First-Order da Costa Logic

Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls ‘Da Costa Logic’. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa’s own C_ω, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.     

Logic of primary-conditionals and secondary-conditionals

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establish the minimum system C2L _m of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of C2L _m, introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of C2L _m by the canonical model. Finally, according to the technical results of the minimum system C2L _m, the authors discuss some of the important problems concerning primary-conditionals and secondary-onditionals. __________ Translated from Luoji Yu Renzhi 逻辑与认知 (Logic and Cognition) (online journal), 2004 (3)     

Socratic Proofs and Paraconsistency: A Case Study

This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schütte's maximally paraconsistent logic Φ_v. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.     

Paraconsistency in classical logic

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.     

Models for normal intuitionistic modal logics

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic.     

Non-Adjunctive Inference and Classical Modalities

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A∧B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented in Chellas, 1980). The unified semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowski's system D₂ into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski.     

Completeness Theorems via the Double Dual Functor

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ideal Paraconsistent Logics

We define in precise terms the basic properties that an ??ideal propositional paraconsistent logic?? is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.     

Valuation Semantics for First-Order Logics of Evidence and Truth

This paper introduces the logic QLET_F, a quantified extension of the logic of evidence and truth LET_F, together with a corresponding sound and complete first-order non-deterministic valuation semantics. LET_F is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (FDE) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘A entails that A behaves classically, ∙A follows from A’s violating some classically valid inferences. The semantics of QLET_F combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of FDE, K3, and LP, we show how these tools, which we call here the method of anti-extensions + valuations, can be naturally applied to a number of non-classical logics.

     

Paraconsistent algebras

The prepositional calculiC _n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra forC _n . C. Mortensen settled the problem, proving that no equivalence relation forC _n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC _n , as well as the concept of paraconsistent closure system, are introduced in this paper.We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.     

Minimally inconsistent LP

The paper explains how a paraconsistent logician can appropriate all classical reasoning. This is to take consistency as a default assumption, and hence to work within those models of the theory at hand which are minimally inconsistent. The paper spells out the formal application of this strategy to one paraconsistent logic, first-order LP. (See, Ch. 5 of: G. Priest, In Contradiction, Nijhoff, 1987.) The result is a strong non-monotonic paraconsistent logic agreeing with classical logic in consistent situations. It is shown that the logical closure of a theory under this logic is trivial only if its closure under LP is trivial.     

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency (LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8] and [9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8] and [9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert?s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B∧¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACIarithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.     

Anti-intuitionism and paraconsistency

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson's dual calculus and ending up with Gödel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued logics (Iⁿ)_nω we show that the anti-intuitionistic hierarchy (I^n*)_nω obtained from (Iⁿ)_nω does coincide with the hierarchy of the many-valued paraconsistent logics (Pⁿ)_nω. Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them.     

On The Computational Consequences of Independence in Propositional Logic

Sandu and Pietarinen [Partiality and Games: Propositional Logic. Logic J. IGPL 9 (2001) 101] study independence friendly propositional logics. That is, traditional propositional logic extended by means of syntax that allow connectives to be independent of each other, although the one may be subordinate to the other. Sandu and Pietarinen observe that the IF propositional logics have exotic properties, like functional completeness for three-valued functions. In this paper we focus on one of their IF propositional logics and study its properties, by means of notions from computational complexity. This approach enables us to compare propositional logic before and after the IF make-over. We observe that all but one of the best-known decision problems experience a complexity jump, provided that the complexity classes at hand are not equal. Our results concern every discipline that incorporates some notion of independence such as computer science, natural language semantics, and game theory. A corollary of one of our theorems illustrates this claim with respect to the latter discipline.     

AI,ME AND LEWIS (ABELIAN IMPLICATION,MATERIAL EQUIVALENCE AND C I LEWIS 1920)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). Meyer was supported in this work as a Visiting Fellow in the College of Engineering and Computer Science, ANU.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Generic generalized Rosser fixed points

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR. This paper supersedes: Franco Montagna, Extremely undecidable sentences and generic generalized Rosser's fixed points, Rapporto Matematico, No. 95, Siena, 1983.     

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core” of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.     

Cut-free sequent and tableau systems for propositional Diodorean modal logics

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae involved are always bounded by a finite set of formulaeX* _L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive.Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hintikka and Rautenberg.Presented byDov M. Gabbay     

A finite model theorem for the propositional μ-calculus

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.Supported by NSF grant DCR-8602663