Non-Boolean classical relevant logics II: Classicality through truth-constants

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence.

Equality Logic

In this paper, we introduce and study a corresponding logic toequality-algebras and obtain some basic properties of this logic. We provethe soundness and completeness of this logic based on equality-algebrasand local deduction theorem. Then we introduce the concept of (prelinear)equality-algebras and investigate some related properties. Also, westudy -deductive systems of equality-algebras. In particular, we provethat every prelinear equality-algebra is a subdirect product of linearly orderedequality-algebras. Finally, we construct prelinear equality logicand prove the soundness and strong completeness of this logic respect toprelinear equality-algebras.

A duality theorem

This chapter presents a duality theorem providing, for each geometric theory, a natural bijection between its geometric theory extensions (also called ‘quotients’) and the subtoposes of its classifying topos. Two different proofs of this theorem are provided, one relying on the theory of classifying toposes and the other, of purely syntactic nature, based on a proof-theoretic interpretation of the notion of Grothendieck topology. Via this interpretation the theorem can be reformulated as a proof-theoretic equivalence between the classical system of geometric logic over a given geometric theory and a suitable proof system whose rules correspond to the axioms defining the notion of Grothendieck topology. The role of this duality as a means for shedding light on axiomatization problems for geometric theories is thoroughly discussed, and a deduction theorem for geometric logic is derived from it.

Note on Deduction Theorems in Contraction-Free Logics

In this short paper we present a finer analysis of the variants of Local Deduction Theorem in contraction-free logics. We define some natural generalisations called Implicational Deduction Theorems and study their basic properties. The hierarchy of classes of logics defined by these theorems is presented.

Propositional Linear Temporal Logic with Initial Validity Semantics

Summary In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].

Many-valued logics—implications and semantic consequences

In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication. In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.