with cases that deliver classical logic. In short, alethic pluralism yields logical pluralism. (Pedersen,2014, p. 262) Lynch, similarly, argues that, although there is no direct unavoidable argument from truth pluralism to domain-specific logical pluralism, one can indirectly make the connection like this: If there is more than one way to manifest truth, and some of the manifesting properties are epistemically defined properties like superwarrant, and some not, then different domains will admit of different manifestations of the consequence relation. And this means, among other things, that argument forms that are valid in some domains may not be so in others.All this of course, assumes that there is more than one way to play the truth-role. If there is not, then there may still be more than one consequence relation, but this will presumably be motivated by other things than a view about the nature of truth. (Lynch,2008, pp. 134–135) 26 Logical Localism So, according to Pedersen and Lynch, the thing that characterizes a domain and distinguishes it from other domains is how the truth property is manifested within that domain or which truth property a domain has. This, in turn, might be the factor that determines correct reasoning within a domain, making it possible that correct reasoning and, therefore, logic, varies from one domain to another. Since the argument from alethic pluralism to localism has been one of the most popular in the literature, especially in the literature having to do with theories about truth, we will dive into the details given by Pedersen and Lynch later on in section 2.1.4. Let me finish, now, by adding that this way of individuating domains, by looking at the truth property, could be developed more by linking truth properties with ontological/metaphysical properties, as Pedersen (2014) does, for instance. If we follow this path, we might end up in something not so far from da Costa’s idea of domains individuated by kinds of objects. Thus, Pedersen argues that if one grants that there is both a correspondence domain and a superwarrant domain [i.e., a domain with an epistemically constrained truth property], one should also grant that there is a domain with respect to which one is committed to metaphysical realism and a domain for which one is not thus committed. But, if one is not committed to metaphysical realism, one must be committed to some other metaphysical view on the entities in the relevant domain. All in all, this amounts to a form of metaphysical pluralism—or at the very least, it seems to be very congenial to a form of metaphysical pluralism. (Pedersen,2014, p. 271) Therefore, domains would be ultimately individuated by the ontological/metaphysical properties, which would then make truth manifest in different ways and, finally, this different truth manifestations would impose different consequence relations. One can see that there are important localist proposals in the literature. Despite the fact that the philosophy of logic has not been very interested in logical localism (maybe, as I said, because the dominant position has been Beall and Restall’s logical pluralism and other authors have tried to challenge it) we can find, already from the beginning of pluralistic proposals about logic, historical and relevant contributions with a localist spirit. Moreover, the literature on theories of truth and, in particular, on alethic pluralism, has clearly had a tendency to favour localist implications with respect to logic, 27 Logical Localism as the natural philosophical position for an alethic pluralist. However, one could be a localist about logic without committing to pluralism about truth. In the next section, I want to propose a redefinition of the theoretical options and the conceptual map regarding the debate around logical pluralism, broadly understood. Redefining the conceptual map What I have presented so far yields a picture of the debate around logical pluralism that allows, I believe, for a new conceptualization. I have tried to frame the localist thesis by contrasting it to globalism and distinguishing it from Beall and Restall’s pluralism, which, in turn, I have contrasted with monism. Thus, the localist thesis states that there is a multiplicity of domains of discourse, with possibly different criteria of correct reasoning, that require adopting different logics. Globalism, on the other hand, is the position defending that the application of logic is global, in the sense that logical laws and valid arguments must be applicable regardless of the content, the subjectmatter or the domain of reasoning. Under the assumption that there is a canonical application of logic and that this is the application to reasoning (assumption that I made following Priest (2006)), this means that localism implies, contrary to globalism, that there are sub-canonical applications (since different domains of reasoning require different logical theories). Now, if we allow pluralism and monism to be theses about the plurality or uniqueness of legitimate logics with respect to a (sub)canonical application, we get a richer conceptual framework. As far as I know, Haack (1978) is the first (and only?) to conceptualize something similar (but only combining localism/globalism with pluralism and with slightly different senses). In this case, we get four theoretical positions: •Global Monism: there is just one correct logic and it is neutral with respect to the domain to which it is applied. •Global Pluralism: there are a variety of logics that are equally correct and their application is global, i.e. independent of the objects of reasoning. •Local Monism: Different domains of discourse require different logics, but there is only one correct logic for each domain. 28 Logical Localism can be true neither in Inor in any state of information extending I, because we are not allowing for inconsistent states of information. So, ¬¬(p∨ ¬p) is warranted in Iand in any state of information extending I. Therefore, we get that, for that epistemically constrained domain, there is a proposition p and a state of information Isuch that ¬¬(p∨ ¬p)is superwarranted, i.e., true, but p∨ ¬pis not. Therefore, the logic of that epistemically constrained domain should give us 0α∨¬αand ¬¬α0α. This argument, developed by Lynch and Pedersen, strongly suggests that the logic that goes with superwarrant is not classical logic but intuitionistic logic. But, on the contrary, if we also have the epistemically unconstrained, complete and consistent domain, with correspondence truth property and a strong form of the principle of bivalence (i.e., every sentence is either true or false but not both), then, we will have a domain whose logic is, most likely, classical logic. This would show that the legitimacy of different truth properties for different domains justifies that there are different domains that require different logics. Hence, localism. We have just considered two kinds of domains with different truth properties but, arguably, one could make a defence of paraconsistency and the virtues of adopting a paraconsistent logic, by a similar strategy. For instance, one could argue that in a domain of discourse like humour, being true might be related to the fact that some people take it as true.With such a low standard for truth, it is very likely that both the proposition that some joke is funny and its negation are both true. So, it seems at least plausible that some interesting logic systems, the ones that are usually invoked as being good candidates to be canonically applied, can be motivated for different domains with a similar strategy, namely, in virtue of being the logic that better fits a given truth property of a domain. We will see in the next section that this connection of alethic pluralism and localism works for motivating localism, but also carries over to the problems and challenges of these philosophical theories. That is, there is a strong analogy, if not identity, between the problems that are usually attributed to alethic pluralism and localism. While this might itself be problematic, I take it as a possible advantage since I might kill two birds with one stone. Here I am aiming just at localism but, if the other bird falls, it will have been a happy accident. 35 Logical Localism 2.2 Main Challenge: The Problem of Mixed Inferences We have just argued for the plausibility of logical localism and have given reasons in favour of it. However, there is a crucial empirical fact that would seem to count in favour of globalism and that localism has to account for, namely, that we do reason across domains. Thus, localist theses, intuitive as they might be, have to face an important challenge; a challenge that Priest (2006) raises and that I will summarise as follows: The Problem of Mixed Inferences: one might defend that there are a variety of domains that require different logics. But there are cases in which one reasons about the interaction of different domains, with premises about different kinds of objects coming from those domains. What kind of logic do we use, then? An underlying logic for both domains? This would give reasons for thinking that there is a logic of global application. Maybe a new logic specific for that domain of interaction? But which one? The intersection of the logics involved in each of the interacting domains might be too weak to be of any utility. Moreover, it should be taken into account that if we start trying to mix the connectives of different logics some of them may collapse. The intuitionistic conditional, for instance, collapses into the classical conditional under the presence of classical negation (Priest, 2006, p. 199). This, I believe, encompasses all the problems that localism has to answer, specially, for the technical approach of combining logics that I will be taking and that is threatened by the collapse of connectives. The more specific problems that we find by dissecting this main challenge, are the problems of mixed compounds and collapse theorems. Let me clarify that the problem of collapse theorems does not usually appear, even mentioned, in the more philosophical literature10. But, since the approach that I will be taking for answering the challenge is that of the methods for combining logics and these methods are threatened by collapse theorems, I include it in characterizing the problem of mixed inferences. 10In fact, even the more logically oriented authors, including Priest, despite mentioning the collapse theorems, take them as knockdown arguments, obviating, or ignoring, the fact that there are methods for combining logics designed to avoid the collapse. Hopefully, this dissertation is also useful for closing the gap between those seemingly isolated worlds. 36 Logical Localism With respect to the problem of mixed compounds, one could argue that it is a sub-problem of mixed inferences. As we present the problems it will be clearer why, but simply noticing that mixed inferences can have as premises or conclusions mixed compounds makes it clear enough11. Moreover, it is worth pointing out, as Lynch does, that [the problem of mixed inferences] does not arise solely for those who have come to DLP [domain-specific logical pluralism, a.k.a., localism] via truth pluralism. The issue of how to deal with mixed inference and compounds is an issue for any logical pluralist who takes it that distinct logics operate in different domains of discourse. (Lynch,2008, p. 137) The first one who put forward the problem of mixed compounds, as a challenge to alethic pluralism, was T. Williamson in Williamson (1994), where he reviews Wright’s Truth and Objectivity, though some people also refer to Tappolet (2000) since she raised the same problem in a notorious reply to Beall (2000). The problem of mixed compounds for alethic pluralism goes as follows: a sentence like ‘Alex killed 13 people and killing for fun is wrong’ can well be true. However, it is not obvious how the alethic pluralist could explain the truth of the conjunction. Surely, she can take the first conjunct to be true1,T1, (maybe in a correspondentist sense) and the second conjunct to be true2,T2, (maybe a notion of truth grounded in social agreement). But, then, in which sense is the conjunction true? Which is the truth predicate that applies to it? It seems that it can be neither T1nor T2, so maybe there is another truth predicate that applies to the conjunction. But, it is plausible to think that the conjunction will be true in that further way if and only if the conjuncts are true in that very same way too. Why do we need then the other truth predicates T1and T2? The version of the problem, as applied to logical localism, is analogous. Basically, the challenge is to answer which the logic of a compound proposition should be. This might seem innocuous, but consider a compound proposition like pc∨ ¬qp, where pcis a classical proposition and qpis a paraconsistent proposition. If the logic that governs the domain of pcis classical 11Another option is to consider mixed compounds as a problem for alethic pluralism only, arguing that the question about the correct logic only makes sense for inferences, not for propositions. In any case, giving a logic for mixed inferences (with mixed compounds) will, most likely, require solving how the semantic value of a mixed compound is determined or explaining which rule is governing the introduction or elimination of the main connective in a mixed compound. 37 Logical Localism logic and the logic that governs the domain of qpis Priest’s logic of paradox, LP, which is the logic of the compound? The answer is not trivial at all, since the way we answer it will affect, say, whether we can use disjunctive syllogism as a valid inference or not (because it is not valid under LP). This is the sense in which the problem of mixed compounds is a sub-problem of the problem of mixed inferences, in the versions directed against localism. Responding to the problem of mixed inferences requires having an account of the logic of mixed compounds, to begin with12. The problem of mixed inferences was, indeed, first put forward by Tappolet in Tappolet (1997) and it is based on what she takes to be a central platitude about truth: ‘truth is what is preserved in valid inferences’ (Tappolet,1997, p. 210). But, then, she proceeds to consider the following valid inference: Wet cats are funny This cat is wet This cat is funny Tappolet’s objection, then, goes like this: The validity of an inference requires that the truth of the premises necessitates the truth of the conclusion. But how can this inference be valid if we are to suppose with Crispin Wright that two different kinds of truth predicates are involved in these premises? For the conclusion to hold, some unique truth predicate must apply to all three sentences. But what truth predicate is that? And if there is such a truth predicate, why isn’t it the only one we need? (Tappolet,1997, p. 210) Thus, Tappolet challenges the truth pluralist with a trilemma: (a) either one denies that mixed inferences are valid, (b) accepts that there is a generic truth property that all domain-specific truth properties have in common (which would make this specific ones redundant) (c) or rejects the standard account of validity as necessary truth preservation. 12Again, one might prefer to reserve the naming ‘problem of mixed compounds’ to the version affecting alethic pluralism. Then, the version of the problem affecting localism would be just a sub-problem of the problem of mixed inferences. I believe this is just a terminological issue but the reader should keep in mind that the problem of mixed compounds is generally regarded as a problem for alethic pluralism. 38 Logical Localism Many authors have replied to Tappolet by trying to make the case for a possible satisfactory way out of the trilemma. Especially interesting responses are those of Beall (2000); Cotnoir (2013); Lynch (2008,2009) and Yu (2017). But, as I said before, I am more interested in solving the problem in its logical localist version. That is the aim of all the combination mechanisms that I will be presenting later on. If, as a side effect, we get an interpretation of the solution that is also satisfactory for alethic pluralism, it would be the icing on the cake. So, the version of the problem of mixed inferences that is directed against localism goes, roughly, as follows: suppose that there are (at least) two components, within the premises or conclusion of an argument, belonging to different domains whose logics are L1and L2, respectively. Then, which is the criterion of validity for the argument? That is, which is the logic that captures correct reasoning for that mixed domain? 2.2.1 Some attempts to solve the problem of mixed inferences Although not much, there have been some authors who have attempted to meet the challenge. However, very few of them have been systematic enough in their effort, with the exception of (maybe among others that I am not aware of) Cotnoir (2013); Lynch (2008,2009); Wrenn (2018) and Yu (2017, 2018). Let me present their accounts and explain why I take them to be unsatisfactory. Cotnoir’s algebraic account The aim of A. Cotnoir in Cotnoir (2013) is, first and foremost, to solve the problem of mixed inferences in the version directed against alethic pluralism that Tappolet raises. However, after providing an account for that, he goes on to extend the idea to make room for variation in each domain’s logic. So, Cotnoir starts by modelling the idea that each domain has a distinct truth property. In order to do this, he defines semantic values, V, to be ntuples, nbeing the number of domains and each element of the tuple being either 1 or 0: V={ha1, ..., ani| each ai∈ {1,0}} 39 Logical Localism So, having a 1 in the i-th position means that the proposition is in the i-th domain and it is truei. While having a 0 in the i-th position means either that the proposition is not in the i-th domain or that it is falsei. Further, assume that atomic propositions can have at most one truth property, so there can only be a 1 in the tuple representing the semantic value of an atom. He, then goes on to provide a classical account of connectives, in the sense that for each component of the tuple, the negation inverts the values, while conjunction and disjunction are minimum and maximum operations respectively13. That is, •For all proposition A, if v(A) = ha1, ..., anithen v(¬A) = h1−(a1), ..., 1− (an)i. •For all proposition Aand B, if v(A) = ha1, ..., aniand v(B) = hb1, ..., bni then v(A∧B) = hmin(a1, b1), ..., min(an, bn)i. •For all proposition Aand B, if v(A) = ha1, ..., aniand v(B) = hb1, ..., bni then v(A∨B) = hmax(a1, b1), ..., max(an, bn)i. As Yu (2017) rightly notices, these valuation functions already yield unwelcome results. First, notice that if, say, an atomic proposition is true, in the sense specific to its domain, the negation of the proposition will be true in the other domains represented in the tuple. For instance, suppose we are considering a mathematical domain and an ethical domain. Take the proposition A= ‘1 + 1 = 2’. The value of this atomic proposition will be v(A) = h1,0iassuming that the first element of the tuple represents the value in the mathematical domain and the second the value of the ethical domain. It is already questionable that the ethical truth-value of that proposition is false, but it is even worst that the negation of the proposition, i.e., v(¬A) = h0,1i, is ethically true. If mathematical propositions are not apt for the ethical truth predicate, why should the negation of a true mathematical sentence be ethically true? And how could one even make sense of the fact that ‘one plus one does not equal two’ is ethically true? A second unwelcome result comes from conjunction. Again, consider the mathematical proposition A= ‘1 + 1 = 2’ and the ethical proposition B= ‘Killing babies for fun is wrong’ with valuations v(A) = h1,0iand v(B) = h0,1i. The conjunction of these propositions, according to Cotnoir’s 13Here I am following Yu’s characterization of Cotnoir’s account, in Yu (2017), since it is a good summary of the proposal. 40 Logical Localism proposal, is false! It is false in both of the relevant senses, i.e., v(A∧B) = hmin(1,0), min(0,1)i=h0,0i. So the valuation making each of the conjuncts true in their relevant domains makes the conjunction false in every sense. Cotnoir, proposes a solution to the problem with negation, by introducing a third truth-value, 1 2, and moving to a non-classical logic14. The idea is that atomic propositions get at most a 1 in the tuple and for every other place they get the value 1 2, capturing the idea of being undefined for those domains. With this correction, the problem with negation is alleviated, because if a proposition has value 1 2in the i-th place, it will also get that value if we negate the proposition. But, this repair comes at the expense of having renounced to classical logic and, in any case, it does not solve the problem with conjunction. If we have again, A=‘1+1=2’andB= ‘Killing for fun is wrong’, now with valuations v(A) = h1,1 2iand v(B) = h1 2,1i, the conjunction of these propositions is undefined, i.e., v(A∧B) = hmin(1,1 2), min(1 2,1)i=h1 2,1 2i. This is equally counterintuitive and unsatisfactory. On top of these problems with the truth-value functions, there is a further limitation in Cotnoir’s account that Yu does not identify, since it is a limitation of his account too, as I will explain in a moment. So, what Cotnoir tries to do is to give a solution to the problem of mixed inferences for alethic pluralism. The reason for switching to tuples in order to represent truth-values, is that he wanted to be able to handle different truth-properties, each represented by a different position in the tuple, within a single logic system. First, the consequence relation that results, is classical logic (see (Cotnoir,2013, pp. 570–571)). Then, he introduces a third truth-value and gets a paracomplete and paraconsistent system. And, finally, he introduces Heyting algebras in the tuples and gets intuitionistic logic (see (Cotnoir,2013, pp. 575–577) for the details). In that way, he claims that localism is accommodated in the broader picture of alethic pluralism. But this is only halfway true. He has attempted to handle different truth-properties with different logics, but with a single logic each time! there is no mention of mixed inferences and which the criterion of validity might be when we need to combine different logic systems. Even a situation as simple as the following does not get an answer: suppose pcis a classical proposition and qia proposition belonging to an epistemically constrained domain. Take the proposition that ¬pc∨qi. Do we 14We do not need to enter into many details. Just know that the resulting logic is the intersection of intuitionistic logic and Priest’s logic of paradox. 41 Logical Localism have that ¬pc∨qi`pc→qi? We find no criterion in Cotnoir’s account and, therefore, no solution to the problem of mixed inferences in its version directed against localism. In fact, Cotnoir says that with his proposal, truth pluralists ‘can allow that the logic of unmixed inferences can sometimes be domain dependent’ (Cotnoir,2013, p. 577, my emphasis). Yu’s logic for alethic and logical pluralists We have just seen that A. Yu, in Yu (2017,2018), rightly identifies some of the unwelcome results in Cotnoir’s proposal, so his solution avoids those problems in a quite nice way. Yu’s fundamental idea is that there is an isomorphism between domains, truth properties and falsity properties, that behave in the following way: there is a one-to-one correspondence between domains, domain-specific truth properties, and domain-specific falsity properties. Pure domains are associated with exactly one subject matter, while impure domains are associated with two or more subject matters. Where domains are either pure or impure, pure domains generate all domains. Each atomic sentence is assigned to exactly one domain. Negations are always assigned to the same domain as the negand, while conjunctions and disjunctions may or may not be assigned to the same domain as each operand, depending on whether or not the operands are assigned to the same domain. Each atomic sentence is then assigned a domain-specific truth value, where the relevant domain is the one it is assigned. The domain-specific truth values of negations, conjunctions, and disjunctions are determined by the domain-specific truth values of the operands. Logical consequence necessarily preserves domainspecific truth. (Yu,2018, pp. 413–414) In order to avoid distraction with the details of the proposal,15 let me give a concrete example to roughly illustrate how it works: suppose we have a proposition about the physical middle-sized domain and a proposition about the ethical domain, say pc=‘the dog is at home’ and qi=‘torturing is wrong’. Considering that these are pure domains, they can be combined in order to produce the impure domain of ‘physical middle-sized and ethical’. Equally, 15To clarify, the details are not important because the proposal is not designed for the problem of mixed inferences ‘localism style’ but for ‘alethic pluralism style’. Otherwise, obviously, the details would be important, as happens with Lynch’s and Wrenn’s accounts. 42 Logical Localism the truth-values and falsity-values present the same structure. The physical proposition will take the values Tcor Fc, while the ethical proposition will be either Tior Fi. However, if we make the conjunction of both propositions to get pc∧qithat compound proposition will have the truth-values that correspond to its impure domain of ‘physical middle-sized and ethical’. Let us call them Tci or Fci. There are a number of worries with this type of proposal already known in the literature. In Edwards (2008) we find a similar view, which also relies on the intuition that compounds are true in a derivative sense (i.e., ‘impurely true’, using Yu’s terminology) and, for instance, Cotnoir (2009) makes the case against the proliferation of truth-properties that such a view requires. One could argue back, following the literature on the metaphysics of fundamentality, that there is no problem with multiplying the truth properties as long as they are derivative, since a sparse ontology is only relevant at the fundamental level. In any case, I reckon that Yu’s account still has problems. Let me point out a very obvious one. Take pcto be the previous proposition. By Yu’s account, ¬pcis also from the physical domain and, therefore, ¬pc∨pctoo. In fact, since it is a tautology, its truth-value is Tcfor any interpretation. By an analogous reasoning ¬qi∧qi, which is a contradiction belonging to the ethical domain, always gets value Fi. Now, make the disjunction of these two propositions to get (¬pc∨pc)∨(¬qi∧qi). According to Yu, this is a proposition belonging to the impure domain of ‘physical middle-sized and ethical’ and whose truth-values can only be Tci or Fci. However, this is a very odd result, since there is no doubt that this impure proposition is true in virtue of the same facts as ¬pc∨pcand, therefore, should be true in exactly the same way. In other words, the ethical proposition does not contribute anything to the truth of the impure proposition and, yet, it makes the impure proposition take a different truth property to that of the physical proposition in virtue of which is true. Be that as it may, this would be a problem for his proposed solution to the problem of mixed inferences as applied to alethic pluralism. That is, it is a way of accommodating a variety of truth values, within a single consequence relation and criterion of validity. In Yu (2017) the underlying logic is classical and in Yu (2018), it is extended to allow for a non-classical underlying logic too. But, this is far from being a solution to mixed inferences in which we allow the action of more than one underlying logic within the premises through the conclusion. Yu has just given a way in which a non-classical 43 Logical Localism domain can have a non-classical logic, but has not considered which is the logic of an argument with components governed by different logics. This is the same problem that Cotnoir has. They get non-classical logics for some domains with non-classical truth-values, but they do not offer a method for combining the logics of those domains in mixed inferences. Lynch’s modesty criterion As we said above, Lynch is aware of the connection between the problem of mixed inferences in its alethic and logical versions and, moreover, notices that it is a problem for localism even if that localism does not come from alethic pluralism. So, once he has solved the problem for alethic pluralists by means of his functionalist account, or so he believes, he moves on to solve the problem for localism. The solution, Lynch claims, has two parts. First, the localist, being a pluralist after all, will take it that within a domain, what qualifies as the governing logic will be determined by what manifests truth in that domain. (Lynch,2008, p. 137) So, for instance, following Lynch’s analysis, the logic governing the domain in which superwarrant manifests truth will be intuitionistic logic. The second part is the more difficult one. Lynch defends that in order to evaluate the validity of a mixed inference we have to apply a criterion of modesty. To understand why, consider the next argument16: Nix:If it is not the case that offensive jokes are funny, then snow isn’t white Snow is white Offensive jokes are funny Which we can formalize as: ¬pi→ ¬qc qc pi 16It is a modified version of the argument that Lynch and Wrenn also call ‘Nix’. The propositions are different but the structure is the same. I just wanted to use a different exemplar to show the extent of possible instances. 44 Logical Localism modalities. The collapse theorems were also motivated by reflecting on classical logic and non-classical ones. More concretely, on imagining a dialogue between a classical logician and an intuitionist, willing to cooperate in order to understand what the other party means. However, mixed inferences constitute a new philosophical challenge to combination mechanisms, in the sense that, if logical localism is correct, then there are situations of mixed reasoning that might generate interesting interactions between logic systems that have not been considered yet. I will try to show that this is, in fact, the case and that the philosophical problem of mixed inferences might trigger important developments and research lines for the combinations of logics. 51 Chapter 3 Mixed Reasoning and Combining Logics 3.1 Interaction Principles In situations of combined reasoning, such as the ones represented by the cases of mixed inferences, one should expect some interaction between the logical systems that are being combined. If those logical systems aim at capturing the modes of reasoning of given domains, it is reasonable to expect that the combination of the logical systems will capture some interaction of the combined reasoning, otherwise it would not be a case of mixed reasoning in the first place. One of the most common references when dealing with interaction principles is David Hume’s naturalistic fallacy, namely, the thesis that from a factual statement one cannot deduce a normative one, that is, that from ‘what is’ one cannot derive ‘what ought to be’. To put it in terms of modern modal logic, Hume’s thesis constitutes an objection to interaction principles such as, p→ p stating that, if pis the case, then one ought to p1. 1One could say that this is a limit case of an interaction principle, since there is no modal operator in the antecedent. We will see in a moment, though, that it falls under the definition of bridge principle, which is a type of interaction principle. In any case, it is reasonable to argue that the lack of a modal operator is irrelevant, since the antecedent is expressing a factual modality and this is what interacts with the deontic one. 52 Mixed Reasoning and Combining Logics Another historical reference on these matters is the ‘ought-implies-can’ thesis, usually attributed to Immanuel Kant. This is the thesis that if something is obligatory then it must be possible, formalized as p→♦p. And yet another interaction principle that is often times included in epistemic logics comes from Plato’s characterization of knowledge as ‘justified true belief’. Thus, this understanding of what constitutes ‘knowledge’ motivates having an interaction principle capturing that if a subject xknows that p, then pis the case, Kxp→p. These interaction principles fall under the subcategory of ‘bridge principles’, which were understood as principles linking factualities to norms and, more generally, as principles linking different modalities. Thus, we can define more formally a bridge principle as an axiom schema which has at least one occurrence of an schematic letter under the scope of a modal operator, ?, and at least one occurrence outside the scope of ?. One thing to notice is that these bridge principles vary in their analyticity, so while some of them might be highly desired in virtue of their meaning and how the interaction is established, others might be more problematic and in need of some philosophical justification, or even rejected. The usual interdefinition between necessity and possibility, namely, α≡ ¬♦¬α, could be included among the analytic bridge principles. The bridge principle, previously mentioned, connecting the knowledge of some proposition with the truth of that proposition certainly needs some philosophical justification2, and an axiom stating that if pis the case then some finite agent xknows that pis clearly bad and rejected by any reasonable epistemic logic. As we will see later on, some of these bridge principles can be avoided by some of the methods for combining logics. This can be seen as a positive consequence of the combination mechanisms, as one could add, after the combination, the bridge principles that one desires in the combined logic as additional axioms, while avoiding the problematic ones. However, one might take this procedure to be quite ad hoc and, therefore, it is an interesting issue to wonder whether one could combine given logical systems in such a 2One could argue that Newton knew some laws of physics, despite those laws not being strictly true. 53 Mixed Reasoning and Combining Logics way that the desired bridge principles arose in the very combination process (Schurz,1991, p. 46)3. But, let me now elaborate and widen more the notion of bridge principle. For now, we have only considered bridge principles to be interactions between variables under the scope of a modal operator and variables outside their scope. However, I reckon, following Carnielli and Coniglio (2007), that we should think of bridge principles as principles establishing, more generally, connections or interactions between connectives. But not every interaction between connectives will count as a bridge principle. In a sense, we want these interactions to be “new”, meaning that we did not have these in the original logics being combined. Thus, we take bridge principles to be “any interactions (i.e., derivations) among distinct logic operators which are not instances of valid derivations in the individual logics being combined” (Carnielli and Coniglio,2007, p. 8). Take, for instance, the following mixed inference: Wet cats are funny Fitz Roy isn´t the highest mountain or wet cats are not funny Fitz Roy isn´t the highest mountain Assume that we think CL is the logic of the physical domain and that IL is the logic of the domain about humour. So, let us formalize the argument distinguishing the connectives of each logic by the subindexes cand i. pi ¬cqc∨i¬ipi ¬cqc We can see that there is an interaction between connectives from different logics, but this is not a bridge principle, since it is an instance of a valid derivation in IL, namely, Disjunctive Syllogism. However, had we translated the argument like this, pi ¬cqc∨c¬ipi ¬cqc we would have delivered a bridge principle, since this argument captures an interaction between classical and intuitionistic connectives, but it is no longer an instance of a valid argument in IL, nor in CL. 3I will show later on that my solution to mixed inferences meets this desideratum. 54 Mixed Reasoning and Combining Logics Another example in order to further illustrate what characterizes bridge principles among interaction principles is given by Carnielli and Coniglio (2007). Consider the combination of the logic of classical conjunction, L∧, and the logic of classical disjunction, L∨. In this logic the derivation p∧q`(p∧q)∨r is an interaction principle which is not a bridge principle, since it is a substitution instance of p`p∨r that is a valid derivation in L∨. However, the distributive law of conjunction over disjunction, p∧(q∨r)`(p∧q)∨(p∧r) is a bridge principle of L∧∨, since it is not a substitution instance of any valid derivation either in L∧or L∨. In some sense, then, interaction principles are new derivations just because they involve some vocabulary that we lacked before the combination. But, indeed, the interestingly new interactions come from bridge principles. These really are new derivations that appear in the combination process and that were not present before. 3.1.1 Collapse Whoever is familiar with the debate around logical pluralism has surely come across the notion of ‘collapse’, as applied to the pluralist proposals. One of the first collapse arguments, if not the first, appears in Williamson (1988), and other relevant versions can be found in Read (2006), Priest (2006), Keefe (2014) and Stei (2020). Priest´s very well known argument, is directed against Beall and Restall’s pluralism: Let sbe some situation about which we are reasoning; suppose that sis in different classes of situations, say, K1and K2. Should one use the notion of validity appropriate for K1or for K2? We cannot give the answer ’both’ here. Take some inference that is valid K1but not K2,α`β, and suppose that we know (or assume) αholds in s; are we, or are we not entitled to accept that βdoes? Either we are or we are not: there can be no pluralism about this. (Priest,2006, p. 203) 55 Mixed Reasoning and Combining Logics The essence of the argument is that there are two legitimate consequence relations that disagree with respect to a particular argument, i.e. α`1βbut α02β, and, at the same time, there is a subject who knows that αholds and that α`1βbut α02β. It would seem, then, that the subject is entitled to accept that βholds in s, as Priest defends right after the quoted paragraph. So, the rational thing to do for the agent is to accept it and, therefore, these two logics would collapse to the strongest one. Nevertheless, this is not the notion of collapse that we are going to focus on in this section; for there is another type of collapse that has received almost no attention in the philosophical literature in spite of being a crucial challenge to some pluralist (meaning ’pluralist’ in a relaxed, almost informal, sense) proposals such as localism. Before going into the details, let us roughly introduce this type of collapse by saying that it does not depend on a particular argument over which two logics disagree. Neither does it involve any assumptions about the normativity of logic in order for it to work. It is rather a technical result, consisting of a number of theorems, known as collapse theorems, which show that by freely combining different logic systems, each (possibly) with its own stock of connectives, the logics collapse to one of them, because their different connectives end up behaving as mere notational variants. Let me notice that I will be following Schechter in the terminology and distinguish, as he does, collapse and weak collapse. The precise notation and concepts will be presented in the next section. But let us, for the moment, say that a logic, L, with two stocks of logical connectives, collapses when for every formula δ,δ0, exactly alike except for some or all of their subscripts, {δ} ` δ0. Furthermore, let us say that a logic, L,weakly collapses when there is a translation, t, between the set of formulas with connectives from stock 1 and the set of formulas with connectives from stock 2, such that, if Γ`α, then t(Γ) `t(α). The first collapse results can be traced back to Carnap (1943) and Popper (1948), although the most common references are Harris (1982), in the more philosophically oriented literature, and del Cerro and Herzig (1996) and Gabbay (1996), in the literature revolving around fibring. However, despite their different ’traditions’, all these sources share the common feature of dealing uniquely with the case of combining classical and intuitionistic logics. 56 Mixed Reasoning and Combining Logics 3.1.2 Collapse Theorems In his paper ‘What’s So Logical about the “Logical”Axioms?’, J. H. Harris invites us to imagine an intuitionist logician and a classical logician willing to cooperate in order to understand the axioms that the other party deems valid, taking them as syntactical meaning postulates of how the other understands the connectives. Assume, then, that both logicians want to entertain a dialogue in a common logic Lover a shared language L. Harris delivers the following axiom schemata as the ones that both classical and intuitionist logicians would accept4: 1. Deduction Property (DEDL): A1, ..., Ak, A `LBiff A1, ..., Ak`L A→xBfor all L-formulas A1, ..., Ak, A and B. 2. Modus Ponens (MPL): Γ`LAand Γ`LA→xB, then Γ`LB. 3. A, B `LA∧xB. 4. (a) `LA∧xB→xA (b) `LA∧xB→xB 5. (a) `LA→xA∨xB (b) `LB→xA∨xB 6. A→xC, B →xC`LA∨xB→xC 7. A→xB, A →x¬xB`L¬xA 8. ¬xA, A `LB On top of these shared axioms, we also both accept that Γ`LAif A∈Γ (9). Moreover, the classical logician will also want to include the following axiom: 8c`L¬c¬cA→cA 4Let us use subscript xfor either the intuitionist (i) version or the classical (c) version of the connectives. Also, notice that we will only focus on the propositional part. 57 Mixed Reasoning and Combining Logics However, an interesting point is that we will not need this last axiom, 8c, in order to prove the collapse theorems. Taken together, these theorems are meant to establish the collapse of the logic, i.e. that for every L-formula Ax, the intuitionistic and the classical versions are interderivable, `LAi↔xAc. Let us now show some of the most interesting collapse theorems. Theorem 3.1.1.Assume logic Lsatisfies schema 1 - 8 for both x∈ {i, c}. Then for every L-formula A and B and both x∈ {i, c}we have `LA→i B↔xA→cB. Proof. I prove the right to left direction, which is actually more interesting. The other direction is analogous. We have that A→cB`LA→cBby (9). Then, A→cB, A `LBby (1) and A→cB`LA→iBagain by (1).  Theorem 3.1.2.Assume logic Lsatisfies schema 1 - 8 for both x∈ {i, c}. Then for every L-formula A and both x∈ {i, c}we have `L¬iA↔x¬cA. Proof. Again from right to left. ¬cA, A `LBby (8) and ¬cA`LA→iB by (1). ¬cA, A `L¬iBby (8) and ¬cA`LA→i¬iBby (1). Then apply twice DEDLto (7) in order to get `L(A→iB)→i((A→i¬iB)→i¬iA), and, finally, by two applications of MPLwe get ¬cA`L¬iA. As one might expect now, the rest of the collapse theorems for ∧and ∨ work in a similar way. Thus, all these theorems lead to the unsettling result that when I state the law of excluded middle using the classical connectives, this is logically equivalent to the intuitionistic version, i.e., `L(A∨i¬iA)↔x (A∨c¬cA). In fact, one can prove the following stronger theorem which states the collapse of the logic Lover the language L. Theorem 3.1.3.Let Aibe any L-formula whose connectives are intuitionistic and let Acbe the corresponding L-formula in its classical version. Assume L satisfies schema 1 - 8 for both x∈ {i, c}. Then `LAi↔xAc. Proof. By induction on the number of connectives (see Harris (1982), Theorem 8). 3.1.3 Collapse: the limiting case of bridge principles We started the section talking about interaction principles and proceeded to presenting what collapse is. Notice that, indeed, the collapse theorems are nothing more than some results that provide new derivations, i.e., bridge 58 Mixed Reasoning and Combining Logics principles, establishing connections between the corresponding logical connectives of each logic. The problem of these theorems, however, is that there is too much interaction between the logical connectives. In fact, depending on the method for combining logics, one can give sufficient conditions for which new interactions provoke the total collapse. In the case of juxtaposition, for instance, one can show that if we added α→1βa` α→2β5, the logic would collapse (see Schechter (2011), Prop. 7.3.) As we are going to see later on, one of the most difficult, yet at the same time interesting, aspects of combining logics is to calibrate how much interaction is too much interaction and, also, how little interaction might be too little. We know that the collapse of the connectives is at one extreme of interactions, but there is an analogous problem, first noticed by Béziau in Béziau (2004) and later more deeply analysed in Béziau and Coniglio (2011) at the other extreme, namely, the anti-collapse problem. The authors characterize this problem as “the impossibility of obtaining, in the logics obtained by fibring, intended interaction rules which are justified, for instance, by well-known models or sequent rules” (Coniglio,2007, p. 379). In fact, this is not a problem specific to the combination method of fibring, but also for juxtaposition and, in general, for any combination mechanism that, when combining the logics L1and L2seeks the logic L12, which is the minimal conservative extension6of the logics being combined. This means that the combined logic, because of its minimality, will not include among its valid inferences any properly new interactions, that is, bridge principles, nor, a fortiori, the intended justified ones. In Béziau (2004), the anti-collapse problem was illustrated with the case of combining the logic of conjunction, L∧, and the logic of disjunction, L∨. We have already seen that some of the desirable bridge principles for this combination are the distributivity laws, both of conjunction over disjunction and vice versa. One can easily check that if we put together the two-valued truth tables for ∧and ∨we actually get that the distributivity laws are satisfied. Nevertheless, the standard combination mechanisms (like fibring or juxtaposition) will not yield these bridge principles in the combined logic L∧∨, since this logic will be the minimal conservative extension of L∧and L∨. It might be a matter of disagreement whether the logic of conjunction 5As usual, αa` βis an abbreviation for α`βand β`α. 6We will give the formal definitions later on. 59 Mixed Reasoning and Combining Logics and disjunction, L∧∨ is distributive or not7, but there are other applications of combining logics that more obviously require the emergence of bridge principles, for instance, the application that aims at recovering a logic from its fragments. This has been one of the more recent fields of development in the application of combination mechanisms and it has given rise to new methods for combining logics, given how inappropriate the standard mechanisms are in order to go beyond minimality. One of the few methods developed along those lines is meta-fibring, proposed by Coniglio (2007). This method recovers classical logic when combining the logic of (classical) negation with the logic of (classical) conditional. We will see when showing the applications of juxtaposition that this is not possible with it, since we do not get the Principle of Pseudo-Scotus, `α→(¬α→β), for instance. But the emergence of some bridge principles can also be problematic, even if they do not lead to collapse, just in virtue of not being well justified or being philosophically faulty. As Carnielli and Coniglio (2007) recall, if one combines two normal modal logics, L1and L2, with the method of product of modal logics, each one with its own operators iand ♦i, the following bridge principles pop up at the semantic level: •-commutativity: 12α↔21α •♦-commutativity: ♦1♦2α↔♦2♦1α •(1, 2) Church-Rosser property: ♦12α↔2♦1α •(2, 1) Church-Rosser property: ♦21α↔1♦2α But if the logic L1is an alethic modal logic and L2is an epistemic logic, we would have, ♦Kα ↔K♦α with ‘K´standing for ‘knowledge’, which seems highly implausible. Just consider the fact that it is possible that an agent knows that Higgs Boson exists, while the agent not knowing whether it is possible that Higgs Boson exists. This bridge principle clearly has epistemological shortcomings. One should already see that the issue of bridge principles and how to produce them is a very delicate one. They are not good or bad on their 7See Béziau and Coniglio (2011) for an interesting analysis. 60 Mixed Reasoning and Combining Logics Other methods for combining logics that do not belong to the mainstream of categorial fibring are ecumenism, by Prawitz (2015), and chunk and permeate, by Brown and Priest (2004)(see, specially, Priest (2014)). There are some reasons why I will not focus on these methods. One of them is that there is only so much time one has for doing a thesis and some things, the ones that do not seem so relevant for certain purposes, have to be left out. Another reason, this one specific of ecumenism, is that the philosophical motivation of Prawitz when developing the method appears to be substantially independent of mine. The philosophical motivation of ecumenism is to give an inferentialist semantics for classical connectives. My motivation, though, is to provide a system that combines logics with the aim of having a localist reading of the mechanism. This motivation of mine could possibly match one of Priest’s applications of chunk and permeate, indeed. Ironic as it might sound, Priest, together with M. B. Brown (Brown and Priest (2004)) developed a strategy for “handling the application of different logics in combination” (Priest,2014, 333). The problem is that the method of chunk and permeate is not very systematic nor general and, mainly, depends on how one wants to design the mechanism for a specific application. On top of this limitation, there is no metatheoretical result and no preservation theorem that can help us understand more the mechanism for the sake of systematizing it. Maybe it is because of the monopoly that algebraic fibring has on the field of combining logics, but, the truth is that neither ecumenism, nor chunk and permeate, and not even juxtaposition appear mentioned in the entry “Combining Logics” of the Stanford Encyclopedia. I think this is unfortunate, specially for juxtaposition, which is a very well developed method with so much potential as I hope to show. Before moving on to the presentation of some of the methods just mentioned, let me refer to two different approaches to the combination of logics in order to clarify which one of those I am focusing on. The approaches are those of splitting versus splicing logics. With respect to splitting logics, we may think about an analytic procedure that permits us to decompose a given logic into simpler components. [...] A prototypical case of splitting occurs when one succeeds in describing a given logic in terms of simpler components by means of translating the original logic into a collection of simpler, auxiliary logics, using what is called possibletranslations semantics. (Carnielli et.al.,2008, p. 10) 67 Mixed Reasoning and Combining Logics So, splitting is a top-down analytic approach aiming to decompose a logic into simpler fragments. Splicing, on the other hand, is a “process, by which a bunch of logics is synthesized forming a new logic”(Carnielli et.al.,2008, p. 10). So, it is a bottom-up, synthetic approach, by means of which simple logics are combined in order to obtain a more complex system. This is the approach that we are going to be using. So, every method that I am going to present now, and also juxtaposition, are cases of splicing logics. 3.2.1 Fibring by functions The method of fibring, also known as ‘fibring by functions’, was originally proposed by D. Gabbay in Gabbay (1996)12. As we said above, this mechanism only applies to logics with Kripke semantics. Still, it is a powerful method for combining such logics. Let me start by giving some definitions: Definition 3.2.1 (Definition 7 in Coniglio and Fernández (2005)).Amodal signature is a signature Csuch that C1={¬,},C2={→} and Ck= ∅in any other case. A Kripke model (for modal logics) is a triple m= hWm, Rm, hmisuch that Wmis a nonempty set (the set of possible-worlds of m); Rm⊆Wm×Wm(the accessibility relation of m); and hm:V −→ ℘(Wm) is a mapping (the m-valuation). A Kripke semantics is a class Kr of Kripke models. Let us, then, denote the modal logics by the pair L=hCL, Kri. Given two logics, L1and L2we define the fibred language,L(C⊗), which is obtained from the fibred signature,C⊗, namely: C1 ⊗={¬,1,2};C2 ⊗={→};Ck ⊗=∅in any other case. Now, in order to get a fibred logic, we need to perform the fibring of the Kripke models. The fundamental idea is to take fibred Kripke models with distinguished actual worlds and to connect the worlds of one model with the worlds of the other, in such a way that if we are evaluating, say, a formula like 2αin a Kripke model of L1in a world from Wm1, we can move to the corresponding world from Wm2in model m2, in order to check the validity of 2α. Thus, a fibred model of Kr1and Kr2is a triple (f, g, h)such that:13 12In order to present the method of fibring, I will be following Gabbay (1996), but also Carnielli and Coniglio (2020) and Coniglio and Fernández (2005), since these are very clear and concise explanations of the method. 13Udenotes the disjoint union of sets. 68 Mixed Reasoning and Combining Logics f:] m1∈Kr1 Wm1−→ ] m2∈Kr2 Wm2; g:] m2∈Kr2 Wm2−→ ] m1∈Kr1 Wm1; h:V −→ ℘(W) with W:= (Um1∈Kr1Wm1)](Um2∈Kr2Wm2)and fand gas transfer mappings: ffrom the set of worlds of the class of models Kr1of L1into the class of models Kr2of L2, and gfrom the set of worlds of the class of models Kr2 of L2into the class of models Kr1of L1. The fibred structure Kr1⊗Kr2is the class of all the fibred models of Kr1and Kr2. The satisfaction of a formula αby the fibred model (f, g, h) in the world w, denoted (f, g, h)wα, is defined recursively as usual when the main connective of αis Boolean, i.e., ¬or →, and when the modal operator and the world correspond to the same logic (that is, in a situation of standard modal logic). The relevant cases are those in which the modal operator and the world of evaluation have different origins. For instance, when evaluating the formula 1αin the fibred model (f, g, h)and the world w2. The satisfaction clause goes as follows: the model (f, g, h)satisfies 1α in w2∈Wm2,(f, g, h)w21αiff (f, g, h)w0 1αfor every w0 1∈Wm1such that g(w2)Rm1w0 1. The other case, with the subindexes of the box and the world interchanged, works analogously. Thus, with this notion of satisfaction we characterize logical consequence in the usual way, as preservation of satisfaction from premises to conclusion. The fibred consequence relation is, `Kr1⊗Kr2⊆℘(L(C⊗)) ×L(C⊗)and the logic L⊗=hC⊗,`Kr1⊗Kr2iis the fibring of L1and L2. Again, the method of fibring is limited by its applicability to modal logics with Kripke semantics, although, we will soon see that plain fibring is a natural way of extending the method beyond that limit. In any case, fibring is crucial in the history of combining logics, since it inspired the development of new, more general, methods. Among them, categorial fibring has been the one around which almost the whole field of combining logics has orbited. 3.2.2 Categorial (or Algebraic) Fibring Since categorial fibring was presented in Sernadas et.al. (1999), many developments have emerged around it. So, this brief and rough presentation of 69 Mixed Reasoning and Combining Logics the method should not be taken to represent or be faithful to the whole picture. In fact, even the method itself, leaving aside subsequent improvements around it, poses quite a challenge to be summarized in a simple way, since it involves some ideas from category theory. This is why I will be following Schechter (2011), on top of Carnielli et.al. (2008), and his way of presenting categorial fibring; a way that better lends itself to be compared with juxtaposition, which is the main method that I will be using. Thus, I shall confine myself to the semantics of fibring. Another reason why focusing on the semantics is interesting for us is that, as I pointed out above, the collapse of the algebraic fibring of CL and IL occurs at the semantic level, not when fibring their Hilbert calculi (see Rasga et.al. (2002)). So, it is interesting for my purposes to show a method for combining logics whose semantics collapse, since this is something that I want to avoid when trying to solve the problem of mixed inferences appealing to a combination mechanism. Following Carnielli et.al. (2008) the semantic unit of algebraic fibring is an algebra, hB, Φi, understood as a tuple with a set, the carrier set, B, and a family of operations, Φ. But, “in order to ensure the preservation of some properties by fibring, it is convenient to consider enriched algebras, called interpretation structures” (Carnielli et.al.,2008, p. 92). An interpretation structure over a signature C, is a tuple hB, ≤,Φ,>i, where hB, ≤,>i is a partial order with a top element >and hB, Φiis an algebra over C. In the terminology of Schechter (2011), this interpretation structures are partially ordered unital structures, the ‘unital’ referring to the fact that the >is the unique designated value. Moreover, the relation ≤allows us to compare the truth values in B, which makes possible to define two different notions of entailment, although we will only focus on one; global entailment. Given a class of partially ordered unital structures, i.e., a class of interpretation systems, B, we say that Γglobally entails αin Biff for every interpretation structure in Band valuation, i.e., for every model, if every γ∈Γgets value >, then αgets value >. Now, take an interpretation structure B=hB, ≤,Φ,>i over a signature C. Suppose that C0is a subsignature of C,C0≤C. We define the reduct of Bto C0as the tuple B|C0=hB, ≤,Φ|C0,>i, where Φ|C0is the restriction of Φ to the connectives in C0. Reducts play an important role in algebraic fibring, since when doing the algebraic fibring of two classes of partially ordered unital structures, B1and B2, over the signatures C1and C2, respectively, what we get is a class of partially ordered unital structures and, for each 70 Mixed Reasoning and Combining Logics structure in the class, we must have that B|C1∈B1and B|C2∈B2. That is, the algebraic fibring of B1and B2is the class of partially ordered unital structures, B12 ={B :B|C1∈B1and B|C2∈B2}over the fibred signature C1∪C2. This way of constructing the fibred semantics has some limitations, though. Limitations that are crucial for the philosophical problem that I am dealing with. This is because, as I pointed out above, algebraic fibring is not well suited for combining the semantics of classical and intuitionistic logics. The reason is the following: classical logic is sound and complete with respect to the class of Boolean algebras, BA. Intuitionistic logic is sound and complete with respect to the class of Heyting algebras, HA, which generalize Boolean algebras. Then, the algebraic fibring of the semantics of CL and IL is the class of partially ordered unital structures, let us call it Bci, over the fibred signature Cc∪Ci, with Cc={¬c,∧c,∨c,→c,↔c}and Ci={¬i,∧i,∨i,→i,↔i}, such that Bci ={B|B|Cc∈BA and B|Ci∈HA}. But, if one looks at the definition of reduct, one can easily see that the carrier sets of B|Ccand B|Ci coincide. Thus, for every interpretation structure in the fibred class of interpretation structures to be a Boolean algebra and a Heyting algebra for its relevant reducts, every structure has to be a Boolean algebra, i.e., every B|Ciis a Heyting algebra that is also Boolean. Hence, the intuitionistic connectives will behave exactly like the classical ones and, so, they will be intersubstitutable. This means, according to the definitions given above, that the algebraic fibring of CL and IL collapses and, a fortiori, weakly collapses. Recall, though, that responding to Wrenn’s challenge is an important part of giving a solution to the problem of mixed inferences and that, this challenge, essentially involves being able to give a criterion of validity for mixed inferences with components coming from domains governed by classical and intuitionistic logics. If we are not able to implement a combination mechanism for those logics while avoiding the collapse of the connectives, we will not be able to meet the challenge. But, then, algebraic fibring does not seem a good candidate in order to account for the problem of mixed inferences. We know, however, that there are other options that work better. 3.2.3 Direct Union and Plain Fibring Direct union and plain fibring are an extension of Gabbay’s original notion of fibring to another class of logics, namely, logics characterized by matrix semantics. These methods were proposed in Coniglio and Fernández (2005) 71 Mixed Reasoning and Combining Logics and, although they can still be developed more and further metalogical results could be obtained, they are interesting as a continuation of Gabbay’s work and also because of the similarities with juxtaposition. In fact, I should clarify that they are not two totally independent methods. Direct union is the method that one applies when we are in the simple and smooth scenario of combining two logics in “which the domain and designated values of the matrices involved are the same. In such cases, the combined logic can be simply obtained by putting together both matrices” (Coniglio and Fernández,2005, p. 1596). With plain fibring, however, we can face the more difficult scenario in which the domains and the values designated are different. The idea, in this case, is to build, with appropriate functions, a bigger matrix that encompasses those of the logics being combined. Once we have this general matrix for both logics, we do their direct union. In this sense, direct union is the final stage of plain fibring. Let me begin by laying out some definitions and, then, we will see the methods with a little bit more detail. Definition 3.2.2 (Definition 5 in Coniglio and Fernández (2005)).Given a signature C, a C−matrix is a pair M=hA, Diwhere A=hA, Diis an algebra over Cand D⊆Ais the set of designated values of M. Definition 3.2.3 (Definition 6 in Coniglio and Fernández (2005)).Let C be a signature and let Kbe a class of C-matrices. The matrix semantics induced by K(denoted by `K) is defined by: Γ`Kαiff for every C-matrix M=hA, Dibelonging to Kand every valuation v, if v(Γ) ⊆Dthen v(α)∈ D. We consider, first, the simpler case in which the domains and the designated values of matrices are the same. In this case, the direct union consists of putting together both matrices in the following way: Definition 3.2.4 (Definition 9 in Coniglio and Fernández (2005)).Let L= hCi, Mii(with i∈ {1,2}) be two matrix logics, where each Mi=hAi, Diiis a Ci-matrix. Assume that A1=A=A2and D1=D=D2. The direct union of L1and L2is the logic L1+L2=hC1]C2,`M1+M2iwhere `M1+M2is the consequence relation defined by the C1]C2-matrix M1+M2=hA, Disuch that, if c∈Ck iand a1, ..., ak∈A, then cM1+M2(a1, ..., ak) = cMi(a1, ..., ak) (i∈ {1,2}). 72 Mixed Reasoning and Combining Logics So, one can see that those types of combinations are pretty straightforward. Since the domains and the designated values are the same, the denotation of a connective from Ciis just the same in the original matrix Mi and in the direct union of matrices, M1+M2. That is, we will not get the more difficult case in which the argument of a truth-functional connective is a semantic value for which the truth-function was not defined. We now consider the more interesting case of combining logics characterized by matrix semantics with different domains. This is the method of plain fibring and the reader will immediately see the similarities with fibring, which is a special case of this more general approach. In Coniglio and Fernández (2005), they treat, first, the case in which there is no restriction to the transfer mappings, i.e., unrestricted plain fibring. However, I directly consider the situation in which we restrict the mappings in a way specified below. Definition 3.2.5 (Definition 14 in Coniglio and Fernández (2005)).Let L=hC, Mibe a matrix logic, where each M=hA, Diis a C-matrix with domain Aand set of designated values D. Let A0and D0be two sets such that D0⊆A0. Suppose, without loss of generality, that A∩A0=∅. Finally, let f:A0−→ Abe a mapping. The C-matrix Mfis defined as follows: its domain is A]A0; its set of designated values is D]D0and for c∈Ck i and a1, ..., ak∈A]A0,cMf(a1, ..., ak) = cM(a1, ..., ak)where, for every aj (j= 1, ..., k): - If aj∈A, then aj=aj. - If aj∈A0, then aj=f(aj).14 Definition 3.2.6 (Definitions 13, 15 in Coniglio and Fernández (2005)).Let L=hCi, Mii(with i= 1, 2) be two matrix logics, where each M=hAi, Dii is a Ci-matrix with domain Ai. The fibred signature is given by C1]C2and the fibred language is L(C1]C2). A fibred valuation is a triple (f, g, v), where (f, g)∈AA1 2×AA2 1, such that (f, g)is admissible and v∈(A1]A2)V(Vbeing the set of propositional variables). A pair (f, g)∈AA1 2×AA2 1is admissible if it satisfies: f(x)∈D2iff x∈D1, for every x∈A1; and g(y)∈D1iff y∈D2, for every y∈A2. Given φ∈L(C1]C2)and a fibred valuation (f, g, v), we define (f, g, v)(φ)∈A1]A2by recursion on the complexity of φ: - If φ∈ V then (f, g, v)(φ) = v(φ); 14We will see an illustration of this kind of matrices in Example 3.2.1. 73 Mixed Reasoning and Combining Logics - If φ=c(β1, ..., βk)then (f, g, v)(φ) = c((f, g, v)(β1), ..., (f, g, v)(βk)) where for every formula βj(j= 1, ..., k): •If c∈Ck iand (f, g, v)(βj)∈Aithen (f, g, v)(βj) = (f, g, v)(βj) for (i= 1,2); •If c∈Ck 1and (f, g, v)(βj)∈A2then (f, g, v)(βj) = g((f, g, v)(βj)); •If c∈Ck 2and (f, g, v)(βj)∈A1then (f, g, v)(βj) = f((f, g, v)(βj)); We say that a fibred valuation (f, g, v)satisfies φif (f, g, v)(φ)∈D1]D2. The plain fibred consequence relation `M1M2⊆℘(L(C1]C2))×L(C1]C2)is defined as follows: Γ`M1M2φif, for every fibred valuation (f, g, v)satisfying simultaneously all the formulas of Γ, we have that (f, g, v)satisfies φ. The plain fibring of L1and L2is the pair L1 L2=hC1]C2,`M1M2i. Now, this is already a much more interesting combination mechanism, because there is a wide variety of logics, or fragments of logics, that we can combine. Moreover, we know by Proposition 10 in Coniglio and Fernández (2005), that the plain fibring of two matrix logics, L1 L2, is a conservative extension of both L1and L2. What this means, is that C1⊆C1]C2, C2⊆C1]C2and Li= (L1 L2)|Cifor (i= 1,2). That is, when we restrict the plain fibring to each of the signatures, we get the original logics. Notice that this is an important feature, since it shows already that, if the sets of valid inferences of the logics are different, then the plain fibring of them does not weakly collapse and, therefore, does not collapse either. To see this, think, for instance, about the case of doing the plain fibring of CL and LP. We know that in CL we have modus ponens but in LP we don’t. This means that in CLLP we will have that p→cq, p `CLLP q, but p→pq, p 0CLLP q, which is enough to show that CLLP does not weakly collapse, because we have found a valid inference in the plain fibring, whose translation to the inference with corresponding connectives is no longer valid. Since I want to avoid excessive detail until we plunge into the method of juxtaposition, let me conclude the section by further illustrating the method of plain fibring with an example, given in Coniglio and Fernández (2005) which, I hope, will help to get a better picture of the method. Example 3.2.1.Take the negation fragment of the paraconsistent matrix logic P1.15 Let, then, L1be the fragment of P1defined over the signature {¬P1}, 15This is a paraconsistent logic introduced in Sette (1973). 74 Mixed Reasoning and Combining Logics given by the matrix M1with domain A1={T, T1, F}and set of designated values D1={T, T1}. ¬P1 T F T1T F T Let also L2be the fragment of CL defined over the signature {→c}given by the matrix M2with domain A2={1,0}and set of designated values D2={1}. →c1 0 1 1 0 0 1 1 Now, we start fibring the matrices by taking the disjoint unions of the domains and designated values. So, let the domain be A={T, T1, F, 1,0} and set of designated values D={T, T1,1}and let (f, g)∈AA1 2×AA2 1be the following admissible valuation f(T) = f(T1) = 1, f(F) = 0, g(1) = T and g(0) = F. Then, (M1)gand (M2)fare given by the following matrices, respectively: 16 ¬ T F T1T 1F F T 0T →T T11F0 T1 1 1 0 0 T11 1 1 0 0 1 1 1 1 0 0 F1 1 1 1 1 0 1 1 1 1 1 16These matrices are the extensions of the original matrices with the semantic values of each other. This is possible, because we have the valuations, i.e., the transfer mappings, in order to know how to translate the semantic values of one matrix into the other, so as to be able to determine the semantic matrix of a connective when having as arguments possibly new semantic values that were not in its original definition. 75 Mixed Reasoning and Combining Logics Thus, let Lbe the logic over {¬,→} characterized by the matrix M(f,g)= (M1)g+ (M2)fgiven by the two tables above, with {T, T1,1}as the set of designated values. Notice, though, that there is an additional admissible pair (f, g0)such that g0(1) = T1and g0(0) = F. Therefore, the fibred logic L1L2 is characterized by the set of matrices: M1M2={(M1)g+ (M2)f,(M1)g0+ (M2)f} We can check that, for instance, the mixed formula (p→q)→ ¬¬(p→q) is not valid in L1 L2, since for the pair (f, g0)and a valuation of the propositional variables such that v(p) = v(q)=1, the (f, g0, v)(p→q)=1 and, since g0(1) = T1, by the truth-table of negation ¬(T1) = Tand ¬(T) = F. But, given that the pair is admissible, f(F) = 0 and, therefore, with p→qbeing 1 and ¬¬(p→q)being 0, (p→q)→ ¬¬(p→q)is going to be 0. So, the formula is not valid in L1 L2. Clearly, plain fibring constitutes a big step forward with respect to Gabbay’s fibring and, also, with respect to algebraic fibring (at least concerning the collapse theorems). I, certainly, believe that plain fibring is also a good candidate to successfully be applied in addressing the problem of mixed inferences. However, I will be sticking to juxtaposition since, although maybe being more marginal than any proposal in the fibring tradition, it is more developed than plain fibring. Modulated fibring and cryptofibring where designed to avoid the collapses but also to be applied to an even larger variety of logic systems, which makes them, definitely, very attractive if the ultimate goal of combination mechanisms is to arrive at a universal methodology for combining any given logic systems, independently of how they are presented, e.g., algebraically, axiomatically, with non-deterministic semantics, etc. The reason for not focusing on them and going with juxtaposition as my preferred method for meeting the challenge, then, is that the cases of mixed inferences that I could potentially address with juxtaposition and the method itself were challenging enough. Moreover, I have been able to come up with ways in which juxtaposition could be improved, in directions that are quite illuminating for thinking about interaction principles and combinations of logics, as I will try to show. Hence, I am not claiming that juxtaposition, or any of my improvements thereof, are going to be the best solution and the one that encompasses the greater diversity of logics. But, in order to solve any difficult problem it is usually a good idea to divide it to, at some point, conquer it. My modest goal 76 Mixed Reasoning and Combining Logics Proposition 3.3.3 (Strong Conservativeness).Suppose C1and C2are disjoint signatures. Suppose each of `1and `2is consistent and has no mere followers. Suppose `12 is the juxtaposition of `1and `2. Then `12 is a strong conservative extension of each of `1and `2. Proposition 3.3.4 (Preservation of Consistency).Suppose C1and C2are disjoint signatures. Suppose each of `1and `2is consistent and has no mere followers. Suppose `12 is the juxtaposition of `1and `2. Then `12 is consistent. If Γ⊆Sent(Ci,Pi) is consistent with respect to `i, then Γis consistent with respect to `12. The preservation of consistency from each consequence relation `1and `2to the juxtaposed one, relies on the fact that `12 is a strong conservative extension of `1and `2. Let us now advance to the more challenging completeness results. Schechter’s strategy is based on a modification of the Lindenbaum-Tarski construction. The idea is to provide direct proofs for strong completeness that apply in a variety of cases by finding suitable equivalence relations in order to build the Lindenbaum-Tarski models. Here we focus on the essential results. Let ∼be an equivalence relation on Sent(C,P). We say that ∼is a congruence over C−whenever: •For every c−∈C−n, α1, ..., αn, β ∈Sent(C,P) and k∈ {1, ..., n}, if αk∼β, then c−α1...αk...αn∼c−α1...β...αn. ∼is compatible with `and Γ⊆Sent(C,P) whenever: •For every α, β ∈Sent(C,P), if α∼βthen Γ`αjust in case Γ`β. We say that ∼is strongly compatible with `and Γ⊆Sent(C,P) just in case ∼is a compatible with `and Γand: •For every α, β ∈Sent(C,P), if both Γ`αand Γ`βthen α∼β. 83 Mixed Reasoning and Combining Logics We say that ∼is suitable for C−,`and Γjust in case ∼is a congruence over C−compatible with `and Γ. We say that ∼is unital suitable for C−, `and Γjust in case ∼is a congruence over C−strongly compatible with ` and Γ. In order to construe the Lindenbaum-Tarski models, Schechter makes use of suitable and unital suitable equivalence relations. Let us give some additional definitions before moving to the results. Suppose Γis a nonempty subset of Sent(C12,P12) consistent with respect to `12. For each i∈ {1,2}, suppose ∼Γ iis an equivalence relation on Sent(C12,P12) suitable for Ci,`12 and Γ. We define: •|α|Γ i={β|α∼Γ iβ} •BΓ i={|α|Γ i|α∈Sent(C12,P12)} •DΓ i={|α|Γ i|Γ`12 α} •If ci∈Cn i,ΦΓ i(ci)(|α1|Γ i, ..., |αn|Γ i) = |ciα1...αn|Γ i •BΓ i=hBΓ i,DΓ i,ΦΓ ii •BΓ 12 =hBΓ 1,BΓ 2i •If αis an i-atom, VΓ i(α) = |α|Γ i •MΓ 12 =hBΓ 1,VΓ 1,BΓ 2,VΓ 2i Then, BΓ 12 and MΓ 12 are the Lindenbaum-Tarski juxtaposed structure and the Lindenbaum-Tarski juxtaposed model for C1,C2,`12 and Γ, built with ∼Γ 1and ∼Γ 2. Now, suppose for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12,∼Γ iis an equivalence relation on Sent(C12, P12) suitable for Ci,`12 and Γ. Let us define: B∼ 12 ={BΓ 12|Γ⊆Sent(C12,P12) is nonempty and consistent with respect to `12 and BΓ 12 is built with ∼Γ 1and ∼Γ 2} B∼ 12 is the Lindenbaum-Tarski class of juxtaposed structures for C1,C2 and `12 built with ∼Γ i. 84 Mixed Reasoning and Combining Logics Theorem 3.3.1 (Strong Completeness).Suppose `12 has no mere followers. Suppose for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12,∼Γ iis an equivalence relation on Sent(C12,P12) suitable for Ci,`12 and Γ. Then `12 is strongly complete with respect to B∼ 12. Theorem 3.3.2 (Strong Soundness).Suppose `12 is the juxtaposition of `1 and `2. Suppose for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12,∼Γ iis an equivalence relation on Sent(C12, P12) suitable for Ci,`12 and Γ. Then `12 is strongly sound with respect to B∼ 12. Proposition 3.3.5.Suppose `12 has no mere followers. Suppose for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12, ∼Γ iis an equivalence relation on Sent(C12,P12) suitable for Ci,`12 and Γ. Then if `12 is consistent, there is a coherent nontrivial juxtaposed model based on B∼ 12. Summarizing the results: Theorem 3.3.3.Suppose `12 has no mere followers. Suppose for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12, ∼Γ iis an equivalence relation on Sent(C12,P12) suitable for Ci,`12 and Γ. Then: 1. `12 is strongly complete with respect to B∼ 12. 2. If `12 is the juxtaposition of `1and `2, then `12 is strongly sound with respect to B∼ 12. 3. If `12 is consistent, then there is a coherent nontrivial juxtaposed model based on B∼ 12. 4. B∼ 12 is a class of juxtaposed unital structures just in case for every i∈ {1,2}and nonempty Γ⊆Sent(C12,P12) consistent with respect to `12,∼Γ iis unital suitable for Ci,`12 and Γ. 3.3.5 Applying the results to Classical and Intuitionistic Logics After obtaining this plethora of metalogical results, Schechter himself makes use of the method of juxtaposition in order to apply it to the cases of combining classical and intuitionistic logics, since, as already pointed out, it is 85 Mixed Reasoning and Combining Logics for those cases that the collapse theorems have been proved in the literature. Thus, he starts developing the juxtaposition of classical and intuitionistic logics as follows. Let P1=P2=P12 be a countably infinite set of sentence symbols and for i= 1,2let Cibe the signature containing these sets of connectives: •C1 i={¬i} •C2 i={∧i,∨i,→i,↔i} So, C12 is the set containing two copies of each of the propositional connectives. Let `i 1and `c 2be the intuitionistic and classical consequence relations for Sent(C1,P1) and Sent(C2,P2) respectively. We say that `ic is the intuitionist-classical juxtaposed consequence relation for Sent(C12,P12). We can also have consequence relations for languages with two copies of classical connectives or two copies of intuitionistic connectives. We call these, `cc and `ii, the bi-classical and bi-intuitionist consequence relations respectively. In the section about interaction principles we advanced some rough definitions of collapse and weak collapse. Let us, now, give some more precise definitions needed in order to study `cc,`ii and `ic. A consequence relation, `12, for Sent(C12,P12)collapses just in case for every δ, δ0∈Sent(C12,P12) exactly alike except perhaps for some or all of their subscripts, {δ} `12 δ0. Let fbe a bijection from Sent(C1,P12) to Sent(C2,P12) that maps each sentence α∈Sent(C1,P12) to the sentence that results from uniformly substituting each connective in αwith the corresponding connective from C2. We say that a consequence relation, `12, for Sent(C1,P12)weakly collapses just in case for every Γ⊆Sent(C1,P12) and α∈Sent(C1,P12), Γ`12 αjust in case f(Γ) `12 f(α). We consider here the case of intuitionist-classical consequence relation, `ic. On one hand, for any nontrivial Boolean algebra, hB, ≤i, there is a corresponding unital structure, hB, {1},Φi, where Bis the set of semantic values, {1} is the greatest element of the Boolean algebra given by the order ≤, and for every a, b ∈B: Φ(¬)(a) = −a Φ(a, b)(∧) = aub Φ(a, b)(∨) = atb 86 Mixed Reasoning and Combining Logics Φ(a, b)(→) = −atb Φ(a, b)(↔) = (−atb)u(−bta) With −,u,tas the complement, infimum and supremum operations in the Boolean algebra. We call these structures “Boolean structures” and the classical consequence relation is strongly determined, i.e., is sound and complete, with respect to the class of Boolean structures. It is consistent, has theorems and it is left-extensional. On the other hand, for any nontrivial Heyting algebra, hB, ≤i, there is a corresponding unital structure, hB, {1},Φi, where Bis the set of semantic values, {1} is the greatest element of the Heyting algebra and for every a, b ∈B: Φ(¬)(a) = a⇒0 Φ(a, b)(∧) = aub Φ(a, b)(∨) = atb Φ(a, b)(→) = a⇒b Φ(a, b)(↔) = (a⇒b)u(b⇒a) With u,t,⇒as the infimum, supremum and implication operations in the Heyting algebra and 0as its least element.We call these structures “Heyting structures” and the intuitionist consequence relation is strongly determined with respect to the class of Heyting structures. It is also consistent, has theorems and it is left-extensional. By Proposition 3.3.4 `ic is consistent and by Proposition 3.3.3 `ic is a strong conservative extension of `iand `c. An interesting point that Schechter just mentions without getting into the details is that the juxtaposed consequence relation `ic can be axiomatized “using a copy of any natural deduction-style axiomatization for intuitionist logic and a copy of any natural deduction-style axiomatization for classical logic, each restricted so that the rules for one stock of connectives cannot be applied within any subderivation used in the application of a metarule governing a connective from the other stock” (Schechter,2011, p. 595). 87 Mixed Reasoning and Combining Logics AHeyting-Boolean structure is a juxtaposed unital structure hB1,B2i such that B1is a Heyting structure and B2is a Boolean structure. By Proposition (Proposition 6.34 or corollary 6.33 in Schechter (2011)), `ic is strongly determined with respect to the class of Heyting-Boolean structures. The first non-collapse result that one can easily check is that the intuitionistclassical juxtaposed consequence relation does not collapse. Just notice that since `ic is a strong conservative extension of both `cand `iwe will have 0ic p∨1¬1pbut `ic p∨2¬2p. Moreover, we can prove that no corresponding connectives of C12 are interderivable in `cc, so neither in the weaker relations `ic and `ii. Proposition 3.3.6 (Proposition 7.1 in Schechter (2011)).In `cc, no pair of corresponding connectives are intersubstitutable. In particular: •{¬1p}0cc ¬2p •{p∨1q}0cc p∨2q •{p→1q}0cc p→2q •{p↔1q}0cc p↔2q •{¬2(p∧1q)}0cc ¬2(p∧2q) Proof. In order to prove this, we have to build a coherent juxtaposed countermodel that invalidates each of the previous entailments. We look, thus, for a model based on bi-Boolean structures. As it is usually done, we represent the Boolean algebras using Hasse diagrams. Consider the following Boolean algebras B1and B2: ≤1 1B1 0B1 ≤2 1B2 0B2 ab 88 Mixed Reasoning and Combining Logics Let us choose the following valuations. V1(p1) = V1(p2) = V1(p3) = 0B1, V2(p1) = V2(p2) = a,V2(p3) = band let Vi(p) = 1Bifor every other p∈ P12. Since M1=hB1, V1iand M2=hB2, V2idesignate the same sentence symbols, we know by Proposition 3.3.1 that there is a coherent juxtaposed model M12. Since `cc is strongly sound with respect to the class of biBoolean structures, we make use of the model, M12, in order to invalidate the entailments above. In M12,||¬1p1||1= 1B1and ||¬2p1||2=b, so {¬1p1}0cc ¬2p1.||p1∨2 p3||2= 1B2but ||p1∨1p3||1= 0B1so {p1∨2p3}0cc p1∨1p3and by symmetry {p1∨1p3}0cc p1∨2p3.||p1→1p3||1= 1B1and ||p1→2p3||2=b, therefore {p1→1p3}0cc p1→2p3.||p1↔1p3||1= 1B1and ||p1↔2p3||2= 0B2, so {p1↔1p3}0cc p1↔2p3. And the last one, which involves an embedded connective. ||¬2(p1∧1p2)||2=−2||p1∧1p2||2. Since the 1-value of p1∧1p2 is 0B1and it is a 2-atom, we choose in the construction of the coherent juxtaposed model V2(p1∧1p2) = 0B2, therefore ||¬2(p1∧1p2)||2= 1B2. But, ||¬2(p1∧2p2)||2=−2(a) = b, so {¬2(p1∧1p2)}0cc ¬2(p1∧2p2). As already said, since `ii and `ic are weaker relations than `cc the result applies to them as well. Notice that despite ∧1and ∧2not being intersubstitutable in general, they are intersubstitutable as main connectives even in `ii. This is because {p∧1q} `ii p,{p∧1q} `ii qand {p, q} `ii p∧2q, so {p∧1q} `ii p∧2q. Since `ic and `cc are stronger than `ii, this also holds for them. Another interesting result is that `cc is not left-extensional, therefore, `ii and `ic are not left-extensional either. Proposition 3.3.7.`cc is not left-extensional. Proof. Let B1and B2be the Boolean structures of the previous proof. Take V1(p1) = V1(p2)=1B1,V2(p1) = V2(p2)=1B2and Vi(p)=0Bifor every other p∈P12. Since M1=hB1, V1iand M2=hB2, V2idesignate the same sentence symbols, we know by Proposition 3.3.1 that there is a coherent juxtaposed model M12. We construe the coherent model in such a way that we can have ||p2||2,||p1||2,||¬2¬1p1||2∈D2while ||¬2¬1p2||2/∈D2. Since V1(p1) = V1(p2) = 1B1,||¬1p1||1=||¬1p2||1= 0B1. But both ¬1p1 and ¬1p2are 2-atoms, so take V2(¬1p1) = 0B2and V2(¬1p2) = arespecting coherence. Now we have that ||¬2¬1p1||2= 1B2and ||¬2¬1p2||2=b, so ||p2||2,||p1||2,||¬2¬1p1||2∈D2and ||¬2¬1p2||2/∈D2as desired. Therefore, p1, p2,¬2¬1p10cc ¬2¬1p2and, so, `cc is not left-extensional.  89 Mixed Reasoning and Combining Logics Let us now focus on the case of weak collapse. It is clear from the definition that both `ii and `cc weakly collapse. Notice also that if a consequence relation `collapses, then it weakly collapses. Thus, since we know that the fibring of `iand `ccollapses, it does also weakly collapse. However, the juxtaposition of `iand `cdoes not weakly collapse, because `ic is a strong conservative extension of both `iand `c. That is, we will have, for instance, ¬c¬cp`ic pbut ¬i¬ip0ic p. As I said before, there is a wealth of results that Schechter proved in his paper and here we have just seen some of the most relevant ones, but Schechter’s paper contains other interesting results. My next step, though, is to present some new, further applications of juxtaposition. 3.3.6 Further applications of juxtaposition Since Schechter’s seminal paper on juxtaposition, no further developments have been made on it (I believe that it has not even been applied in the literature). In this section, we apply the method of juxtaposition in order to obtain further results. Some of these have already been explored in the literature (see Béziau and Coniglio (2011) and Coniglio (2007)), but not under the method of juxtaposition. We start with a result that I have already mentioned above. Proposition 3.3.8.The juxtaposition of the logic of conjunction, L∧, and the logic of disjunction, L∨, is not distributive. Furthermore, L∧∨ is not the logic of lattices since absorption does not hold. Proof. Let P1=P2=P12 be a countably infinite set of sentence symbols and let C1and C2be the signatures containing these sets of connectives: •C2 1={∧} •C2 2={∨} Let, then, `∧and `∨be the consequence relations for Sent(C1,P1) and Sent(C2,P2) respectively. Thus, `∧∨ is the juxtaposed consequence relation for the set of sentences Sent(C12,P12). In order to prove this, let us make use again of the algebras we employed in the proof of Proposition 3.3.6, since, as Boolean algebras, they are both join and meet-semilattices. 90 Mixed Reasoning and Combining Logics ≤1 1∧ 0∧ ≤2 1∨ 0∨ ab We have to build coherent juxtaposed models in which nor distributivity neither absorption hold. That is: •Distributivity of ∧over ∨:p∧(q∨r)0∧∨ (p∧q)∨(p∧r) •Distributivity of ∨over ∧:p∨(q∧r)0∧∨ (p∨q)∧(p∨r) •Absorption-1: p∨(p∧q)0∧∨ p •Absorption-2: p∧(p∨q)0∧∨ p Let us take the following valuation V1(p) = 1∧,V1(q) = V1(r) = 0∧,V2(p) = 1∨,V2(q) = aand V2(r) = b. We know by Proposition 3.3.1 that there is a coherent juxtaposed model M12. In this model, ||p∧(q∨r)||1= 1∧uV1(q∨r). Since ||q∨r||2= 1∨,V1(q∨r) = 1∧, therefore, ||p∧(q∨r)||1= 1∧. But, in this very same model, ||(p∧q)∨(p∧r)||2=||p∧q||2t ||p∧r||2. Since ||p∧q||1=||p∧r||1= 1∧u0∧= 0∧, we can take V2((p∧q)) = V2((p∧r)) = 0∨ and we get that the premise is designated in the model while the conclusion is not. Then, the splicing of the logic of conjunction and the logic of disjunction by juxtaposition does not prove distributivity of ∧over ∨. Let us now show that absorption-1 does not hold either. Take V1(p) = V1(q)=0∧and V2(p) = a. In this model, ||p∨(p∧q)||2=atV2(p∧q). Since ||p∧q||1= 0∧, take V2(p∧q) = b. Thus, we get ||p∨(p∧q)||2= 1∨, while the conclusion is not designated. So, the juxtaposition of the logics of a meet-semilattice and a join-semilattice does not result in the logic of a lattice.  The reader can easily check by playing around with the valuations that there are juxtaposed countermodels for distributivity of ∨over ∧and absorption2 as well. 91 Mixed Reasoning and Combining Logics The next results have to do with recovering a logic from its fragments. More specifically, we deal with the case of classical logic. Proposition 3.3.9.The juxtaposition of the logic of (classical) negation, L¬, and the logic of (classical) conditional, L→, does not recover classical logic. Proof. We prove this proposition by showing that for the juxtaposed consequence relation, `¬→, despite Ex Contradictione Quodlibet holding as a rule, i.e. p, ¬p`¬→ q, we do not get the Principle of Pseudo-Scotus, `¬→ p→(¬p→q), which is valid in CL. Let P1=P2=P12 be a countably infinite set of sentence symbols and let C1and C2be the signatures containing these sets of connectives: •C1 1={¬} •C2 2={→} Let, then, `¬and `→be the consequence relations for Sent(C1,P1) and Sent(C2,P2) respectively. Thus, `¬→ is the juxtaposed consequence relation for the set of sentences Sent(C12,P12). We make use, again, of the previous Boolean algebras: ≤1 1¬ 0¬ ≤2 1→ 0→ ab In order to prove 0¬→ p→(¬p→q)we look for a coherent juxtaposed countermodel based on the above pair of structures. Thus, let us take a valuation such that V1(p) = 1¬,V1(q) = 0¬,V2(p) = 1→and V2(q) = 0→. Since ||¬p||1= 0¬, respecting coherence we take V2(¬p) = ain order to have ||p→(¬p→q)||2=−1→t(−at0→) = b, which is not designated, therefore, 0¬→ p→(¬p→q)as desired. However, we do have p, ¬p`¬→ q as announced. Just notice that since juxtaposition is a strong conservative extension and p, ¬p`¬qholds, we will also have that inference in `¬→. Moreover, there is no juxtaposed model satisfying both pand ¬p, so the argument is trivially valid.  92 Towards a Solution to the Problem of Mixed Inferences difference is that we distinguish, now, the sets of sentence symbols. So, Pc6=Piand Pc∪Pi=Pic. Let us start with the easiest case, namely, Mix, to see how it is done. First, we formalize the argument in the juxtaposed language and, then, we apply the method of juxtaposition as the validity criterion. 1Mix:Wet cats are funny Either snow is white or wet cats are not funny Snow is white Recall the assumption that the discourse about humour is an evaluative discourse in which reasoning is best captured by intuitionistic logic, while discourse about middle-sized objects, such as snow, is a discourse in which reasoning is best captured by classical logic. Under this assumption, let us formalize the argument by translating it into our juxtaposed language. pi qc∨x¬ipi qc (x=i, c) There is a first difficulty right away. It seems natural to translate the negation as the intuitionistic one, since it is being applied to an intuitionistic proposition1, i.e., ‘wet cats are funny’. However, which disjunction should we use in order to formalize the argument? Since it is a mixed sentence with one disjunct from the classical domain and another disjunct from the intuitionistic, it is not obvious whether the disjunction should be classical or intuitionistic. Remember, though, that Mix is among the arguments that are intuitively valid and, as we will see, the choice of the disjunction is not innocuous in this respect. 1This is not a strict rule and the context usually plays a major role in determining which the correct interpretation is, as it also happens with a single stock of connectives when, for instance, the use of an ‘if’ in a sentence might be ambiguous and further context might be needed in order to translate it as a conditional or a biconditional. Thus, there might be cases in which what the natural expression is trying to convey is, say, a classical negation applied to an intuitionistic proposition. We will discuss more these issues of translation later on. 99 Towards a Solution to the Problem of Mixed Inferences Indeed, if we translate the disjunction as the classical disjunction, the argument is invalid. An easy way to see this is noticing that, since the disjunction is classical, ¬ipiis a c-atom (classical atom) and so, the logical form of the argument is this: pi, qc∨cr`ic qc. ‘r’ is not a totally independent variable because if ||pi||x∈Dx, then ||¬ipi||x=||r||x6∈ Dx, but this constraint is not enough to make the argument valid. Let us offer a juxtaposed countermodel to show it. Proposition 4.1.1.pi, qc∨c¬ipi`ic qcis invalid.2 Proof. We build a coherent juxtaposed countermodel that invalidates the argument. We look, then, for a model based on a Heyting-Boolean structure. Consider the following algebras: 1B 0B abc 1H 0H We want to find valuations such that ||pi||x∈Dx,||qc∨c¬ipi||x∈Dxand ||qc||x6∈ Dx3. Since ||pi||x∈Dx,Vi(pi)=1Hand Vc(pi)=1B, so ||¬ipi||i= 0H. Given coherence, Vc(¬ipi)6∈ Dc, so take Vc(¬ipi) = b. Now we can choose Vc(qc) = a. With these valuations we get ||pi||c= 1B,||qc∨c¬ipi||c= atcb= 1Band ||qc||c=a. So, we have built a coherent juxtaposed model in which ||pi||x∈Dx,||qc∨c¬ipi||x∈Dxand ||qc||x6∈ Dxas desired.  But this cannot be the end of the story with Mix, otherwise juxtaposition would not be even a partial solution to the problem of mixed inferences because it would not be able to explain the validity of the easiest of the cases that Lynch and Wrenn consider. In fact, we have a way of explaining the intuitive validity of Mix, and this requires that we translate the disjunction as the intuitionistic one. That way, the logical form of the argument is that of 2This could already be problematic for juxtaposition. In fact, I will try to correct this when improving the method. 3Of course, the valuations that we are going to take have to respect coherence, in order to comply with the sufficient condition for the existence of coherent nontrivial juxtaposed models (Proposition 3.3.1). 100 Towards a Solution to the Problem of Mixed Inferences an intuitionistic disjunctive syllogism and, so, the intuitively valid argument will be validated by juxtaposition. Proposition 4.1.2.pi, qc∨i¬ipi`ic qcis valid. Proof. We want to show that for every coherent juxtaposed model that designates the premises, the conclusion will also be designated. So, suppose that ||pi||x∈Dxand ||qc∨i¬ipi||x∈Dx. For ||pi||x∈Dx,Vi(pi)=1H, so ||¬ipi||i= 0H. Now, ||qc∨i¬ipi||i∈Di, so ||qc∨i¬ipi||i= 1H, but having ||¬ipi||i= 0Hthe disjunction is designated only if ||qc||i∈Di. Thus, if the premises are designated the conclusion has to be designated too. Then, the argument is validated by juxtaposition.  Let us consider, now, the mixed inference dubbed Essential Mix: 2Essential Mix: Either it’s not the case that snow isn’t white or wet cats aren’t funny Wet cats are funny Snow is white We formalize the argument as: ¬c¬cpc∨x¬iqi qi pc Here, again, we have a similar situation. If we translate the disjunction as the classical disjunction, it turns out that the argument is invalid. The previous structures together with valuations Vc(pc) = a,Vi(qi) = 1H, Vc(¬ipi) = byield a coherent juxtaposed countermodel. However, we can explain the intuitive validity of the argument if we translate the disjunction as the intuitionistic one. Proposition 4.1.3.¬c¬cpc∨i¬iqi,qi`ic pcis valid. Proof. Let us prove the validity of the argument, in this case using the juxtaposed natural deduction calculus, in the way that Schechter describes it. 101 Towards a Solution to the Problem of Mixed Inferences 1¬c¬cpc∨i¬iqi 2qi 3¬c¬cpc 4¬c¬cpcIdentity, 3 5¬iqi 6⊥i¬iE, 2,5 7¬c¬cpc⊥iE, 6 8¬c¬cpc∨iE, 1–8 9pc¬c¬cE, 8 So, we have a derivation of the conclusion from the premises showing that the argument is valid.  Remember that Wrenn’s improvement on Lynch’s modesty criterion was able to explain the intuitive validity of these arguments we have just considered too. However, Wrenn comes up with the argument ‘Disjunctive Mix’, which is allegedly the knockdown mixed inference against localist proposals. He claims that, if the proposal is modest enough, which needs to be in order to invalidate arguments like ‘Nix’, then it will not be able to account for the validity of ‘Disjunctive Mix’. We will show, now, that this is not in fact the case. 3Disjunctive Mix: Either it’s not the case that snow isn’t white or wet cats are funny Either snow is white or wet cats are funny We formalize the argument as: ¬c¬cpc∨xqi pc∨xqi (x=i, c) 102 Towards a Solution to the Problem of Mixed Inferences Notice that in this mixed inference the translation that would preserve the logical form of a valid argument is the one with a classical disjunction. Under this translation ||¬c¬cpc||c=||pc||cand, so, ||¬c¬cpc∨cqi||c=||pc∨cqi||c, which makes the argument valid, contrary to what Wrenn predicts (given that we are able to explain the intuitive invalidity of ‘Nix’, as we will see below). Nevertheless, let us see that the argument would not be valid if we translated it as: ¬c¬cpc∨iqi pc∨iqi Proposition 4.1.4.¬c¬cpc∨iqi`ic pc∨iqiis invalid. Proof. We build a coherent juxtaposed countermodel that invalidates the argument. We look, then, for a model based on a Heyting-Boolean structure. Consider, for instance, the following algebras: 1B 0B ab 1H 0H cd We want valuations such that ||¬c¬cpc∨iqi||i∈Diand ||pc∨iqi||i6∈ Di. Take the following valuation, Vi(qi) = d,Vi(pc) = 0H,Vi(¬c¬cpc) = c,Vc(pc)=0Band Vc(qi) = a. It is easy to check that those structures together with these valuations give a coherent juxtaposed countermodel for the argument.  Up to this point we have shown that juxtaposition is not too modest with respect to accounting for the intuitive validity of the mixed inferences that Wrenn poses as a challenge to localism. Now, we have to see that juxtaposition is not too immodest. That is, we have to be able to explain the intuitive invalidity of Nix. 103 Towards a Solution to the Problem of Mixed Inferences 4Nix: If it is not the case that offensive jokes are funny, then grass is not green Grass is green Offensive jokes are funny We formalize the argument as: ¬ipi→x¬cqc qc pi (x=i, c) There is an important difference to notice in this case. We cannot simply find a translation in which the argument is (in)valid, as we did in the previous cases. Now, both translations of the conditional have to be invalidated by juxtaposition, otherwise juxtaposition would leave open a way of validating an argument that seems to be intuitively invalid4. Let us see how to find countermodels for each case. Proposition 4.1.5.¬ipi→c¬cqc,qc`ic piis invalid. Proof. We build a coherent juxtaposed countermodel based on the following Heyting-Boolean structure: 4This asymmetry between what is required for validating an argument (having a translation for which the argument is valid) and invalidating an argument (that no translation makes it valid) is not a bizarre feature of juxtaposition. Take the typical argument concluding Socrates’ mortality, which, obviously, is intuitively valid. If we were to translate it to propositional logic, the resulting argument would not be valid, but we can translate it to first-order predicate logic in order to account for its validity. On the contrary, if we have an argument in natural language which is intuitively invalid (say, ‘2 = 2, therefore, there is a McDonald’s on the dark side of the moon’) we would expect that any reasonable formalization to the language of a legitimate logic system would invalidate the argument. So, the problem that juxtaposition might have with this asymmetry is not because of the asymmetry per se, but because the other possible translations also seem intuitively valid and juxtaposition does not account for it. 104 Towards a Solution to the Problem of Mixed Inferences 1B 0B a 1H 0H Choose Vc(qc) = 1B,Vi(pi) = a. Then, ||¬cqc||c= 0Band ||¬ipi||i= 0H. Given coherence, since ||¬ipi||i6∈ Di,Vc(¬ipi)6∈ Dc, so Vc(¬ipi) = 0B. Therefore, ||¬ipi→c¬cqc||c= 1B(0B→c0B)and the argument is not valid, because ¬ipi→c¬cqc∈Dx,qc∈Dxbut pi/∈Dx. Proposition 4.1.6.¬ipi→i¬cqc,qc`ic piis invalid. Proof. Consider the same Heyting-Boolean structure and, in order to build the model, take Vc(qc)=1B,Vi(pi) = a. Then, ||¬cqc||c= 0B, so 6∈ Dc. Given coherence, Vi(¬cqc)6∈ Di, so choose Vi(¬cqc) = a. Since, Vi(pi) = a,||¬ipi||i= 0H. So, ||¬ipi→i¬cqc||i= 1H(0H⇒a)5. Therefore, for the valuation Vi(pi) = a,Vc(qc) = 1Band Vi(¬cqc) = a, the premises are designated while the conclusion is not, making the argument not valid as desired.  This concludes the application of the method of juxtaposition to the mixed inferences that Wrenn presents against localist proposals. Let me summarize the results: •Mix:qc∨i¬ipi,pi`ic qcand qc∨c¬ipi,pi0ic qc •Essential Mix:¬c¬cpc∨i¬iqi,qi`ic pcand ¬c¬cpc∨c¬iqi,qi0ic pc •Disjunctive Mix:¬c¬cpc∨cqi`ic pc∨cqiand ¬c¬cpc∨iqi0ic pc∨iqi •Nix:¬ipi→x¬cqc,qc0ic pi(x=i, c) 5Recall that a⇒bis the relative pseudo-complement of a with respect to b, which is the greatest element xsuch that a∧x≤b. Since ¬a=a⇒ ⊥ the value of ¬ais the greatest xsuch that a∧x=⊥. 105 Towards a Solution to the Problem of Mixed Inferences Remark 4.1.1.Let me briefly comment on these results. First, I reckon that the results make it clear that Wrenn’s claim against localism is, at least, too hasty. The application of the method of juxtaposition has allowed us to account for the (in)validity of the mixed inferences respecting our intuitions about them. This should be more than enough to avoid the direct conclusion that the problem of mixed inferences rules localism out. Even if it is not the perfect and definitive solution, juxtaposition is a first stepping stone towards a more satisfactory explanation of how we might combine different logics and systematise what follows from what in a mixed discourse, while adhering to a localist philosophy of logic. There is a good reason for juxtaposition to be in a good level of (im)modesty: the juxtaposition of two consequence relations is the minimal conservative extension of each of them. Therefore, the juxtaposed consequence relation will contain every inference that was already valid in each logic for its own stock of connectives and will not create new interactions or inferences for those stocks of connectives. As we saw when presenting juxtaposition, this is what guarantees that the combined consequence relation does not collapse. However, recall that this minimality and conservativeness was also the reason for blocking the appearance of bridge principles, such as the distributivity of conjunction over disjunction when juxtaposing the logics L∧and L∨. Now, I believe that we are in a situation in which juxtaposition, despite being a first step towards a solution, falls short of being a conclusive answer, precisely due to the incapacity for allowing interesting bridge principles. Which bridge principles are those? Luckily enough we already have them at hand. Consider again the mixed inferences that were supposed to be intuitively valid according to Wrenn. Take the case of Essential Mix, for instance, and try to think about the verdict given by juxtaposition (i.e., ¬c¬cpc∨i¬iqi, qi`ic pcand ¬c¬cpc∨c¬iqi,qi0ic pc) from a localist point of view. That is, from the point of view of a person (most likely a philosopher) that claims that the logic of evaluative discourse is intuitionistic logic and the logic of middle-sized objects is classical logic. Knowing that the set of valid inferences of IL is included in the set of valid inferences of CL, that the semantics of IL (Heyting algebras, Kripkean possible world semantics,...) generalizes that of CL (Boolean algebras, truth-conditional two-valued semantics, ...) and that the notion of ‘construction’, which is at the center of IL semantics, sets a higher epistemological standard than that of ‘truth’, which is crucial to CL semantics, how can we possibly justify that our method validates ¬c¬cpc∨i¬iqi,qi`ic pcbut invalidates the argument when changing the 106 Towards a Solution to the Problem of Mixed Inferences intuitionistic disjunction by the classical one? Look at it from this perspective: if we had a construction for qiwe would know that the value of ¬iqiis the bottom element in the Heyting algebra, and so that if we had a construction for ¬c¬cpc∨i¬iqiit must be because we had a construction for ¬c¬cpc. Now, if one is a localist and its intuitionistic semantics is telling you that you have a construction for the proposition qi how can you not assign to ¬iqithe bottom value of your classical semantics? And similarly, if when having the intuitionistic disjunction, the disjunction is designated because we have a construction for the disjunct ¬c¬cpc, how can it be that changing the disjunction to the classical one allows you to have ¬c¬cpcnot designated? It is an awkward situation to say the least. The fact is that ¬c¬cpc∨c¬iqi,qi`ic pcis a bridge principle, i.e. a new interaction principle between connectives of IL and CL that seems to be intuitively valid on the face of the examples that we have been considering, together with the philosophy of the logics in play in the mixed inference and a localist standpoint. And an identical reasoning works for the intuitive validity of qc∨c¬ipi,pi`ic qc. The case for ¬c¬cpc∨iqi`ic pc∨iqiis not so clear, though. Since, the intuitionistic standards for designation are higher than those of classical logic, it is not so obvious that the classical fact that a proposition and its double negation have the same semantic value should be preserved once we switch a classical connective into an intuitionistic one. Moreover, given that the set of valid inferences of IL is included in that of CL, perhaps it is not so awkward that an inference that was valid with a CL connective turns out to be invalid when changed by its corresponding intuitionistic one. Nevertheless, I reckon that there are legitimate localist positions that could make a case for the validity of ¬c¬cpc∨iqi`ic pc∨iqitoo. Later on, I will try to present a modified method of combination in order to be able to account for it as well. For the moment, let us try to make room for the more obviously valid bridge principles of Mix and Essential Mix. 107 Towards a Solution to the Problem of Mixed Inferences 4.2 Improving on the method of juxtaposition: coordinating logics for mixed inferences Let me very informally suggest the motivation behind this improvement with an analogy. Imagine you are trekking in an alpine terrain. You carry a map of the area and a compass in case you get lost. There are situations in which, only with the map, one could find the way back home. Suppose you are heading towards a shelter to spend the night over there and suddenly you stop seeing the milestones that mark the trail. You look around trying to look for something to orient yourself. Far in the distance you recognise the shape of a summit and look for it in the map. Your shelter is at the foot of the mountain, so you know which direction you must take. Now, there are also situations in which just the compass will do, for instance, if you know that the shelter is to the south-west of where you are, you might be able to find your way out. But there are other situations in which, neither the compass nor the map alone will be enough. That is, situations in which you need to use the map and the compass in combination because each of them can have more applications when used coordinated with the other. When used in coordination, you can calculate the exact course you must follow, even if you do not know much about the geographical features surrounding you. Following the analogy, juxtaposition is like having the map and the compass in your backpack but using them as if they where independent instruments. You have, say, intuitionistic and classical logic, you have every valid inference for their respective languages, but you do not allow properly new interactions. However, there seem to be situations, in our case, contexts in which we reason across domains employing mixed inferences, that require new interactions. That is, the logics in combination can potentially generate more valid inferences than only by themselves, just like the map and the compass are useful in more situations when they are used in combination than by their own. In fact, the very name of ‘juxtaposition’ suggests this situation of having two or more things together but not interacting. That is why, even if it is a modification of juxtaposition, I will call the improved method ‘coordination’ (C) in order to make clear that the combination we are looking for allows for the emergence of bridge principles. The goal, then, is to modify the combination mechanism in such a way 108 Towards a Solution to the Problem of Mixed Inferences 4Preservation theorems for Coordination Now that we know the combination mechanism, let us offer some metatheoretical results. Some of them are direct consequences of those given by Schechter and some other’s require more or less significant modifications to be proved. We begin by proving the existence of coordinated nontrivial models. First, we define the semantic notion of a coordination of two models just as Schechter defines a juxtaposition of two models. Suppose Mi=hBi, Viiis a model over Ciand Piand Mc=hBc, Vci is a model over Ccand Pc. A coordination of the models Miand Mcis a coordinated model hBi, V + i,Bc, V + ciover Ci,Ccand Pic such that: •If p∈Pi, V + i(p) = Vi(p)and •If p∈Pc, V + c(p) = Vc(p) Notice that if MC ic is a coordination of Miand Mc, for any α∈Sent(Cx, Px) (for x=i, c), ||α||MC ic x=||α||Mx. Therefore, MC ic αjust in case Mxα. We now provide a necessary and sufficient condition for the existence of coordinated models. Proposition 4.2.1 (Existence of Coordinated Nontrivial Models).Suppose Ciand Ccare disjoint signatures. Suppose Mi=hBi, Viiis a model over Ci and Piand Mc=hBc, Vciis a model over Ccand Pc, satisfying the condition that if the cardinality of the set of semantic values of Biis two, |Bi|= 2, then |Bc|= 210. Then there is a coordinated nontrivial model, MC ic, over Cic and Pic based on Bic, just in case for every p∈Pi∩Pc,Mipjust in case Mcpand if ||p||Mi=⊥i, then ||p||Mc=⊥c. Proof. Suppose there is some p∈Pi∩Pcsuch that, either Mipand Mc2por Mi2pand Mcpor ||p||Mi=⊥ibut ||p||Mc6=⊥c, then there is no coordination of Miand Mc. Now, suppose that for every p∈Pi∩Pc,Mipjust in case Mcpand if ||p||Mi=⊥i, then ||p||Mc=⊥c. We show that there is a weak coordination of Miand Mc. 10This is because, if we allowed |Bc|>2when |Bi|= 2, if α∈Sent(Cc,Pc) and ||α||Mcis non-designated but not-bottom, when doing the coordination, we would have to evaluate this intuitionistic atom as ⊥i, since it is the only non-designated value in the two-element Boolean algebra. But, then, we would not get a coordinated model, since we would have ||α||MC ic i=⊥iand ||α||MC ic c=bc6=⊥c, where bcis a non-designated non-bottom value. 115 Towards a Solution to the Problem of Mixed Inferences Let >xbe the top value of Bx,⊥xbe the bottom value of Bxand let bx be an element of Bx− {>x}. We inductively define, [ ]x, the function from Sent(Cic,Pic) to Bxsuch that: •If p∈Px,[p]x=Vx(p); •If p∈Pi−Pc,[p]c=>cif Vi(p) = >i,[p]c=⊥cif Vi(p) = ⊥iand [p]c=bcotherwise; •If p∈Pc−Pi,[p]i=>iif Vc(p) = >c,[p]i=biif Vc(p) = ⊥cand [p]i=bi(6=⊥i)otherwise; •If c∈Cn x,[cα1...αn]x= Φx(c)([α1]x...[αn]x); •If c∈Cn i,[cα1...αn]c=>cif Φi(c)([α1]i...[αn]i) = >i,[cα1...αn]c=⊥c if Φi(c)([α1]i...[αn]i) = ⊥iand [cα1...αn]c=bcotherwise; •If c∈Cn c,[cα1...αn]i=>iif Φc(c)([α1]c...[αn]c) = >c,[cα1...αn]i=biif Φc(c)([α1]c...[αn]c) = ⊥cand [cα1...αn]i=bi(6=⊥i)otherwise; If αis an x-atom, let V+ x(α) = [α]x. Let Mic =hBi, V + i,Bc, V + ci. We show that for every α∈Sent(Cic,Pic), ||α||Mic i=>iiff ||α||Mic c=>cand if ||α||Mic i=⊥i, then ||α||Mic c=⊥c. That is, [α]i=>iiff [α]c=>cand if [α]i=⊥ithen [α]c=⊥c. If p∈Pi∩Pc,[p]i=Vi(p). So, [p]i=>iiff Vi(p) = >iiff Vc(p) = >ciff [p]c=>c, and [p]i=⊥iiff Vi(p) = ⊥i. But, if Vi(p) = ⊥i, then Vc(p) = ⊥c and, so, [p]c=⊥c. If p∈Pi−Pc,[p]i=>iiff Vi(p) = >iiff [p]c=>c. Also, [p]i=⊥iiff Vi(p) = ⊥iand if Vi(p) = ⊥i, then [p]c=⊥c. If p∈Pc−Pi,[p]c=>ciff Vc(p) = >ciff [p]i=>i. Also, if [p]i=⊥ithen Vc(p) = ⊥c= [p]c11. If c∈Cn iand α1...αn∈Sent(Cic,Pic), [cα1...αn]i=>iiff Φi(c)([α1]i...[αn]i) = >iiff [cα1...αn]c=>c12. Also, [cα1...αn]i=⊥iiff Φi(c)([α1]i...[αn]i) = ⊥i and [cα1...αn]c=⊥cif Φi(c)([α1]i...[αn]i) = ⊥i. 11Notice that there is no other option. If [p]i=⊥ithen it can only be that Vc(p) = ⊥c. The reasoning is by contraposition. If it had any other non-designated classical value then pcould not have the intuitionistic bottom value. 12Since Ciand Ccare disjoint, the only reason for having [cα1...αn]c=>cis because Φi(c)([α1]i...[αn]i) = >i. Again, reasoning with the contrapositive might be helpful. 116 Towards a Solution to the Problem of Mixed Inferences If c∈Cn cand α1...αn∈Sent(Cic,Pic), [cα1...αn]c=>ciff Φc(c)([α1]c...[αn]c) = >ciff [cα1...αn]i=>i. If [cα1...αn]i=⊥ithen Φc(c)([α1]c...[αn]c) = [cα1...αn]c= ⊥c. Since we are doing the coordination of intuitionistic and classical models, it is obvious that this coordinated model will be nontrivial (because there are α1, α2∈Sent(Cic,Pic) such that MC ic α1and MC ic 2α2).  Now, we proceed with the proof of strong soundness of the coordinated consequence relation (axiomatized by the coordinated natural deduction calculus), `ic C, with respect to the class of coordinated Heyting-Boolean structures, Bic. Before proving the main result we need two auxiliary lemmas (similar to Lemmas 5.3 and 5.4 in Schechter (2011)). Lemma 4.2.1.Suppose Bic is the class of coordinated structures over Ciand Cc. Then, Bic Cis a consequence relation for Sent(Cic,Pic). Proof. We have to show that Bic Csatisfies Identity, Weakening, Cut and Uniform Substitution. For the first three the proof is basically that of Schechter in Lemma 5.3. We focus, then, on Uniform Substitution, i.e., if ΓBic Cαthen Γ[β/p]Bic Cα[β/p], to extend and clarify what Schechter does. We prove its contrapositive. So, suppose Γ[β/p]2Bic Cα[β/p]. This means that there is a coordinated model, MC ic, such that MC ic Γ[β/p]and MC ic 2α[β/p]. With the help of this model, we build another coordinated model, M0C ic =hBi, V 0 i,Bc, V 0 ci, by letting V0 x(δ)=||δ[β/p]||MC ic x whenever δis an x-atom. We show that ||δ||M0C ic x=||δ[β/p]||MC ic xfor every δ∈ Sent(Cic,Pic), by induction on the complexity of the formulas: •Base case: we have defined V0 x(δ) in such a way that if δis an x-atom the equality holds. •Inductive Hypothesis (IH): Assume that the equality holds for all formulas less complex than α. We show that the equality holds for any possible α. Assume that α=¬xγ. By IH, we know that ||γ||M0C ic x= ||γ[β/p]||MC ic x. So, clearly ||¬xγ||M0C ic x=||¬xγ[β/p]||MC ic x. Assume now that α=γ∨xω. By IH, we know that ||γ||M0C ic x= ||γ[β/p]||MC ic xand ||ω||M0C ic x=||ω[β/p]||MC ic x. Again, it is clear that the 117 Towards a Solution to the Problem of Mixed Inferences same operation, namely, ∨x, over the same semantic values will yield the same semantic value. So, ||γ∨xω||M0C ic x=||(γ∨xω)[β/p]||MC ic x. One can easily check that the same holds for α=γ∧xωand α=γ→xω. Therefore, M0C ic is a coordinated model such that M0C ic Γand M0C ic 2 α. Lemma 4.2.2.Suppose Bic is the coordination of Biand Bc. If ΓBiαor ΓBcα, then ΓBic Cα. Proof. Similar to the proof of Schechter for Lemma 5.4. The idea is that if ΓBxα, with Γ∪α⊆Sent(Cx,Px), then we take a coordinated model MC ic =hBi, Vi,Bc, Vcisuch that MC ic Γand Mx|Px=hBx, Vx|Pxiis the restriction of Mxto Px, making ||β||Mx|Px=||β||MC ic xfor every β∈Sent(Cx, Px). With this and knowing that Mx|Pxis based on Bx, we get that MC ic α. So, ΓBic Cα. Now we have the ingredients to prove the main result. Theorem 4.2.1 (Strong Soundness).The coordinated natural deduction calculus is strongly sound with respect to the class of Heyting-Boolean structures. That is, Γ`ic Cα⇒ΓBic Cα. Proof. By the previous lemmas, we know that Bic Cis a consequence relation and that if ΓBiαor ΓBcα, then ΓBic Cα. We also know that the juxtaposed natural deduction of intuitionistic and classical logics is the minimal conservative extension of them and that IL and CL are strongly sound w.r.t. the classes of Heyting and Boolean algebras respectively. From there, we get that, for the juxtaposed natural deduction derivation, for any Γ and α, if Γ`ic αthen ΓBic α. Therefore, we know that Γ`ic α⇒Γ`ic Cα (because we have all the previous rules and some of them have fewer restrictions), we also know that Γ`ic α⇒ΓBic α(from Schechter’s proof of Soundness) and, finally, also that ΓBic α⇒ΓBic Cα(because every coordinated model is a juxtaposed model but not vice versa). Thus, now we just need to consider the rules that we have changed, i.e., relaxed, for giving the coordinated natural deduction calculus of intuitionistic and classical logics, and check that they are strongly sound with respect to the coordinated semantics. 118 Towards a Solution to the Problem of Mixed Inferences The proof is, as usual, by induction on the length of the derivation. Let us start with the base case, namely, proofs of size k= 1: •Base case (k= 1): if Γ`ic Cαand the length of the derivation is k= 1, then α∈Γ. Since α∈Γ, for every coordinated model, MC ic, if MC ic Γ, then MC ic α. Therefore, ΓBic Cα. •Inductive Hypothesis (IH): assume that if Γ`ic Cαand the length of the derivation is ≤k, then ΓBic Cα. We have to show that this also holds for the new rules with derivations of length k+ 1. Let me start with ¬cI: (¬cI)γ1 . . . γj Π1 α Π2 k⊥i k+ 1 ¬cα Recall that, within Π2only IL rules are allowed or repetitions of φsuch that (Γ`ic Cφ)∈Π113. The derivation ends with an application of ¬cI in line k+ 1, but previously, we get to ⊥ifrom undischarged assumptions Γ∪ {α}, in ≤klines. Therefore, Γ∪ {α} `ic C⊥iand, by IH, Γ∪ {α}Bic C⊥i.14 Now, we want to show that for every model MC ic designating every formula in Γ,|| Vx(Γ)||MC ic x=>x, the value of αhas to be bottom, ||α||MC ic i=⊥i, and, also, that within Π1we could have used rules both from CL and IL 13Of course, this includes any of the γ∈Γand, in fact, in most of the cases in which I will apply the rules, what is going to be repeated within Π2is one of the premisses in Γ. 14Notice that with Γ∪ {α} `ic C⊥i⇒Γ∪ {α}Bic C⊥iwe are in the same scenario as in a standard intuitionistic negation introduction rule, since in Π2we have only applied IL rules. So, the reasoning for showing that ||α||i=⊥ishould be the same. 119 Towards a Solution to the Problem of Mixed Inferences natural deductions. So, notice that for whatever rule we applied in Π1, if || Vx(Γ)||MC ic x=>xand ΓBic Cφ(because (Γ`ic Cφ)∈Π1in ≤klines), then ||α||MC ic x=>x15. So, if some formula φwas used together with αfor obtaining ⊥i, that formula takes the value >iin the Heyting algebra for the models that make || Vi(Γ)||MC ic i=>i. Since from assuming αonly IL rules were used in Π2and these are sound with respect to the class of Heyting algebras, where for any ∆and ω,∆HA ω iff || V(∆)||HA ≤ ||ω||HA, then, Γ∪ {α}Bic C⊥iiff || Vi(Γ)∧iα||MC ic i≤ ||⊥i||MC ic i for every MC ic. So, take any coordinated model Mjsuch that || Vi(Γ)||Mj i= >i. Given that for every MC ic,|| Vi(Γ) ∧iα||MC ic i=⊥i, for Mjin particular, || Vi(Γ) ∧iα||Mj i=⊥i. That is, ||α||Mj i=⊥iand, by coordination, ||α||Mj c= ⊥c. Therefore, for every MC ic, if || Vi(Γ)||MC ic i=>i, then ||¬cα||MC ic c=>c. Hence, ΓBic C¬cαas desired. The next rule we have to consider is →cI. If no intuitionistic rule is applied within the subderivation, we know that the rule is sound because the natural deduction calculus for CL is strongly sound with respect to the class of Boolean algebras. When intuitionistic rules are applied within →cI, then the rule is as follows and it is sound with respect to the coordinated semantics: (→cI)γ1 . . . γj Π1 α Π2 ⊥i k δ k+ 1 α→cδ 15This shows that whatever we used by repetition after assuming α(any φsuch that (Γ`ic Cφ)∈Π1) has value >xin the coordinated models that make || Vx(Γ)||MC ic x=>x. 120 Towards a Solution to the Problem of Mixed Inferences Just as with ¬cI, notice that within Π2only IL rules are allowed or repetitions of φsuch that (Γ`ic Cφ)∈Π1. In this case, the derivation ends with an application of →cI in line k+1, but previously, we get to ⊥iand δfrom undischarged assumptions Γ∪ {α}, in ≤klines. Therefore, Γ∪ {α} `ic C⊥i and, by IH, Γ∪ {α}Bic C⊥i. Now, notice that what we need to do is exactly what we did in the previous case, which is to prove that for every model MC ic designating every formula in Γ,|| Vx(Γ)||MC ic x=>x, the value of αhas to be bottom, ||α||MC ic i=⊥i. So, by the same argument as before, which relied on the fact that Γ∪ {α} `ic C ⊥i⇒Γ∪ {α}Bic C⊥i, we know that for every MC ic, if || Vi(Γ)||MC ic i=>i, then ||α||MC ic i=⊥iand, by coordination, ||α||MC ic c=⊥c. Given the truth conditions of the classical conditional, if in every coordinated model in which Γis designated, the value of αis ⊥c, then for every coordinated model, if MC ic Γthen MC ic α→cδ. Hence, ΓBic Cα→cδas desired. We conclude by showing the soundness of the rule of ∨cE when IL rules are applied within the subderivations: 121 Towards a Solution to the Problem of Mixed Inferences (∨cE)γ1 . . . γj Π1 α∨cβ α Π2 ⊥i δ β Π3 δ k+ 1 δ Like in the previous cases, within Π2only IL rules are allowed or repetitions of φsuch that (Γ`ic Cφ)∈Π1. The derivation ends with an application of ∨cE in line k+ 1, but previously, we get to ⊥iand δfrom undischarged assumptions Γ∪ {α}, in ≤k, and we also get δfrom undischarged assumptions Γ∪ {β}, in ≤k. Therefore, Γ∪ {α} `ic WC ⊥iand Γ∪ {β} `ic WC δ and, by IH, Γ∪ {α}Bic C⊥iand Γ∪ {β}Bic Cδ. And, again, notice that the same argument that we used above can be applied here to conclude that for every MC ic, if || Vi(Γ)||MC ic i=>i, then ||α||MC ic i=⊥iand, by coordination, ||α||MC ic c=⊥c. Now, we also have that Γ∪{β}Bic Cδ, from which we know that every coordinated model that satisfies Γ∪{β}also satisfies δ. Thus, we can conclude that every model satisfying Γ∪ {α∨cβ}is a model satisfying Γ∪ {β}and, therefore, also δ16. Hence, for every coordinated model, if MC ic Γ∪{α∨cβ} 16Notice that here the semantic clause of coordination is crucial, as in the previous rules. This is what guarantees that, since we concluded that the intuitionistic semantic value of 122 Towards a Solution to the Problem of Mixed Inferences then MC ic δ, so, Γ∪ {α∨cβ}Bic Cδ, as desired. This concludes the strong soundness proof for weak coordination.  4.2.2 Applying coordination to mixed inferences: a better reply to Wrenn One of the motivations for developing coordination was to allow for the emergence of bridge principles when combining intuitionistic and classical logics, especially, those cases of bridge principles belonging to Mix and Essential Mix. If we are capable of doing this while assuming a localist stand towards logic, then we are in a good position for meeting Wrenn’s challenge against localism and, in general, for giving a localist account of mixed inferences. Let us show that we can, in fact, account for the validity of Mix and Essential Mix when they are formalized as bridge principles. 1Mix:Wet cats are funny Either snow is white or wet cats are not funny Snow is white Let us formalize the argument as a bridge principle by translating it into our coordinated language and recall the assumption that the discourse about humour is an evaluative discourse in which reasoning is best captured by intuitionistic logic, while discourse about middle-sized objects, such as snow, is a discourse in which reasoning is best captured by classical logic. pi qc∨c¬ipi qc I showed in Proposition 4.1.1 that this argument was invalidated by juxtaposition. Now I will show that coordination makes the argument valid, as desired. Proposition 4.2.2.pi, qc∨c¬ipiic Cqcis valid. αhas to be ⊥i, its classical semantic value, under the scope of ∨c, has to be ⊥c. 123 Towards a Solution to the Problem of Mixed Inferences Proof. We want to check that for every coordinated model based on Bic, if the premises are designated, then the conclusion is designated too. Suppose then, that ||pi||x∈Dxand ||qc∨c¬ipi||x∈Dx.||pi||x∈Dxjust in case ||pi||i=>i. If ||pi||i=>i, then ||¬ipi||i=⊥iand Vc(¬ipi) = ⊥c(by the second condition of coordination, i.e. if ||α||i=⊥ithen ||α||c=⊥c). Thus, in order for qc∨c¬ipito be designated, ||qc||c=>c. Therefore, ||qc||c∈Dc and, so, the argument is validated by coordination.  2Essential Mix: Either it’s not the case that snow isn’t white or wet cats aren’t funny Wet cats are funny Snow is white We formalize the argument as: ¬c¬cpc∨c¬iqi qi pc This argument too was invalidated by juxtaposition. Let us see that we can now account for its validity applying coordination. Proposition 4.2.3.¬c¬cpc∨c¬iqi,qiic Cpcis valid. Proof. We prove it, this time, using our coordinated natural deduction, since we know that it is strongly sound with respect to the class of HeytingBoolean structures. 124 Towards a Solution to the Problem of Mixed Inferences 1c¬ip 2c p 3⊥i¬iE, 1,2 4¬cp¬cI, 2–3 5¬ip→c¬cp!!→cI, 1–4  The problem with this derivation is that it does not respect the restrictions that we have established in order to apply IL rules within CL subderivations. On one hand, the rules allow the application of IL rules within CL subderivations but, in that case, every rule applied within the CL subderivation must be intuitionistic. However, in this derivation both ¬iE and ¬cI are applied within →cI. On the other hand, if IL rules are applied within →cI, the only way of closing is with ⊥iand δin the outermost subderivation (in this case the one having ¬ipas an assumption). But we do not get that in this subderivation and, therefore, the rule →cI is badly applied. Notice, though, that once pis forced to have a semantic value that reduces the possible translations between structures (i.e. ||p||i=>iiff ||p||c=>c or if ||p||i=⊥ithen ||p||c=⊥c) and, therefore, the flexibility for getting countermodels, the argument might be valid. For instance, we can put pin the premises to get a valid argument: •pic C¬ip→c¬cp Proof. Suppose ||p||i=>i. Then, ||¬ip||i=⊥i, so ||¬ip||c=⊥c. This is enough to see that the conclusion will be designated too, i.e. ||¬ip→c ¬cp||c=>c. And with the natural deduction, 131 Towards a Solution to the Problem of Mixed Inferences 1p 2c¬ip 3⊥i¬iE, 1,2 4¬cp⊥iE, 3 5¬ip→c¬cp→cI, 1–4  •¬ip, q, ¬ir, (p∨ir)∨c(q→is)ic Cs Proof. Suppose that ||¬ip||x=||q||x=||¬ir||x=||(p∨ir)∨c(q→is)||x= >x. Since, ||¬ip||i=||¬ir||i=>i, then ||p||i=||r||i=⊥iand, so, ||p∨ir||i= ⊥i. By coordination, ||p∨ir||c=⊥cwhenever ||p∨ir||i=⊥i. Therefore, ||(p∨ir)∨c(q→is)||x=>xiff ||q→is||x=>xiff ||s||x=>x(because we have supposed ||q||x=>x). And by the coordinated natural deduction, 132 Towards a Solution to the Problem of Mixed Inferences 1¬ip 2q 3¬ir 4 (p∨ir)∨c(q→is) 5¬ip∧i¬ir∧iI, 1,3 6¬i(p∨ir)De Morgan, 5 7c p ∨ir 8⊥i¬iE, 6,7 9q→is⊥iE, 8 10 c q →is 11 q→isIdentity, 10 12 q→is∨cE, 4,7–11 13 s→iE, 2,12  I will end up with a couple more cases to make sure that different types of inferences are available to the reader in order to enhance the comprehension of the method. •p, q ic C(p→i¬iq)→c¬cq 133 Towards a Solution to the Problem of Mixed Inferences Proof. 1p 2q 3c p →i¬iq 4¬iq→iE, 1,3 5⊥i¬iE, 2,4 6¬cq⊥iE, 5 7 (p→i¬iq)→c¬cq→cI, 3–6  Let me conclude with an argument which I believe might be quite suggestive, since it is the ‘mirror image’ of a version of Mix. Consider the following argument: Snow is white Either wet cats are funny or snow isn’t white Wet cats are funny So, we formalize the argument like this: pc qi∨x¬cpc qi (x=i, c) In this case, the disjunctive syllogism is occurring with the classical part of the logic while in the original Mix, we had an intuitionistic disjunctive syllogism. In this second version, if we translate the argument with a classical disjunction, the argument is clearly valid (already in juxtaposition and a fortiori in coordination), since it is an instance of a classically valid argument. However, if we translate the argument with an intuitionistic disjunction, we can give a coordinated countermodel to it. So, •pc, qi∨i¬cpc2ic Cqi 134 Towards a Solution to the Problem of Mixed Inferences Proof. Take the pair of structures that we have been using in this section and valuations Vc(pc) = >c,Vi(qi) = c.||¬cpc||c=⊥c, so by coordination, ||¬cpc||i/∈Di, so take Vi(¬cpc) = d. Then, ||qi∨i¬cpc||i=>i, so ||pc||x∈Dx, ||qi∨i¬cpc||x∈Dx, while ||qi||x/∈Dx. And, from the natural deduction side, it is quite easy to see that we cannot make the derivation because that requires the application of a classical rule within an intuitionistic subderivation. However, the important question is whether our combined logic system of classical and intuitionistic logic should make this inference valid or not. So, is the argument intuitively valid? Or are there philosophical and localist reasons that could be provided in favour of the validity of the argument? Or, another way of putting it, should our combined system aim for this kind of bridge principles too? Is coordination still too modest? To my mind, it is not obvious that it is a bridge principle for which there is good enough justification. In fact, some of the reasons that we gave for the clause ‘if ||α||i=⊥ithen ||α||c=⊥c’ and so, for letting the emergence of bridge principles like qc∨c¬ipi,piic Cqc, would speak against accepting this new bridge principle. The intuitionistic epistemic standard is higher than the classical, intuitionistic logic is weaker than (i.e. it is included in) classical logic, having a proof for the absurdity of a proposition seems to imply the falsity of that proposition, etc. So, there might be localists who will find these reasons appealing enough as to stick with coordination and avoid going beyond it by letting more suspicious bridge principles emerge. Yet, there might be some other localists who find the bridge principle appealing for other reasons. To start with, it is true that the epistemic standard is higher for intuitionism than for classical logic, but that might have less to do with the top and bottom values of the algebras and more with the structural features of the intermediate ones. That is, with the fact that not for every semantic value, x, in a Heyting algebra, A,¬¬x=xand ¬x∨x=>. In fact, despite the different epistemic standards and semantic conceptions of what it takes for a proposition to get the top value, the localist defending coordination is already accepting that ||α||i=>iiff ||α||c=>c, so why not accept also that ||α||i=⊥iiff ||α||c=⊥c? This is enough for bridge principles like pc, qi∨i¬cpcic Cqito emerge, indeed. In the following section I will explore this new strengthening of juxtaposition that I call ‘strong coordination’. I hope it is clear enough that I am not using ‘strong’ or the lack of an adjective as a value judgement.For 135 Towards a Solution to the Problem of Mixed Inferences the moment, I am exploring the different possibilities and analysing how to make them technically viable while assessing their possible philosophical consequences. I reckon it is a complex and delicate matter how to weigh the technical and philosophical virtues and shortcomings of each of them, but hopefully the analysis will shed some light on this new issue before us. 4.2.4 Strong Coordination (SC) Let me introduce a further strengthening of the juxtaposition, which is also a strengthening of coordination. From the syntactic point of view, strong coordination is a consequence relation and, like coordination, it is a particular case of juxtaposed consequence relation. That is, the strongly coordinated consequence relation, `ic SC, is a juxtaposed consequence relation that extends `i,`c,`ic and `ic C. Since we cannot appeal to minimality, as it is done with the juxtaposed consequence relation, in order to discriminate `ic SC from other juxtaposed consequence relations, we will characterize it by means of its semantic and syntactic properties. 1Semantics of Strong Coordination With respect to the semantics the approach is almost the same. The strong coordination of the structures Bi, over Ci, and Bc, over Cc, is the juxtaposition of the structures, hBi,Bci, and the strong coordination of the classes of structures Biand Bcis the Cartesian product Bi×Bc, which is the juxtaposition of the classes of structures. Astrongly coordinated model, MSC ic =hBi, Vi,Bc, Vciover Cic and Pic, based on the strongly coordinated structure hBi,Bci(or, more generally, based on the class of strongly coordinated structures Bic) is a coherent juxtaposed model satisfying the following property: •Strong Coordination: A model is strongly coordinated when for every α∈Sent(Cic,Pic), 1. ||α||i=>iiff ||α||c=>c 2. ||α||i=⊥iiff ||α||c=⊥c Again, part of the motivation for the strengthening of the second clause has come from the fact that bridge principles like pc, qi∨i¬cpc`qiwere 136 Towards a Solution to the Problem of Mixed Inferences invalidated by coordination. We have seen some reasons for this argument to be invalid and for the clause of coordination, but I have also pointed out some reasons that might justify strong coordination. In any case, after presenting the method and some of its results, we will come back to its adequacy as a localist solution to mixed inferences and to its justification. 2A Natural Deduction calculus for Strong Coordination As one might expect, just as with coordination the natural deduction calculus reflected the asymmetry of the semantic clauses, now the strongly coordinated natural deduction calculus has to be modified in such a way that we get the symmetry of the semantic clauses of strong coordination in the calculus. Therefore, what we need to do is just to relax the intuitionistic rules opening subderivations in the same way that we relaxed the classical ones. That is, on top of the coordinated natural deduction calculus, we allow the application of classical rules within ¬iI, →iI and ∨iE with the same caveats that had their analogous classical rules. Let us consider each intuitionistic rule opening subderivations and specify, just in case, how to apply classical rules within them. Consider, first, the rule of intuitionistic negation introduction: (¬iI)α Π ⊥i ¬iα If no application of a classical rule occurred within ¬iI, the rule is just the standard intuitionistic rule. If there is an application of a classical rule within ¬iI, then the subderivation has to be closed like this: (¬iI)α Π ⊥c ¬iα 137 Towards a Solution to the Problem of Mixed Inferences Analogously, here we require that the only rules that can be applied in the derivation Πare classical rules, together with repetitions of formulas derived from the premisses of the argument. For →iI and ∨iE, let me jump directly to the relaxed versions of the standard rules. The details for applying the rules mirror exactly those of the coordinated natural deduction calculus. (→iI)α Π ⊥c β α→iβ (∨iE)α∨iβ α Π1 ⊥c δ β Π2 δ δ 3Preservation theorems for Strong Coordination Just as we did in the case of coordination, let me present now some meta-theoretical results concerning strong coordination. We start, as we did before, proving the existence of strongly coordinated nontrivial models. This time, we skip the definition of a strong coordination of the models Miand Mc, since it remains the same as before. 138 Towards a Solution to the Problem of Mixed Inferences Proposition 4.2.4 (Existence of Strongly Coordinated Nontrivial Models). Suppose Ciand Ccare disjoint signatures. Suppose Mi=hBi, Viiis a model over Ciand Piand Mc=hBc, Vciis a model over Ccand Pc, satisfying the condition that |Bi|= 2 just in case |Bc|= 220. Then there is a strongly coordinated nontrivial model, MSC ic , over Cic and Pic based on Bic, just in case for every p∈Pi∩Pc,Mipjust in case Mcpand ||p||Mi=⊥iiff ||p||Mc=⊥c. Proof. Suppose there is some p∈Pi∩Pcsuch that, either Mipand Mc2por Mi2pand Mcpor ||p||Mi=⊥iand ||p||Mc6=⊥cor ||p||Mc=⊥c and ||p||Mi6=⊥i, then there is no strong coordination of Miand Mc. Now, suppose that for every p∈Pi∩Pc,Mipjust in case Mcp and ||p||Mi=⊥ijust in case ||p||Mc=⊥c. We show that there is a strong coordination of Miand Mc. Let >xbe the top value of Bx,⊥xbe the bottom value of Bxand let bx be an element of Bx− {>x,⊥x}. We inductively define, [ ]x, the function from Sent(Cic,Pic) to Bxsuch that: •If p∈Px,[p]x=Vx(p); •If p∈Pi−Pc,[p]c=>cif Vi(p) = >i,[p]c=⊥cif Vi(p) = ⊥iand [p]c=bcotherwise; •If p∈Pc−Pi,[p]i=>iif Vc(p) = >c,[p]i=⊥iif Vc(p) = ⊥cand [p]i=biotherwise; •If c∈Cn x,[cα1...αn]x= Φx(c)([α1]x...[αn]x); •If c∈Cn i,[cα1...αn]c=>cif Φi(c)([α1]i...[αn]i) = >i,[cα1...αn]c=⊥c if Φi(c)([α1]i...[αn]i) = ⊥iand [cα1...αn]c=bcotherwise; •If c∈Cn c,[cα1...αn]i=>iif Φc(c)([α1]c...[αn]c) = >c,[cα1...αn]i=⊥i if Φc(c)([α1]c...[αn]c) = ⊥cand [cα1...αn]i=biotherwise; If αis an x-atom, let V+ x(α) = [α]x. Let Mic =hBi, V + i,Bc, V + ci. We show that for every α∈Sent(Cic,Pic), ||α||Mic i=>iiff ||α||Mic c=>cand ||α||Mic i=⊥iiff ||α||Mic c=⊥c. That is, [α]i=>iiff [α]c=>cand [α]i=⊥i iff [α]c=⊥c. 20The reason is analogous to that given for the existence of coordinated nontrivial models. 139 Towards a Solution to the Problem of Mixed Inferences If p∈Pi∩Pc,[p]i=Vi(p). So, [p]i=>iiff Vi(p) = >iiff Vc(p) = >ciff [p]c=>c, and [p]i=⊥iiff Vi(p) = ⊥iiff Vc(p) = ⊥ciff [p]c=⊥c. If p∈Pi−Pc,[p]i=>iiff Vi(p) = >iiff [p]c=>c. Also, [p]i=⊥iiff Vi(p) = ⊥iiff [p]c=⊥c. If p∈Pc−Pi,[p]c=>ciff Vc(p) = >ciff [p]i=>i. Also, [p]i=⊥iiff Vc(p) = ⊥c= [p]c. If c∈Cn iand α1...αn∈Sent(Cic,Pic), [cα1...αn]i=>iiff Φi(c)([α1]i...[αn]i) =>iiff [cα1...αn]c=>c. Also, [cα1...αn]i=⊥iiff Φi(c)([α1]i...[αn]i) = ⊥iiff [cα1...αn]c=⊥c. If c∈Cn cand α1...αn∈Sent(Cic,Pic), [cα1...αn]c=>ciff Φc(c)([α1]c...[αn]c) =>ciff [cα1...αn]i=>i. Also, [cα1...αn]c=⊥ciff Φc(c)([α1]c...[αn]c) = ⊥c iff [cα1...αn]i=⊥i. Since we are doing the strong coordination of intuitionistic and classical models, it is obvious that this strongly coordinated model will be nontrivial (because there are α1, α2∈Sent(Cic,Pic) such that MSC ic α1and MSC ic 2 α2).  In order to prove the strong soundness of the strongly coordinated natural deduction calculus, we rely on the same lemmas that we did for coordination, appropriately adapted for strong coordination. Lemma 4.2.3.Suppose Bic is the class of strongly coordinated structures over Ciand Cc. Then, Bic SC is a consequence relation for Sent(Cic,Pic). Lemma 4.2.4.Suppose Bic is the strong coordination of Biand Bc. If ΓBiα or ΓBcα, then ΓBic SC α. The proofs for these lemmas are almost identical to the ones given for coordination. Since we have just relaxed three more rules compared to the natural deduction calculus that we had for coordination, it is enough to prove that, for the inferences that those new rules allow in the calculus, we have designation preservation in the semantics. Theorem 4.2.2 (Strong Soundness).The strongly coordinated natural deduction calculus is strongly sound with respect to the class of HeytingBoolean structures. That is, Γ`ic SC α⇒ΓBic SC α. Proof. Immediate from the proof of Strong Soundness for coordination. It is enough to change the subindexes ‘c’ and ‘i’. 140