Intermediate quantifiers and valid syllogisms on EQ-algebras

Journal of Algebraic Hyperstructures and Logical Algebras Volume 5, Number 1, (2024), pp. 95-105 Intermediate quantifiers and valid syllogisms on EQ-algebras V. Nov´ak 1 1University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic Vilem.Nov[email protected] “This paper is dedicated to Professor Anatolij Dvureˇcenskij on the occasion of his 75th birthday” Abstract Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra). Article Information Corresponding Author: V. Nov´ak; Received: April 2024; Accepted: Invited; Paper type: Original. Keywords: EQ-algebra, intermediate quantifiers, logical syllogisms, fuzzy natural logic. 1 Introduction This paper addresses the concept of EQ-algebra as an algebraic basis of higher-order fuzzy logic called Fuzzy Type Theory (FTT). EQ-algebras were introduced in [18, 22] and since then studied by many authors (see [1, 3, 7, 8, 9, 27]). The motivation comes from the classical type theory in which the basic connective is equality (identity) [2, 12]. Note that the endeavor to develop logic on the basis of equality as the principal connective can be traced to G. W. Leibniz, L. Wittgenstein, and F. P. Ramsey (cf. [4, 26]). The mathematical fuzzy logic initiated in [10] and later developed by many authors (cf. [11, 24] and the citations therein) is based on the concept of residuated lattice which is a lattice endowed by additional operations of multiplication ⊗and residuation →tied by the adjointness property. The main connective in the corresponding fuzzy logic is implication and the equivalence is derived. Thus, the following question arises: can we introduce an algebra with fuzzy equality as the main operation which would be a counterpart to residuated lattice? The answer is the mentioned EQ-algebra. Then the corresponding fuzzy logic has implication as a derived connective. This https://doi.org/10.61838/kman.jahla.5.1.9 96 V. Nov´ak logic [5, 6, 21] has, moreover a non-commutative strong conjunction and, consequently, unlike residuated lattice based non-commutative logics, one implication only. It should be emphasized that EQ-algebras are more general than residuated lattices. Namely that every residuated lattice is an EQ-algebra but not vice-versa. One of the applications of mathematical fuzzy logic is the theory of intermediate quantifiers and their syllogisms. Informal analysis of them was provided by Peterson in the book [25]. Its formalization in the frame of higher-order mathematical fuzzy logic was suggested by Nov´ak in [19] and further developed in many papers (see, e.g., [13, 14, 15, 16, 17] and others). In [23] we proved that all 105 valid syllogisms originally introduced in [25] are a consequence of only two algebraic inequalities and one equality. This result is obtained in a fuzzy type theory in which the algebra of truth values is an MV-algebra. In this paper we will show that this result is more general and, in fact, can be proved if we assume that the algebra of truth values is an involutive EQ-algebra (IEQ-algebra). 2 EQ-algebras Definition 2.1. A non-commutative EQ-algebra is an algebra E=⟨E, ∧,⊗,∼,1⟩(1) of type (2, 2, 2, 0) fulfilling the following axioms for all a, b, c ∈E: (E1) ⟨E, ∧,1⟩is a commutative idempotent monoid (∧-semilattice with the top element 1). We put a≤biff a∧b=a, as usual. (E2) ⟨E, ⊗,1⟩is a monoid such that ⊗is isotone w.r.t. ≤. (E3) a∼a=1,(reflexivity) (E4) ((a∧b)∼c)⊗(d∼a)≤c∼(d∧b),(substitution) (E5) (a∼b)⊗(c∼d)≤(a∼c)∼(b∼d),(congruence) (E6) (a∧b∧c)∼a≤(a∧b)∼a,(monotonicity) (E7) a⊗b≤a∼b.(boundedness) EQ-algebra is commutative if ⊗is commutative. We put:  a→b= (a∧b)∼a, (implication)  If Econtains a bottom element 0then ¬a=a∼0, (negation)  ˜a=a∼1. Definition 2.2. An EQ-algebra Eis:  separated if a∼b=1implies a=bfor all a, b ∈E. Then a∼b=1if and only if a=b.  good if a∼1=afor all a∈E.  residuated if (a⊗b)∧c=a⊗bif and only if a∧((b∧c)∼b) = afor all a, b, c ∈E. Intermediate quantifiers and valid syllogisms on EQ-algebras 97  involutive (IEQ-algebra) if ¬¬a=a.  complete if inf(K)exists for every K⊆E. Clearly, complete EQ-algebras are complete lattices.  lattice ordered if it has also a binary operation ∨such that ⟨E, ∧,∨⟩ is a lattice.  lattice EQ-algebra (ℓEQ-algebra) if it is lattice ordered and the following additional substitution axiom holds ((a∨b)∼c)⊗(d∼a)≤((d∨b)∼c)for all a, b, c, d ∈E. Many properties of EQ-algebras were proved in the cited papers. Let us remind here only a few properties that hold in good EQ-algebras. Lemma 2.3. Let Ebe a good EQ-algebra. Then the following holds for all a, b, c ∈E. (a) a≤(a∼b)∼b, (b) a≤(a→b)→b, (c) a⊗(a→b)≤b, (d) a→(b→c) = b→(a→c). (e) a≤b→ciff b≤a→c, (f) Vi∈I(ai→b) = Wi∈Iai→b, provided that both infimum as well as supremum exist. Proof. See [7, 8]. In the rest of this paper, we assume that the algebra of truth values is a good, linearly ordered, complete IEQ-algebra. 3 Intermediate quantifiers and their syllogisms For the definition of intermediate quantifiers we need the following: (i) The concept of evaluative linguistic expression, which are special expressions of natural language such as “very deep, roughly big, more or less good”, etc. Formalization of their semantics was developed in [20]. For this paper, it is enough to take them as special formulas of type oo (i.e., they are interpreted as fuzzy sets on on the set of truth values). We will consider evaluative expressions Bi ∆ ∆ ∆ (“big”), Bi Ex (“extremely big”), Bi Ve (“very big”) and ¬Sm (“not small’). (ii) The operation cut of a fuzzy set y∈Formoα w.r.t. a fuzzy set z∈Formoα: y|z≡λxα·zx& & &∆ ∆ ∆(Υ(zx)⇒ ⇒ ⇒(yx ≡zx)).(2) Interpretation of this formula in a model is the following: for given fuzzy sets B, Z ⊂ ∼M, B|Zis a “cut” of Bsuch that (B|Z)(m) = Z(m) if there is such m, otherwise (B|Z)(m) = 0, m∈M. If there is no such mthen B|Z=∅. We can thus take various fuzzy sets Zto “pick up proper elements from B” (together with their membership degrees). 98 V. Nov´ak (iii) Measure µ∈Formo(oα)(oα). A compound formula (µB)Ais interpreted as a measure of a fuzzy set Aw.r.t. the fuzzy set B. Definition 3.1. Let Ev ∈Formoo be a formula representing some evaluative linguistic expression, z∈Formoα,x∈Formαbe variables and A, B ∈Formoα be formulas where Brepresents a measurable fuzzy set. An intermediate quantifier is one of the following formulas: (Q∀ Ev xα)(Boα, Aoα)≡(∃z)[(∀x)((B|z)x⇒ ⇒ ⇒Ax)∧ ∧ ∧Ev((µB)(B|z))],(3) (Q∃ Ev xα)(Boα, Aoα)≡(∃z)[(∃x)((B|z)x∧ ∧ ∧Ax)∧ ∧ ∧Ev((µB)(B|z))].(4) Either of the quantifiers (3) or (4) construes the sentence ⟨Quantifier⟩B′sare A(5) where ⟨Quantifier⟩is a quantifier in a linguistic form. Note that in this definition, we replaced cardinality of a fuzzy set (originally suggested by Zadeh in [28]) by its measure. Formula Boα in (i)–(iii) represents a universe of quantification. Definition 3.2. Let Nbe a set and A, B ∈ F(N). Let us denote by FR Ev (B, B|Z)interpretation of the formula Ev((µB)(B|z)). Then a semantic interpretation of the intermediate quantifiers (3) and (4) are truth values given by the respective formulas Q∀ Ev (B, A) = _(^ u∈N ((B|Z)(u)→A(u)) ∧FR Ev (S, S|Z)|Z∈ F(N)),(6) Q∃ Ev (B, A) = _(_ u∈N ((B|Z)(u)∧A(u)) ∧FR Ev (B, B|Z)|Z∈ F(N)).(7) The measure and evaluation of the size of B|Zis hidden in the function FR Ev . If we replace the metavariable Ev in (3) or (4) by a formula representing a specific evaluative linguistic expression then we obtain the following algebraic representations of the specific intermediate quantifiers. Definition 3.3. (A)“All B’s are A”: Vu∈N(B(u)→A(u)), (E)“No B’s is A”: Vu∈N(B(u)→ ¬A(u)), (P)“Almost all B’s are A”: WZ∈F(N)Vu∈N((B|Z)(u)→A(u)) ∧FR BiEx(B, B|Z), (B)“Almost all B’s are not A”: WZ∈F(N)Vu∈N((B|Z)(u)→ ¬A(u)) ∧FR BiEx(B, B|Z), (T)“Most B’s are A”: WZ∈F(N)Vu∈N((B|Z)(u)→A(u)) ∧FR BiVe(B, B|Z), (D)“Most B’s are not A”: WZ∈F(N)Vu∈N((B|Z)(u)→ ¬A(u)) ∧FR BiVe(B, B|Z), (K)“Many B’s are A”: WZ∈F(N)Vu∈N((B|Z)(u)→A(u)) ∧FR ¬Sm (B, B|Z), (G)“Many B’s are not A”: WZ∈F(N)Vu∈N((B|Z)(u)→ ¬A(u)) ∧FR ¬Sm (B, B|Z), (I)“Some B’s are A”: Wu∈N(B(u)∧A(u)), (O) “Some B’s are not A”: Wu∈N(B(u)∧ ¬A(u)). We will denote various kinds of quantifiers by QX(B, A) where X∈ {A, E, P, B,T, D, K, G, I, O}. Intermediate quantifiers and valid syllogisms on EQ-algebras 99 4 Algebraic analysis of valid intermediate syllogisms 4.1 Formalization of syllogisms Asyllogism is a triple of formulas P1,P2,Cwhere P1and P2are major and minor premises and C is a conclusion. Let QP1, QP2, QCbe quantifier symbols from (3) or (4) occurring in both premises and conclusion, respectively, and S, P, M be formulas representing properties of elements. The formula Sis a subject,Pis a predicate and Mis a middle formula. As usual, syllogisms are gathered into the following four figures: Figure I QP1Mare P QP2Sare M QCSare P Figure II QP1Pare M QP2Sare M QCSare P Figure III QP1Mare P QP2Mare S QCSare P Figure IV QP1Pare M QP2Mare S QCSare P If all QP1, QP2, QC∈ {∀,∃} then the corresponding syllogism is classical. We say that it is valid if TIQ ⊢ P1& & &P2⇒ ⇒ ⇒ C.(8) By the completeness theorem, syllogism (8) is valid iff M(P1)⊗ M(P2)≤ M(C) (9) holds in any model M |=TIQ. Validity of (8) for all syllogisms presented in [?] was proven syntactically in [13, 15]. 4.2 Fundamental algebraic inequalities In the rest of this paper, we consider a complete IEQ-algebra E. Theorem 4.1 ([7], Theorem 3).Let Ψbe an inequality in the language of EQ-algebras. Let ¯ Ψ be an inequality obtained from Ψby replacing all occurrences of ⊗by its reverse ¯ ⊗. Then Ψis universally valid iff ¯ Ψis valid. By this theorem, we may consider the inequalities presented below from one side of ⊗only. Lemma 4.2. Let a, b, c ∈E. Then the following relations hold true: (a) (b→c)⊗(a→b)≤a→c, (b) (b→c)⊗(a∧b)≤a∧c, (c) (a→b)=(¬ ¬ ¬b→ ¬ ¬ ¬a), Proof. (a) was proved in [22], Theorem 1(c) and in [7], Lemma 2(c) and Lemma 6(k). (b) was proved in [22], Proposition 1 and Lemma 10(d). (c) follows from [7], Lemma 7(a). The following inequalities are consequences of the previous lemma. Lemma 4.3. Let a, b, c, d, f ∈E. Then (a) Let a⊗b≤c. Then a⊗(b∧f)≤c∧f, 100 V. Nov´ak (b) (b→c)⊗((a→b)∧f)≤(a→c)∧f, (c) (b→c)⊗a⊗(a→b)≤a∧c, (d) (b→c)⊗b⊗(b→a)≤a∧c, (e) ((b→c)∧f)⊗b⊗(b→a)≤a∧c, (f) ((b→c)) ⊗b⊗((b→a)∧f)≤a∧c, (g) (b→c)⊗a⊗((a→b)∧f)≤a∧c, (h) ((b→c)∧f)⊗a⊗((a→b)∧f′)≤a∧c, (i) ¬¬a=a. (j) ((b→c)∧f)⊗((b′→a)∧f′)⊗(b⊗b′)≤a∧c. (k) a∨Wj∈Jbj=Wj∈J(a∨bj). Proof. (a) follows from the properties of semilattice and (E2). (b) is a consequence of (a) and Lemma 4.2(a). (c) By Lemmas 2.3(c), 4.2(a) and (E2) we obtain (b→c)⊗a⊗(a→b)≤a⊗(a→c)≤c, a and, hence, we obtain (b→c)⊗a⊗(a→b)≤a∧c. (d) Let us denote the left-had side of (d) by L. Then it is a consequence of Lemma 4.2(b), the properties of good EQ-algebra and the properties of semilatice. Indeed, we have L≤(b→c)⊗a≤a as well as L≤(b→a)⊗c≤cwhich gives L≤a∧c. (e) and (f) are consequences of (d). (g) is a consequence of (c) and general properties of good EQ-algebras. (h) is a consequence of (g). (i) holds in each IEQ-algebra. (j) As a consequence of (d), we have ((b→c)∧f)⊗b⊗((b→a)∧f′)≤a∧c, as well as ((b′→c)∧f)⊗b′⊗((b′→a)∧f′)≤a∧c. From both inequalities, using (E2) and the inequality r⊗s≤r∧s≤r, s we obtain (j). (k) a≤Wj∈J(a∨bj) as well as Wj∈Jbj≤Wj∈J(a∨bj) from which we conclude that a∨Wj∈Jbj≤ Wj∈J(a∨bj). Conversely, from bj≤Wj∈Jbjwe conclude that Wj∈J(a∨bj)≤Wj∈J(a∨Wj∈Jbj) = a∨Wj∈Jbj. In the sequel, we will consider properties M, S, P. In fact, they are determined by a context (or a possible world) in which they are used. For simplicity, however, we will take them as simple fuzzy sets in some universe. Lemma 4.4. Let Ube a universe and M, S, P ⊂ ∼Nbe properties of elements of N. (a) Inequality (m→p)⊗(s→m)≤s→pof Lemma 4.2(a) implies validity of syllogism (AAA-I), whose algebraic form is ^ u∈N (M(u)→P(u)) ⊗^ u∈N (S(u)→M(u)) ≤^ u∈N (S(u)→P(u)). Intermediate quantifiers and valid syllogisms on EQ-algebras 101 (b) Inequality (m→p)⊗((s→m)∧f)≤(s→p)∧fof Lemma 4.3(b) implies validity of syllogism (APP-I), whose algebraic form is ^ u∈N (M(u)→P(u)) ⊗_ Z∈F(N) ^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!≤ _ Z∈F(N) ^ u∈N ((S|Z)(u)→P(u)) ∧FR Bi Ex(S, S|Z)!. (c) Inequality (m→p)⊗s⊗(s→m)≤s∧pof Lemma 4.3(c) implies validity of syllogism (A∗ AI-I), whose algebraic form is ^ u∈N (M(u)→P(u)) ⊗_ u∈N S(u)⊗^ u∈N (S(u)→M(u)) ≤_ u∈N (S(u)∧P(u)). (d) Inequality (m→p)⊗s⊗((s→m)∧f)≤s∧pof Lemma 4.3(g) implies validity of syllogism (A∗ PI-I), whose algebraic form is ^ u∈N (M(u)→P(u))⊗ _ Z∈F(N) _ u∈U (S|Z)(u)⊗ ^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!!≤ _ u∈N (S(u)∧P(u)). (e) Inequality (m→p)⊗(s∧m)≤s∧pof Lemma 4.2(b) implies validity of syllogism (AII-I), whose algebraic form is ^ u∈N (M(u)→P(u)) ⊗_ u∈N (S(u)∧M(u)) ≤_ u∈N (S(u)∧P(u)). Proof. Ad (a): We start with the following instance of Lemma 4.2(a): (M(u)→P(u)) ⊗(S(u)→M(u)) ≤S(u)→P(u), where u∈N. By the properties of infimum and monotonicity of ⊗we have ^ u∈N (M(u)→P(u)) ⊗^ u∈N (S(u)→M(u)) ≤(M(u)→P(u)) ⊗(S(u)→M(u)). From both inequalities, using again the properties of infimum we obtain ^ u∈N (M(u)→P(u)) ⊗^ u∈N (S(u)→M(u)) ≤^ u∈N (S(u)→P(u)).(10) Ad (b): We begin with the following instance of Lemma 4.3(b): (M(u)→P(u)) ⊗(((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)) ≤ ((S|Z)(u)→P(u)) ∧FR Bi Ex(S, S|Z). 102 V. Nov´ak Similarly as in (a) we have ^ u∈N (M(u)→P(u)) ⊗ ^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!≤ (M(u)→P(u)) ⊗((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z). Joining these two inequalities and using the properties of supremum, we obtain ^ u∈N (M(u)→P(u)) ⊗ ^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!≤ _ Z∈F(N) ^ u∈N ((S|Z)(u)→P(u)) ∧FR Bi Ex(S, S|Z)!. Again, using the properties of supremum we obtain _ Z∈F(N) ^ u∈N (M(u)→P(u)) ⊗^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!≤ _ Z∈F(N) ^ u∈N ((S|Z)(u)→P(u)) ∧FR Bi Ex(S, S|Z)!, and using Lemma 4.3(k) we obtain (b). Ad (c): We begin with the following instance of Lemma 4.3(c): (M(u)→P(u)) ⊗S(u)⊗(S(u)→M(u)) ≤(S(u)∧P(u). Then ^ u∈N (M(u)→P(u)) ⊗S(u)⊗^ u∈N (S(u)→M(u)) ≤(M(u)→P(u)) ⊗S(u)⊗(S(u)→M(u)) ≤ _ u∈N (^ u∈N (M(u)→P(u)) ⊗S(u)⊗^ u∈N (S(u)→M(u))) = ^ u∈N (M(u)→P(u)) ⊗_ u∈N S(u)⊗^ u∈N (S(u)→M(u)) ≤_ u∈N (S(u)∧P(u))), using properties of supremum and Lemma 4.3(k). (d) To prove (A∗ PI-I), we start with syllogism (A∗ AI-I) and apply Lemma 4.3(g) to obtain the following provable inequality for any Z⊂ ∼N: ^ u∈N (M(u)→P(u))⊗ _ u∈N (S|Z)(u)⊗ ^ u∈N ((S|Z)(u)→M(u)) ∧FR Bi Ex(S, S|Z)!!→ _ u∈N (S(u)∧P(u)). Intermediate quantifiers and valid syllogisms on EQ-algebras 103 From this, taking supremum over all Z∈ F(N), using the properties of supremum and Lemma 4.3(k) we obtain (d). Ad (e): We begin with the following instance of Lemma 4.2(b): (M(u)→P(u)) ⊗(S(u)∧M(u)) ≤(S(u)∧P(u). From it we prove that ^ u∈N (M(u)→P(u)) ⊗(S(u)∧M(u)) ≤_ u∈N (S(u)∧P(u)), for all u∈N. Then, using the properties of supremum and Lemma 4.3(k) we obtain inequality (e), i.e., syllogism (AII-I). From the previous lemma follows validity of syllogisms (AAP-I), (AAT-I), (AAK-I), (APTI), (APK-I), (AKK-I), (ATT-I) and (ATK-I) (A∗ KI-I) and (A∗ TI-I). The proof of validity of the other syllogisms is similar. Thus, the following theorem holds true. Theorem 4.5. All valid syllogisms with intermediate quantifiers formulated in EQ-algebra based fuzzy type theory are the consequence of two inequalities and one equality of Lemma 4.2. Similar theorem has been proved in [23] for the case that the formal system is MV-algebra based fuzzy type theory. The theorem above is more general since MV-algebras are special EQ-algebras. From our results we conclude that human syllogistic reasoning that uses even non-precisely defined properties is based on fundamental algebraic properties of ordered structures without other special assumptions. 5 Conclusions In this paper we introduced the concept of EQ-algebra and a special theory of intermediate quantifiers based on the EQ-algebra fuzzy type theory. We proved that human syllogistic reasoning has more general roots and does not depend on special properties of the used algebra of truth values. Namely, we proved that validity of intermediate syllogisms is based on two fundamental algebraic inequalities and one equality, that are valid already in very general IEQ-algebras. The paper is a contribution to the concept of Fuzzy Natural Logic. Acknowledgments This article has been supported from the project “Research of Excellence on Digital Technologies and Wellbeing CZ.02.01.01/00/22-008/0004583” which is co-financed by the European Union. Conflict of interest No potential conflict of interest relevant to this article was reported.