Propositions with Negative Predicates in Arabic Logic

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Propositions with Negative Predicates in Arabic Logic © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group Published version Daşdemir, Yusuf Daşdemir, Y. (2024). Propositions with Negative Predicates in Arabic Logic. History and Philosophy of Logic, Early online. https://doi.org/10.1080/01445340.2024.2410107 2024 History and Philosophy of Logic ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/thpl20 Propositions with Negative Predicates in Arabic Logic Yusuf Daşdemir To cite this article: Yusuf Daşdemir (12 Nov 2024): Propositions with Negative Predicates in Arabic Logic, History and Philosophy of Logic, DOI: 10.1080/01445340.2024.2410107 To link to this article: https://doi.org/10.1080/01445340.2024.2410107 © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. Published online: 12 Nov 2024. Submit your article to this journal Article views: 252 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=thpl20 HISTORY AND PHILOSOPHY OF LOGIC https://doi.org/10.1080/01445340.2024.2410107 Propositions with Negative Predicates in Arabic Logic Yusuf Daşdemir Department of Philosophy, University of Jyväskylä, Jyväskylä, Finland ABSTRACT This paper explores a neglected category of propositions in Arabic logic, propositions with negative predicates (s¯ alibat al-mah .m¯ ul), by addressing two pivotal questions concerning this propositional form: first, whether it is possible to defend it as distinct from metathetic and simple negative propositions and second, whether affirmative instances of these propositions have existential import. The paper argues for the existence of two distinct and conflicting theories of existential import frequently implicit in the views of Arabic logicians: one centered on the copula and the other on the predicate. ARTICLE HISTORY Received 20 February 2024 Accepted 25 September 2024 KEYWORDS Arabic Logic; proposition; existential import; negative propositions; negation 1. Introduction Syntactically, the simplest form of a proposition, Avicenna (Ibn S¯ın¯a, d. 1037) remarks,1is the affirmative one, comprising a subject (mawd .¯ u– ), S, and a predicate (mah .m¯ ul), P: SisP.2 This signifies that P is attributed to S in the sense that P exists for S, the relation here being signified by is,thecopula(r¯ abit .a). Next comes the negative one, SisnotP, more complex with the addition of the negative marker, not (in Arabic laysa,l¯ a, and the like), attached to the copula to signify the removal or denial of the affirmative relation in which P stands to S. In this case, the copula becomes is not to signify that P does not occur for S. In addition to this syntactic difference made by the occurrence of the negative marker, affirmative propositions differ from their negative counterparts in another perhaps more crucial way: they have existential import (EI)3in the sense that they are true only if S refers to something existent or if there is something that is S. This requirement does not apply to negative ones because if the affirmative is not true in the absence of S, then its negative counterpart is necessarily true, given the principles of non-contradiction and excluded middle.4 Arabic logicians, following Aristotle’s lead in the De Int.XandAn. Pr. I.46, recognized an alternative way to introduce negation into a proposition, where the negative particle CONTACT Yusuf Daşdemir dasdemir[email protected] Department of Philosophy, University of Jyväskylä, Jyväskylä, Finland 1Avicenna 1970, p. 34, 1982, pp. 51–52, 1992, vol. 1, p. 224. 2In Arabic, a proposition might come with no explicit copula, like SP, which is called ‘twofold’ (thun¯ a» ¯ ı), while those with an explicit copula (SisP) are called ‘threefold’ (thul¯ ath¯ ı). However, in a twofold sentence, an implicit copula must be assumed if needed. For related discussions, see e.g., Avicenna 1970,p.76ff;Zimmermann 1981,p.1,26. 3‘A proposition has existential import if and only if it cannot be true unless its subject refers to some existing object(s)’: Chatti 2016,p. 102. 4See Avicenna 1959, pp. 258–259, 1970, pp. 79–81, 1992, vol. 1, p. 224; Tah .t¯ an¯ ı1948,p.99;also,Hodges 2012,Dasdemir 2019. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 Y. DAŞDEMIR belongs with the subject, the predicate, or both: Non-S is P,Sisnon-P,orNon-S is non-P respectively. The term in these examples combining with the negative particle to form a single expression is called ‘indefinite’ (ghayr muh .as .s .al or ma– d¯ ul),5and hence the proposition involving such a term as its one or both parts is called ma– d¯ ula, literally ‘deviant’. But adhering to the more prevalent nomenclature, I refer to it as ‘metathetic’ and restrict the discussion in the following to metathetic propositions with indefinite predicates (ma– d¯ ulat al-mah .m¯ ul). For the other forms of metathetic propositions do not play such a significant role in the work of Arabic logicians. We have now two forms of proposition which includes negation: (1) Affirmative-metathetic (Ma): Sisnon-P (2) Negative simple (Dn):6SisnotP Obviously,thesetwoareveryclosetoeachother.HenceArabiclogiciansputforthsyntactic and semantic criteria to distinguish between them. In terms of syntax, the negative marker in an Masentence attaches to the predicate rendering it indefinite. Conversely, in a Dn proposition it relates to the copula, negating the relationship between S and P to make the entire proposition negative. Metathetic propositions, of course, can also be made negative by means of a negative copula, resulting in a sentence of the form of Sisnotnon-P. Concerning semantic features, Mapropositions diverge from the Dnby the condition ofEI,implyingthatitistrueofonlyexistentsubjects.Aspointedoutearlier,negativepropositionslackthisstipulationandcanbetrueofnon-existentaswellasexistent subjects.7 Afd .al al-D¯ın al-Kh¯ unaj¯ı (d. 1248),8a prominent logician of the thirteenth century, offered a novel reading of propositions with a negated part.9Interpreting the combination of the negative particle (specifically laysa)withthesubjectand/orpredicatetosignify something different from the metathetic, he came up with ‘propositions with negative subjects’ (s¯ alibat al-mawd .¯ u– ), Not-S is P, ‘propositions with negative predicates’ (s¯ alibat almah .m¯ ul), Sisnot-P,and‘propositionswithtwonegativeparts’(s¯ alibat al-t .arafayn), Not-S is not-P.10 Of these, the most influential in the tradition is the second one, propositions with negative predicates (hereafter SM), and hence the discussion will be devoted to them in the following. 5Although Aristotle speaks of indefinite nouns and indefinite verbs in the De Int. II & III respectively, the latter is ignored by Arabic logicians altogether on the ground that in Arabic, there is no indefinite verbs; see Zimmermann 1981,p.28;Avicenna 1970, p. 28. Regarding indefinite nouns on the other hand, F¯ ar¯ ab¯ ı says that a noun ‘becomes indefinite when the negative particle, i.e. the particle ‘not’, is linked with it in such a way that together the two words assume the shape of a single expression’: Zimmermann 1981, p. 222. 6In Arabic logic, non-metathetic propositions are called ‘simple’ (bas¯ ıt .a) or ‘definite’ (muh .as .s .ala). Hence, they are referred to hereafter as ‘Dn’. 7For a historical debate over the difference between Maand Dnpropositions, see Dasdemir 2019. 8On Kh¯ unaj¯ ı’s life and work as a logician, see El-Rouayheb 2010,2019, pp. 44–47; Street 2014;Zolghadr 2024.Kh¯ unaj¯ ı’s enormous influence on the trajectory of Arabic logic was widely acknowledged in the decades immediately following his death. The renowned historian and social scientist Ibn Khaldun (d. 1406) (1958, vol. 3, p. 143), who was born less than a century later in Andalusia at the other end of the Islamic world, for example, names him alongside Fakhr al-D¯ ın al-R¯ az¯ ı (d. 1210) as the pioneers of the new conception of logic in the Islamic world. 9This is in line with Street’s observation (2014, p. 457) that ‘the profusion of propositional types’ is one the most striking features of Arabic logic in the period. 10 To differentiate between metathetic and negative terms, I prefer to show the former with nonand latter with not-,aswill be clearer shortly. HISTORY AND PHILOSOPHY OF LOGIC 3 We have thus three propositional forms in which negation somehow occurs: alongside (1) and (2) above, (3) Affirmative proposition with negative predicate (SMa): Sisnot-P Kh¯ unaj¯ıdoesnotexpoundindetailonthesemanticsandtruthconditionsofthisnew category of propositions,11 which sparked continuous debates from the thirteenth century to the present day,12 revolving particularly around two issues: (1) The possibility of SMapropositions, i.e. whether it is possible, on syntactic or semantic grounds, to defend this propositional form as distinct from Maand Dnpropositions. This is a problem because if the negative marker in SMapropositions is included in the predicate, as Kh¯ unaj¯ı seems to have accepted, then it is not easy to distinguish them from the Ma.Ifitisnotincludedinthepredicate,ontheotherhand,thenitwillattach to the copula, making the proposition Dn. (2) The EI of SM propositions, namely the question of whether SMapropositions bear EI or, to put otherwise, how defensible to recognize a proposition that is affirmative yet lacking EI. In addressing these two problems, proponents of SM propositions have tried to establish the conjunction that SM propositions constitute an independent form and they do not have EI when affirmative, whereas opponents have challenged either or both conjuncts. They have either directly rejected these propositions through categorical denial, on syntactic or semantic grounds, or indirectly challenged them by positing that, being affirmative, SMa propositions require the subject to refer to something(s). I take this position as a rejection of SM propositions because if they have EI in their affirmative instances, it would be virtually impossible to differentiate them from Mapropositions. More crucially, in that case, they would become redundant, given that SM propositions were initially introduced due to the quest for an affirmative proposition free from EI requirement, as will be seen shortly. This paper scrutinizes, this overlooked category of propositions as expounded in the writings of Arabic logicians from the thirteenth to the sixteenth century, the second section addressing the first problem and the third section the second problem. 2. The Possibility of S¯ alibat al-mah .m¯ ul Propositions In his magnum opus,Kashf al-asr¯ ar,Kh¯ unaj¯ıforthefirsttimetalksaboutpropositionswith negative parts when he offers an exceedingly detailed examination of the contraposition (– aks al-naq¯ ıd .) of categorical propositions. There he classifies propositions according to 11 See e.g., Kh¯ unaj¯ ı2010, pp. 152, 154–55, 185. 12 The discussions of SM propositions are still alive especially in contemporary Iranian philosophy in connection with the notion of possibility along the lines drawn by Mull¯ aS .adr¯ a (d. 1641), who defines (1990, vol. 1, part 1, p. 169) possibility in an SMaproposition as something’s being attributed the negation of both sides of necessity, i.e., the necessity of existence and the necessity of non-existence; for a detailed discussion of the relationship between possibility and SM propositions in S .adr¯ a’s thought, see also Jav¯ ad¯ ı¯ Amol¯ ı1382/2004, vol. 1, part 2, pp. 590–600; Moh .ammad¯ ı1375/1996. The discussion betweenMoh .ammadH .osaynT .ab¯ at .ab¯ a¯ ı(d.1981) andMahd¯ ıH .¯ a» er¯ ıYazd ¯ ı(d.1999) inthe1970sis significant in thisrespect. For their respective stands on the issue, see H .¯ a» er¯ ıYazd ¯ ı1353/1974–5and T .ab¯ at .ab¯ a¯ ı1360/1981–2. 4 Y. DAŞDEMIR their subjects and predicates being simple/definite (S and/or P), metathetic (non-S and/or non-P), or negative (not-S and/or not-P), obtaining at the end nine different propositions.13 ThisexaminationishighlyinnovativeinthatitintroducesSMpropositions(and also in some other aspects the details of which do not pertain to our subject).14 To do so, Kh¯ unaj¯ı first seems to draw a distinction between two Arabic expressions of negation, namely not- (laysa)andnon- (l¯ a). The former renders negative any term it attaches to, while the latter makes it metathetic.15 That is, a part of proposition is negative if it has the form that which is not P (m¯ alaysaB)ornot-P (laysa B) for short, whereas it is metathetic if it is of the form thatwhichisnon-P(m¯ ahuwal ¯ a-B)ornon-P for short. Accordingly, such a proposition as S is [something] that is non-P isapropositionwithametatheticpredicateand hence Ma,whileS is [something] that is not P is an SMaproposition. Second, he suggests on several occasions16 that not-P (laysa B)ismoreinclusivethannon-P (l¯ a-B)becauseit applies to non-existent things as well. This is understandable given the semantic of not-P as that which is not P because it is evident that not only existent but also non-existent things might not be P. To establish the feasibility of such a category as SMapropositions without EI, Kh¯ unaj¯ı does not provide positive theoretical proofs. Instead, he puts these propositions into practice to propose substantiated revisions to Avicennian logic. Two instances of these revisions are noteworthy. First, Kh¯ unaj¯ı criticizes Avicenna’s definition of contraposition as ‘taking what contradicts the predicate to make it the subject and what contradicts the subject to posit it as the predicate’.17 AccordingtoKh¯ unaj¯ı, contraposition should be redefined as an inference in which the original subject or its contradictory is predicated of the original predicate’s contradictory, provided that in the latter case, the original proposition retains its quality, whereasintheformer,itassumestheoppositeone. 18 Thus, All Ss are P,forinstance,implies as its contrapositive either No not-P is S or All not-Ps are not-S.Noticeherethatthesecond contrapositive is an SMabut equivalent to its negative counterpart, No not-P is S.Thisis possible only if the SMais taken without EI because, otherwise, the negative one would be more generally true than, hence not equivalent to, the SMa. AccordingtoKh¯ unaj¯ı, we must recognize a universal SMaproposition without EI if Avicenna’s proof for the contraposition of universal affirmative propositions will go through, which is the following:19 Avicenna’s proof: If all Ss are P, then all not-Ps are not-S. 13 Kh¯ unaj¯ ı2010, pp. 147–194. 14 See Zolghadr 2024. 15 In fact, Avicenna at times (e.g. 1910, p. 66) explicitly acknowledges that a proposition containing a laysa following the copula (as in Zaydun huwa laysa bi-– aqil) is more likely to be interpreted as affirmative rather than negative. Kh¯ unaj¯ ımay have been inspired by such statements of Avicenna when introducing SM propositions. 16 Kh¯ unaj¯ ı2010, p. 90, 152. 17 Avicenna 1964, p. 93. 18 Kh¯ unaj¯ ı2010, pp. 147–148. His account of contraposition is much more complex, but I am simplifying it for the sake of brevity. 19 Avicenna 1964, p. 93. HISTORY AND PHILOSOPHY OF LOGIC 5 (1) All Ss are P (premise) (2) Some not-Ps are not not-Ss (assumed contradictory of the consequent) (3) Some not-Ps are Ss (from2,doublenegationlaw) (4) Some Ss are not-Ps (from3,conversion) (5) ⊥(contradiction between 1 & 4) (6) All not-Ps are not-Ss (from 2 & 5) Kh¯ unaj¯ı argues that if the contrapositive here is to be taken as Marather than SMa,then the inference from (2) to (3) would not be valid. This is because, (i) (2) is more generally true than (3), the reason being that (2), being negative, lacks EI, while(3)possessesit.Moregenerallytruestatementsdonotimplymorespecificones. (ii) Thinking otherwise would result in the impossibility of two contradictories being simultaneously false. For let us assume that (2) is true and implies (3), then its contradictory, i.e. All not-Ps are not-S,wouldbefalse,but(3)alsocouldbefalseatthe same time because in the absence of the subject, All not-Ps are not-S and Some notPisSwouldbebothfalse.Althoughthesetwopropositionsarenotcontradictory,if the latter is false, (2) must be also false because the consequent being false makes the antecedent false.20 Through this argument, Kh¯ unaj¯ı demonstrates that Avicenna must have interpreted the contrapositive of the universal affirmative proposition, All not-Ps are not-S,asanSM a proposition without EI, rather than an Mawith EI.21 AsasecondinstancetowhichKh ¯ unaj¯ı applies his SM propositions, I could mention his approach to affirmative premises in the firstand third-figure syllogisms. According to him, the following syllogism is valid: Everything that is not-existent is not-sensible Thevoidisnot-existent _______________________________________ Therefore, the void is not-sensible Kh¯ unaj¯ı views this syllogism as valid whose minor premise declares the subject to be notexistent, while the major states that everything of which being not-existent is affirmed has thepropertysignifiedbythepredicate.However,thepremisescannotbothbenegative, astwonegativepremiseswouldyieldnoconclusion.Furthermore,theycannotbeM a, either, because they—especially the minor premise, traditionally supposed to be affirmative and therefore possess EI—lack EI, as evident from the propositions themselves stating 20 Kh¯ unaj¯ ı2010, p. 147, also 87; K¯ atib¯ ı2019, p. 448. Kh¯ unaj¯ ı’s theory of contraposition elicited both supportive and opposing arguments from later logicians. However, given the limitations of this paper, a more thorough investigation of this matter will not be pursued further. For a historical and theoretical account of contraposition in Arabic logic, see Fallahi 2019, 2023. 21 The fact that Avicenna (1964, p. 94) justifies the contraposition of particular affirmative propositions by saying ‘there happen to be existent and non-existent things that are outside both J and B’, lends support to this conclusion. His mention of non-existent things could be taken as implying their inclusion in the extension of the subject in the particular SMa proposition. 6 Y. DAŞDEMIR that the subjects do not exist. Therefore, the minor in this first-figure syllogism must be an affirmative premise without EI, that is, I argue,22 an SMaproposition.23 Kh¯ unaj¯ı’s distinction between two negative particles, laysa and l¯ a,toseparateSMand metathetic propositions seems to have found support from his students. Ath¯ır al-D¯ın alAbhar¯ı (d. 1265), for instance, in his Tanz¯ ılal-afk¯ ar, his most ‘revisionist’24 writing on logic, reiterates the distinction. For him, Every S is not-P (Kullu J laysa B), read as an SMaproposition, signifies that the negated predicate, not-P, is affirmed of every individual falling under S.25 It seems, therefore, that in the early generations, SM propositions were conceptualized ashavingthenegativemarkeroflaysain their predicate, setting them apart from metathetic propositions with l¯ ain their predicate. Nas .¯ır al-D¯ın al-T .¯ us¯ı (d. 1274), a card-carrying Avicennian logician, who authored a critical commentary on Abhar¯ı’s Tanz¯ ıl al-afk¯ ar, categorically rejects this newfangled class of propositions of the post-Avicennian tradition. His argument runs as follows: [T1] T .¯ us¯ ı1974, p. 168: If the negation follows the copula [in Arabic], then it indicates metathesis (– ud¯ ul), irrespective of whether the expression not- (laysa) combines with another or the expression non- (l¯ a) merges with another. This is because all these expressions combined or merged with another are taken as a single unit (mufrad) to be predicated [of the subject]. For a proposition could not be predicated of a single subject through a predication of it-is-it (h .amla huwa huwa). T .¯ us¯ı rejects the distinction suggested by Kh¯ unaj¯ıandAbhar¯ı between laysa and l¯ ain determining whether a proposition is M or SM. For T .¯ us¯ı,whatmattersinthiscontextisthe position of these negative markers. If they follow the copula in Arabic, that is, if it attaches to the predicate rather than the copula, the proposition should be deemed metathetic. This is because the compound expression (laysa Porl¯ a-P)istreatedasasingleunit,attributable to S. Otherwise, if this compound expression is not considered a single unit, then it would be a negative sentence with laysa,whichwouldposeanimportantissuegiventhatasentence cannot be predicated of a single subject.26 Therefore, according to T .¯ us¯ı, there is no way to interpret Every J is not-B (Kullu J laysa B)asanSM aproposition as suggested by Abhar¯ı; instead, it must be regarded as an Maproposition. InhiscommentaryonhisownQist .¯ as al-afk¯ ar,Shamsal-D¯ın al-Samarqand¯ı (d. 1322), another influential logician of the period, raises another argument to the same end. For him,thenegativewordoflaysa is utilized in Arabic language to negate the predicate of the subject, regardless of whether it precedes or follows the copula. Hence, the proposition in 22 As the anonymous reviewer of HPL brings to my attention, Kh¯ unaj¯ ı never explicitly labels the minor premise as an SM proposition. However, his mention of ‘negation being truly predicated of the subject’ in the premise, which might count as the definition of SM propositions, makes clear enough that what he means by the affirmative proposition with no EI here is the SM proposition. In addition, as the reviewer also points out, Sir¯ aj al-D¯ ın al-Urmav¯ ı (d. 1283), one of Kh¯ unaj¯ ı’s most influential followers, explicitly states that the minor premise here should be SM with the same justification; see Tah .t¯ an¯ ı, 1393/2014–5, p. 284. It should also be noted that regarding such syllogisms, Najm al-D¯ ın al-K¯ atib¯ ı (d. 1277) (2022, p. 168) requires the minor premise to be a mental proposition, a propositional category not available to Kh¯ unaj¯ ıyet. 23 Kh¯ unaj¯ ı2010, p. 90. Indeed, Aristotle’s argument in the De Caelo I.3 was interpreted as involving a first-figure syllogism composed of two Mapropositions, and since then, logicians including Alexander of Aphrodisias, Boethius, and Avicenna deemed such syllogisms valid, provided that they included Maminors. As such, they were not considered an exception to the rule requiring affirmative premises to have EI. See Alexander of Aphrodisias 2006,p.94;Zimmermann 1981,pp. 239–240, esp. notes on these pages; Avicenna 1964, p. 492, 1970,p.81. 24 El-Rouayheb 2019,p.50. 25 Abhar¯ ı2022, p. 111. Also, Abhar¯ ı does not accept Avicenna’s theory of contraposition and his argument above; see Abhar¯ ı 2022, p. 157. 26 For an explanation, see Siy¯ alk¯ ut¯ ı1288/1871–2,p.46. HISTORY AND PHILOSOPHY OF LOGIC 7 which it occurs is inevitably Dn.Thenegativeexpressionsl¯ aand ghayr,however,hecontends, render the proposition M. Consequently, Arabic syntax makes no room to formulate an SM proposition.27 Qut .bal-D¯ın al-R¯az¯ıal-Tah .t¯an¯ı (d. 1365), who was otherwise rather unhappy with postAvicennian innovations in logic, introduced a new analysis of SM propositions, which would serve as the standard point of departure for subsequent discussions. Tah .t¯an¯ıfirst raises a hypothetical objection, highly reminiscent of T .¯ us¯ı’s argument above, that in, e.g. SisnotP(JlaysaB), if the negation is regarded as part of the predicate, the proposition assumes the form of an Ma. Conversely, if it is considered external to the predicate, the proposition is then Dn.Hence,suchapropositionasSM abecomes inconceivable. In response to this objection, Tah .t¯an¯ıstates: [T2] Tah .t¯ an¯ ı1393/2014–5, p. 286: The negative marker is outside the predicate both in [simple] negative and SM propositions. However, in the case of SM propositions, there is an additional element to consider: in the [simple] negative, we conceptualize the subject, the predicate, and the affirmative nexus between them before negating that nexus. In SM propositions, on the other hand, we [similarly] conceptualize the subject, the predicate, and the affirmative nexus, and subsequently, negate that nexus. Yet after that, we proceed to predicate that negation of the subject. This is because if the predicate’s being affirmed of the subject is not true, then its negation of it must be true. That is, unlike the [simple] negative, the SM proposition involves considering the negation twice. The most significant in this account is Tah .t¯an¯ı’s assertion that the negative marker is outside the predicate. Very probably under the influence of T .¯ us¯ı’s argument in T1,Tah .t¯an¯ı musthavefelttheneedtoexcludethenegativeparticlefromthepredicatesoastodefend the possibility of SM propositions. Yet he seems aware that this move will confront him with the question of how to differentiate them in that case from Dnpropositions. His solution is noteworthy as he accepts four semantic elements in the latter: the conception of the subject, the conception of the predicate, the conception of affirmative nexus between them and its negation. In the SM proposition, there is a fifth element to take into account in addition to the four: the affirmative predication of this negation of the subject. This is whythepropositionisaffirmativeatthefinalanalysis. On the following lines, Tah .t¯an¯ımakesclearerapointinthepassagethatanSMproposition is a result of two proposition-making operations. In the first, one forms a negative proposition, SisnotP, and in the second, one forms an SMaby predicating that negative proposition of the same subject, S is something that is not P. This reading of the SM proposition clearly separates them from both Maand Dnones because, Tah .t¯an¯ıremarks,the Dnsignifies that P is negated of S while the MathatSisnon-P.Noteherethatthecopula of the SMais affirmative, and therefore so is the proposition itself. However, despite being affirmative, it exceptionally lacks EI, according to Tah .t¯an¯ı, too, just like a Dnproposition.28 Nevertheless, even with this two-layered analysis of SM propositions by Tah .t¯an¯ı, it appears, not all the problems were settled, as the question of whether the negation is part of the predicate persisted as a point of contention in subsequent discussions. For instance, Jal¯al al-D¯ın al-Daw¯an¯ı (d. 1502), the influential scholar of the fifteenth-century Iran, addresses the question to criticize an argument saying that the negative particle is not 27 Samarqand¯ ıMS, fol. 39a. 28 Tah .t¯ an¯ ı1393/2014–5, p. 286. 14 Y. DAŞDEMIR thereshouldbeanaffirmativepropositionwithoutEIinorderforthisandthatinference go through. Or they explain this lack of EI merely averring that SMapropositions are essentially equivalent to Dnpropositions.52 However, Jurj¯an¯ı brings up a compact argument for this equivalence. According to him, SMapropositions lack EI because they are in fact reducible (r¯ aji– atun)toD npropositions due to the fact that the absence of something from another necessarily implies that the other is described by the absence of that thing from it and vice versa. The distinction between these two cases, as al-Jurj¯an¯ı asserts, is only a matter of difference in perspectives (bi-li– tib¯ ar).53 An unpacked version of this argument that SMapropositions are equivalent to Dnpropositions is provided by the fifteenth-century Ottoman scholar, Taşköprizâde Kâsım (d. 1513),54 inhistreatiseofmentalexistence(MS, fols. 175b22–176a2), which could be reconstructed as follows: The equivalence argument: IfSisnotP,thenSisnot-P (1) S is not P (premise) (2) S is not not-P (assumed contradictory of the consequent) (3) SisP (from2,doublenegationlaw) (4) ⊥(contradiction between 1 & 3) (5) Sisnot-P (from 2 & 4) If S is not-P, then S is not P (1) Sisnot-P (premise) (2) SisP (assumed contradictory of the consequent) (3) ⊥(contradiction between 1 & 2) (4) S is not P (from 2 & 3) ThisargumentmakesitquiteclearthattheSM a,Sisnot-P,andtheD npropositions, Sisnot P, are equivalent and therefore contradict the same proposition. However, this inevitably leads to the question of why the former should be regarded as affirmative? The only answer wecangiveisthatitisaffirmativeinform due to the occurrence of the affirmative copula is, and Arabic logicians, at least those who regard it as affirmative, seem to have attended to the surface structure or linguistic expression of the proposition. In fact, the equivalence between two propositions does not necessarily render either redundant. Otherwise, a substantial number of propositional and even syllogistic forms would have to be excluded from the logical framework. Furthermore, there is another, more serious, issue with the argument: it is not evident whetheritprovesorpresupposesthelackofEIinSM apropositions. The argument relies on considering the affirmative proposition of SisPas the contradictory of the SMaproposition. However, this assumption holds only if the latter does not have EI. If one were to consider the SMaproposition with EI, this argument could not proceed as, in that case, 52 Abhar¯ ı2022, p. 111; Tah .t¯ an¯ ı1393/2014–5, p. 286. 53 Jurj¯ an¯ ı2020,vol.2,p.84. 54 For the earliest account of his life and work, see Taşköprizâde Ah .med 2019, pp. 616–618. HISTORY AND PHILOSOPHY OF LOGIC 15 Sisnot-Pand SisPwould be contrary propositions—both could be false but not simultaneously true. In the case S refers nothing existent, both would be false. Therefore, the argument seems to be begging the question.55 It is also understood from the argument that Jurj¯an¯ı implicitly assume the predicatebasedtheoryofEI.Otherwise,theargumentwouldnotyieldthesoughtconclusion.This assumption becomes more obvious in Jurj¯an¯ı’sconvictionthatthereareunrealaswellas real affirmations. Only if the predicate is free from any kind of negation in its semantic content, the affirmation could be real and irreducible to a negative proposition. For Jurj¯an¯ı, this criterion effectively distinguishes between SMaand Mapropositions because the formerisaninstanceofunrealaffirmationreducibletoitsD nequivalent, while the latter is not reducible in the same manner. Jurj¯an¯ı elaborates that when one predicates, say, the privation of writing of Zayd, one obtains a Maproposition, Zayd is non-writing.Thisisan affirmative proposition that could entail a negative proposition yet is not reducible to it. An SMaproposition, on the other hand, is obtained through negating writing of Zayd and then predicating that negation of Zayd. This is why it is reducible to its negative equivalent.56 Therefore, for Jurj¯an¯ı, if the predicate is not positive then the proposition cannot be affirmative in the real sense, even if it formally involves an affirmative copula. Jurj¯an¯ı’s distinction of real-unreal affirmation seems to have some influence on later logicians. Taşköprizâde Kâsım, for instance, incorporates this distinction into his argument against the EI of SMapropositions. According to him, affirmation can be understood in two distinct ways: (a) real (h .aq¯ ıq¯ ı) affirmation, signifying the assertion that the nexus between the subject and predicate obtains in reality, and (b) unreal (ghayr h .aq¯ ıq¯ ı)affirmation,which resembles an affirmation in form and linguistic structure. For him, SMapropositions fall into the latter category because these propositions essentially consist of nothing more than the rephrasing of a negative proposition in a condensed manner.57 That is, the Dnproposition, SisnotP, is reconsidered as the predicate of SMapropositioninsuchacompact form as not-P. Given that its predicate is a shorthand for a negative proposition, the SMa proposition could not count as real affirmation and hence cannot have EI, according to the predicate-based theory. Finally, we see the same idea articulated in similar terms by the renowned Ottoman bio-bibliographer Taşköprizâde Ahmed (d. 1561), the nephew of Taşköprizâde Kâsım. He also accounts for the absence of EI in SMapropositions with reference to Tah .t¯an¯ı’s twolayeredanalysis,accordingtowhich,aswealreadysaw,thesepropositionsincorporatea Dnproposition within their predicates. Based on this, Taşköprizâde Ahmed maintains that SMapropositions lack EI only because the Dnpropositions serving as their predicates do not possess it either.58 Furthermore, Taşköprizâde Ahmed argues against the copula-based theory which he ascribes to Avicenna. His argument depends on a distinction we discussed in the previous 55 Actually, Taşköprizâde Kâsım (MS, fol. 176a14), too, notes such an objection to the argument. 56 Jurj¯ an¯ ı2020,vol.2,p.84.Jurj ¯ an¯ ı’s category of unreal affirmation is reminiscent of R¯ az¯ ı’s dismissal of Mapropositions as not affirmative in the real sense discussed above. However, Jurj¯ an¯ ı seems to separate negation from privation such that SM propositions have negative predicates and therefore are not affirmative in the real sense, while the predicate of metathetic propositions is privative, and hence they could count as real affirmation. 57 Taşköprizâde Kâsım MS, fol. 177b8–9. 58 Taşköprizâde Ahmed 2009,p.57. 16 Y. DAŞDEMIR section between the judgment time and the reference time. According to him, the copulabasedtheoryofAvicennacouldbegrantedifconsideredinthejudgmenttime.However,in thereferencetime,whatrequiresthesubjecttoexististhepositivenatureofthepredicate. 59 Notice here that Taşköprizâde Ahmed’s remarks align closely with my assessment above that the copula-based theory is more logic-oriented, while the predicate-based theory is more focused on ontology. However, Taşköprizâde Ahmed’s approach would be notably unfair to the copula-based theory. This is because a concept of existence confined to the judgment time and consequently to the mind does not truly represent existence in the real sense. As emphasized, the presence of judgment components in the mind constitutes a highly trivial interpretation of EI. Therefore, this perspective leads to the problematic notion that Avicenna and his followers are not genuinely seeking EI requirement in affirmative propositions. Moreover, if Avicenna had perceived the EI requirement as limited onlytothejudgmenttime,hewouldhavehadtoimposetheconditionofexistencenot only for subjects of affirmative propositions but also for subjects of negative propositions andevenforthepredicatesofallkindsofpropositions.Thisisbecauseanegativeproposition is as much a judgment as an affirmative one, and the predicate is as much a part of the judgment as the subject. Therefore, while the subject and predicate of all propositions must be present in the mind at the judgment time, Avicenna never discusses them in contexts related to EI. Hence, Taşköprizâde Ahmed’s comment about the copula-based theory does not appear to be accurate. To wrap up, our exploration of the arguments presented by two parties—one affirming and the other denying the EI of SMapropositions—has shown that the root of this disagreement lies in conflicting views about the origin of the EI requirement within propositions. However, it remains uncertain to me whether there are more profound ontological or logical assumptions underpinning these views. 4. Conclusion This paper has explored the diverse stands on SM propositions embraced by Arabic logicians from the thirteenth to the sixteenth centuries. Two central questions have guided this examination: whether it is conceivable to accept an affirmative proposition without EI, such as the SMaproposition introduced by Kh¯ unaj¯ı, and whether it is justified for these propositions to lack EI. In addressing the first question, we have encountered logicians like T .¯ us¯ı, who categorically reject the viability of this propositional category. In contrast, others, such as Kh¯ unaj¯ıandTah .t¯an¯ı, endeavour to accommodate SMapropositions by either incorporating the negative marker into the predicate or excluding it therefrom. Tah .t¯an¯ı’s influential two-layered analysis of these propositions left a lasting impact in subsequent centuries. Regarding the second issue, I have identified two underlying theories of EI assumed by Arabic logicians often implicitly, which I term the copula-based theory and the predicatebased theory. According to the former, the affirmative nature of the copula is crucial in endowing affirmative propositions with EI. However, the predicate-based theory posits that although the copula’s affirmativeness is necessary, it is not sufficient; the predicate must alsobepositiveforanaffirmativepropositiontobearEI.Thecopula-basedtheoryhas 59 Taşköprizâde Ahmed 2009, p. 60. This view, too, seems to have its source in Jurj¯ an¯ ı’s writings: see Jurj¯ an¯ ı2020, p. 84, n. t¯ a» . HISTORY AND PHILOSOPHY OF LOGIC 17 led logicians to argue that SMapropositions must have EI because they are affirmative. Conversely, the predicate-based theory has steered them towards the perspective that these propositions lack EI simply because their predicates are not positive. Disclosure Statement No potential conflict of interest was reported by the author(s). References Abhar¯ı, A. a.-D. 2022.Tenzîlü’l-efkâr fî ta’dîli’l-esrâr,ed.byK.KömürcüandE.S.Gül,Ankara:Tüba. 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