Preservation of Craig interpolation by the product of matrix logics

The product of matrix logics, possibly with additional interaction axioms, is shown to preserve a slightly relaxed notion of Craig interpolation. The result is established symbolically, capitalizing on the complete axiomatization of the product of matrix logics provided by their meet-combination. Along the way preservation of the metatheorem of deduction is also proved. The computation of the interpolant in the resulting logic is proved to be polynomially reducible to the computation of the interpolants in the two given logics. Illustrations are provided for classical, intuitionistic and modal propositional logics.     

Action negation and alternative reductions for dynamic deontic logics

Dynamic deontic logics reduce normative assertions about explicit complex actions to standard dynamic logic assertions about the relation between complex actions and violation conditions. We address two general, but related problems in this field. The first is to find a formalization of the notion of ‘action negation’ that (1) has an intuitive interpretation as an action forming combinator and (2) does not impose restrictions on the use of other relevant action combinators such as sequence and iteration, and (3) has a meaningful interpretation in the normative context. The second problem we address concerns the reduction from deontic assertions to dynamic logic assertions. Our first point is that we want this reduction to obey the free-choice semantics for norms. For ought-to-be deontic logics it is generally accepted that the free-choice semantics is counter-intuitive. But for dynamic deontic logics we actually consider it a viable, if not, the better alternative. Our second concern with the reduction is that we want it to be more liberal than the ones that were proposed before in the literature. For instance, Meyer's reduction does not leave room for action whose normative status is neither permitted nor forbidden. We test the logics we define in this paper against a set of minimal logic requirements.     

Subformula semantics for strong negation systems

We present a semantics for strong negation systems on the basis of the subformula property of the sequent calculus. The new models, called subformula models, are constructed as a special class of canonical Kripke models for providing the way from the cut-elimination theorem to model-theoretic results. This semantics is more intuitive than the standard Kripke semantics for strong negation systems.     

Automated theorem proving for non-classical logics

Automated theorem proving for non-classical logics     

A metacompleteness theorem for contraction-free relevant logics

I note that the logics of the relevant group most closely tied to the research programme in paraconsistency are those without the contraction postulate(A.AB).AB and its close relatives. As a move towards gaining control of the contraction-free systems I show that they are prime (that wheneverA B is a theorem so is eitherA orB). The proof is an extension of the metavaluational techniques standardly used for analogous results about intuitionist logic or the relevant positive logics.The main results of this paper were presented at the Paraconsistent Logic conference in Canberra in 1980. The author wishes to thank the participants in that conference for comments and suggestions made at the time.     

Fregean logics with the multiterm deduction theorem and their algebraization

A deductive system $\mathcal{S}$ (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas $$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$ is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.     

A deduction theorem schema for deductive systems of propositional logics

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) _s q.     

Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.

     

The contraction rule and decision problems for logics without structural rules

This paper shows a role of the contraction rule in decision problems for the logics weaker than the intuitionistic logic that are obtained by deleting some or all of structural rules. It is well-known that for such a predicate logic L, if L does not have the contraction rule then it is decidable. In this paper, it will be shown first that the predicate logic FL_ec with the contraction and exchange rules, but without the weakening rule, is undecidable while the propositional fragment of FL_ec is decidable. On the other hand, it will be remarked that logics without the contraction rule are still decidable, if our language contains function symbols.     

Intuitive semantics for some three-valued logics connected with information,contrariety and subcontrariety

Four known three-valued logics are formulated axiomatically and several completeness theorems with respect to nonstandard intuitive semantics, connected with the notions of information, contrariety and subcontrariety is given.To the memory of Jerzy Supecki     

Criteria for admissibility of inference rules. Modal and intermediate logics with the branching property

The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsD _n; to all intermediate Gabbay-De-Jong logicsD _n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well.The results of this paper were obtained by the author during a stay at the Free University of Berlin with support of the Alexander von Humboldt Foundation in 1992 – 1993.Presented byWolfgang Rauntenberg     

The American plan completed: Alternative classical-style semantics,without stars,for relevant and paraconsistent logics

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa.     

Sahlqvist's theorem for boolean algebras with operators with an application to cylindric algebras

For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V.We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity calledHenkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the first-order correspondents of these two equations are equivalent over the class of cylindric atom structures.Presented byIstván Németi     

A representation theorem for languages with generalized quantifiers through back-and-forth methods

We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.     

The strong programme for the sociology of science,reflexivity and relativism

David Bloor has advocated a bold hypothesis about the form any sociology of science should take in setting out the four central tenets of his ‘strong programme’ (SP). The first section of this paper discusses how three of these tenets are best formulated and how they relate to one another. The second section discusses how reasons can be causes of belief and how such reasons raise a serious difficulty for SP. The third section discusses how SP is committed to a form of relativism about truth. The fourth section discusses how one might deal with the problem of SP applying both to itself and to other sociological theories. In addition there is, throughout, a discussion of how rules of inference, methodologies, and philosophical doctrines either apply to SP or are exempt from applying. It is argued that SP must be a severely limited doctrine impotent to make evaluative claims about the worth of any theory, including itself.