1. The Modal System ZFM

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1 Bulletin of the Section of Logic Volume 14/4 (1985), pp reedition 2007 [original edition, pp ] Lafayette de Moraes ON DISCUSSIVE SET THEORY Abstract This paper was read at the VII Simpōsio Latino Americano de Lōgica Matemātica de Campinas, Campinas, SP, Brasil, August 2, In this note we describe a discussive set theory which is related to a certain modal set theory as discussive propositional logic is related to the system S5 of Lewis. 1. The Modal System ZFM Our starting system is the modal system of da Costa and de Alcântara [1] ZFM. ZFM is a modal set theory based on S5 with quantification and contingent equality S5 [3]. The system S5 The system S5 is a first-order modal logic with contingent equality (see [3]), defined as follows: I. Primitive Symbols: a) Connectives: and (, and are defined as usual); b) The necessity operator: (the possibility operator is defined in terms of as usual); c) The universal quantifier: (the existential quantifier is defined in terms of as usual);

2 On Discussive Set Theory 145 d) The symbol of equality: =; e) A set of predicate variables: ϕ, ψ, χ,...; f) Individual terms: a denumerable family of individual variables and a denumerable family of individual constants; g) Parenthesis: (,). The usual syntactic notions, such as the notions of formula, free and bound occurrences of variables in a formula etc. are introduced as usual. II. Axiom Schemes: A1) α, where α is an instance of a classical (propositional) tautology; A2) x(α β) (α xβ), where the variable x does not occur free in α; A3) xα(x) α(t), where the term t is free fir the variable x in α(x); A4) x = x, where x is any variable; A5) x = y (α(x) α(y)), with the common restrictions for contingent equality; A6) (α β) ( α β); A7) α α; A8) α α. III. Rules: R1) From α and α β to infer β; R2) From α to infer xα; R3) From α to infer α. If α is provable in S5, we write α; Γ α, where Γ is a set of formulas, means that for α 1, α 2,..., α n in Γ, we have that (α 1... α n ) α in S5. As is well known, S5 has a Kripke semantics relative to which it is sound and complete. The System ZPM The system ZPM constitutes a modal set theory (or, better, a theory of attributes). Its languages is the language of S5, whose set of predicate symbols contains only the membership symbol. The axiom schemes and rules of ZFM are those of S5, A1)-A8) and R1)-R3) above plus the following:

3 146 Lafayette de Moraes M0) (x = y) (α(x) α(y)), where y is free for x in the occurrences it substitutes for x in α(x); M1) ( t(t x t y) x = y)); M2) t z(z t (z = x z = y)); M3) z t(t z y(y t y x)); M4) z t(t z y(y x t y)); M5) y x(x y (α(x) x z)); M6) x(( t(t x)) ( y) (y x t(t x t y))). ZFM could be reinforced with the introduction of axioms such as the axiom of infinity, choice and replacement. ZPM so reinforced will be denoted by ZPM. ZFM and ZFM have semantics similar to that of ML p of Gallin [2], relative to which it is sound and complete. The system ZPM (and also ZPM ) is a strong system of attributes, since da Costa and de Alcântara [1] proved the following theorems: Theorem 1. Theorem 2. IL and ML p of Gallin [2] are interpretable in ZPM. Usual ZP and ZPM are equiconsistent. 2. The Discussive System JM Similarly as Jaśkowski defined his discussive system of propositional calculus in [4], by means of S5, we define a corresponding discussive set theory (or, if one prefers, a general theory of attributes), by means of ZFM. By α, where α is a formula, we denote any formula of the form x 1... x n α, where x 1, x 2,..., x n contains all free variables of α. Then we define the discusive system JM as follows: α is a theorem of JM if, and only if, α is a thesis of ZFM. I. Axiom Schemes: JM1) If α is an axiom of ZFM, then α is an axiom of JM. II. Rules: RJM1) From α and (α β) to infer β; RJM2) From α to infer α;

4 On Discussive Set Theory 147 RJM3) From α to infer α; RJM4) From α to infer α; RJM5) From (β α(x)) to infer (β xα(x)); RJM6) From α to infer α (α has no free variable). In order to prove that the above axiomatization is a complete axiomatization of JM, we need the following two lemmas. Lemma 1. Lemma 2. If α is a theorem of ZFM, then α is a theorem of JM. If α is a theorem of ZFM, then α is a theorem of JM. Theorem 3. JM. The above axiomatization is a complete axiom system for Remark 1. Our method of defining JM by means by ZFM can be employed to obtain a discussive set theory starting with any modal set theory. So to every modal set theory there is a corresponding discussive set theory. Remark 2. Our method can ba applied to obtain higher-order discussive logics starting with higher-order modal logics such that of Gallin [2]. This paper ie a summary of the talk given in the Logic Seminar run by Professor N. C. A. da Costa in the Department of Mathematics of the Catholic University of Sao Paulo. The underlying logic of JM was used by Professor da Costa in his studies of pragmatic truth. References [1] N. C. A. da Costa and L.P. de Alcântara, Remarks on Higher-Order Modal Logic, Relatório Interno 217, IMECC-UKICAMP. [2] D. Gallin, Intensional and Higher-Order Modal Logic, North- Holland Publ. Co., Amsterdam, [3] G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic, Methuen, London, 1968.

5 148 Lafayette de Moraes [4] S. Jaśkowski, Rachunek zdań dla systemów dedukcyjnych sprzecznych, Studia Societatis Scientiarum Torunensis, Sectio A, Vol. 1, No. 5, Toruń, 1948 (English version: Propositional calculus for contradictory deductive systems, Studia Logica 24 (1969), pp ). University of Campinas (FE-UNICAMP) Campinas, SP Brazil

RELATIONS BETWEEN PARACONSISTENT LOGIC AND MANY-VALUED LOGIC

Bulletin of the Section of Logic Volume 10/4 (1981), pp. 185 190 reedition 2009 [original edition, pp. 185 191] Newton C. A. da Costa Elias H. Alves RELATIONS BETWEEN PARACONSISTENT LOGIC AND MANY-VALUED