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28Between Int<W)W> and intuitionistic propositional logic1
Vladimir M. Popov
abstract. This short paper presents a new domain of logical investigations.
Keywords: paralogic, paracomplete logic, paraconsistent logic, paranormal logic, intuitionistic propositional logic
The language L of each logic in the paper is a standard propositional language whose alphabet is as follows: {&, V, D, (, ),p1,p2,p3, • • • }• As it is expected, V, D are binary logical connectives in L, - is a unary logical connective in L, brackets (, ) are technical symbols in L and p1,p2,p3, • • • are propositional variables in L. A definition of L-formula is as usual. Below, we say 'formula' instead of 'L-formula' only and adopt the convention on omitting brackets. A formula is said to be quasi-elemental iff no logical connective in L other than - occurs in it. A length of a formula A is, traditionally, said to be the number of all occurrences of the logical connectives in L in A. A logic is said to be a nonempty set of formulas closed under the rule of modus ponens in L and the rule of substitution of a formula into a formula instead of a propositional variable in L.
Let us agree that a and / are arbitrary elements in {0,1, 2, 3, •••w}. We define calculus HInt<a;^>. This calculus is a Hilbert-type calculus, the language of HInt<a>^> is L. HInt<a>^> has the rule of modus ponens in L as the only rule of inference. The notion of a proof in HInt<«;^> and the notion of a formula provable
xThe paper is supported by Russian Foundation for Humanities, projects № 10-03-00570a and № 13-03-00088a.
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Vladimir M. Popov
in this calculus are defined as usual. Now we only need to define the set of axioms of HInt<a;^>.
A formula belongs to the set of axioms of calculus HInt<a>^> iff it is one of the following forms (A, B, C denote formulas):
(I) (A D B) D ((B D C) D (A D C)), (II) A D (A V B), (III) B D (A V B), (IV) (A D C) D ((B D C) D ((A V B) D C)), (V) (A&B) D A, (VI) (A&B) D B, (VII) (C D A) D ((C D B) D (C D (A&B))), (VIII) (A D (B D C)) D ((A&B) D C), (IX) ((A&B) D C) D (A D (B D C)), (X, a) -D D (D D A), where D is formula which is not a quasi-elemental formula of a length less than a, (XI, 3) (E D -(B D B)) D -E, where E is formula which is not a quasi-elemental formula of a length less than 3.
Let us agree that, for any j and k in {0,1,2,3,... w}, Int<J;fc> is the set of formulas provable in HInt<jfc>. It is clear that, for any j and k in {0,1, 2, 3, ...w}, a set Int<J;fc> is a logic. It is proved that Int<o,o> is the set of intuitionistic tautologies in L (that is, the intuitionistic propositional logic in L). By S we denote the set of all logics which include logic Int<W;W> and are included in Int<0;0> and by ParaInt we denote S \ {Int<0;0>}. Note logic Int<W;W> is the intersection of all logics, other than itself, in ParaInt. The set ParaInt is of interest for scholars who study paralogics (paraconsistent or paracomplete logics). The set ParaInt contains (1) a continuous set of paraconsistent, but non-paracomplete logics, (2) a continuous set of paracomplete, but non-paraconsistent logics, (3) a continuous set of paranormal logics. We have some results concerning both logics from ParaInt and classes of such logics. In particular, we have methods to construct axiomatisations (sequent calculus and analytic-tableaux calculus) and semantics (in the sense of Kripke) for any logic Int<j)fc>, where j and k in {0,1,2,3,... w}.
References
[1] Popov, V. M., Two sequences of simple paranormal logics, Modern logic: theory, history and applications in science. The proceedings of the IX All-Russian Scientific Conference, June 22-24, 2006, St.-Petersburg, SPbU Publishers, 2006, pp. 382-385 (in Russian).
[2] Popov, V. M., Intervals of simple paralogies, Proceedings of the V conference "Smirnov Readings in Logic", June, 20-22, 2007, M., 2007, pp. 35-37 (in Russian).
Between Int<u,u> and intuitionistic propositional logic 199
[3] Popov, V. M., Two sequences of simple paraconsistent logics, Logical investigations, 14:257-261, 2007 (in Russian).
[4] Popov, V. M., Two sequence of simple paracomplete logics, Logic today: theory, history and applications. The proceedings of X Russian conference, June, 26-28, 2008, St.-Petersburg, SPbU Publishers, 2008, pp. 304-306 (in Russian).
[5] Popov, V. M., Some intervals between simple paralogics, Logical investigations, 15:182-184, 2009 (in Russian).
[6] Popov, V. M., Semantical characterization of intuitionistically acceptable simple paralogics and their connection with the intuitionistic propositional logic, The proceedings of the research seminar of the logical center of IFRAN, XIX:82-91, 2009 (in Russian).
[7] Popov, V. M., Sequential characterization of simple paralogics, Logical investigations, 16:205-220, 2010 (in Russian).
