Structuralist Logic: Implications, Inferences, and Consequences

On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B20, 03B42, 03B60     

Polymodal Lattices and Polymodal Logic

A polymodal lattice is a distributive lattice carrying an n-place operator preserving top elements and certain finite meets. After exploring some of the basic properties of such structures, we investigate their freely generated instances and apply the results to the corresponding logical systems — polymodal logics — which constitute natural generalizations of the usual systems of modal logic familiar from the literature. We conclude by formulating an extension of Kripke semantics to classical polymodal logic and proving soundness and completeness theorems. Mathematics Subject Classification: 03G10, 06D99, 03B45.     

Classically complete modal relevant logics

A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.     

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

Hierarchical structure and applications of fuzzy logical systems

This paper focuses on hierarchical structures of formulas in fuzzy logical systems. Basic concepts and hierarchical structures of generalized tautologies based on a class of fuzzy logical systems are discussed. The class of fuzzy logical systems contains the monoidal t-norm based system and its several important schematic extensions: the ?ukasiewicz logical system, the Gödel logical system, the product logical system and the nilpotent minimum logical system. Furthermore, hierarchical structures of generalized tautologies are applied to discuss the transformation situation of tautological degrees during the procedure of fuzzy reasoning.     

Local computation in linear logic

This work deals with the exponential fragment of Girard's linear logic ([3]) without the contraction rule, a logical system which has a natural relation with the direct logic ([10], [7]). A new sequent calculus for this logic is presented in order to remove the weakening rule and recover its behavior via a special treatment of the propositional constants, so that the process of cut-elimination can be performed using only “local” reductions. Hence a typed calculus, which admits only local rewriting rules, can be introduced in a natural manner. Its main properties — normalizability and confluence — has been investigated; moreover this calculus has been proved to satisfy a Curry-Howard isomorphism ([6]) with respect to the logical system in question. MSC: 03B40, 03F05.     

On witnessed models in fuzzy logic II

First the expansion of the ?ukasiewicz (propositional and predicate) logic by the unary connectives of dividing by any natural number (Rational ?ukasiewicz logic) is studied; it is shown that in the predicate case the expansion is conservative w.r.t. witnessed standard 1‐tautologies. This result is used to prove that the set of witnessed standard 1‐tautologies of the predicate product logic is Π₂‐hard. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extensions of Hałkowska–Zajac's three-valued paraconsistent logic

 As it was proved in [4, Sect. 3], the poset of extensions of the propositional logic defined by a class of logical matrices with equationally-definable set of distinguished values is a retract, under a Galois connection, of the poset of subprevarieties of the prevariety generated by the class of the underlying algebras of the defining matrices. In the present paper we apply this general result to the three-valued paraconsistent logic proposed by Hałkowska–Zajac [2]. Studying corresponding prevarieties, we prove that extensions of the logic involved form a four-element chain, the only proper consistent extensions being the least non-paraconsistent extension of it and the classical logic. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03B50, 03B53, 03G10 RID="ID=" <E5>Key words or phrases:</E5> Many-valued logic &ndash; Paraconsistent logic &ndash; Extension &ndash; Prevariety &ndash; Distributive lattice Received 12 August 2000 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03B50, 03B53, 03G10 RID=" ID=" <E5>Key words or phrases:</E5> Many-valued logic &ndash; Paraconsistent logic &ndash; Extension &ndash; Prevariety &ndash; Distributive lattice     

Fuzzy propositional logics

Fuzzy logic $\text{L}$_∞9 considered in connection with fuzzy sets theory, is a special theory, is a special many valued logic with truth-value sets [0, 1], which has been studied already by Lukasiewicz. We consider also his versions $\text{L}$_m for m ? 2 with finite truth-value sets. In all cases we add two further propositional connectives, one conjunction and one disjunction. For these logics we give a list of tautologies, consider relations between their sets of tautologies, prove their compactness, and mention some further results.     

Mixed logic and storage operators

In 1990 J-L. Krivine introduced the notion of storage operators. They are -terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using G?del translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. Received: 7 May 1997     

Algorithmic correspondence and canonicity for non-distributive logics

We extend the theory of unified correspondence to a broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as ‘lattices with operators’. Specifically, we introduce a syntactic definition of the class of Sahlqvist formulas and inequalities which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. We also introduce the algorithm ALBA, parametric in each LE-setting, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. Projecting these results on specific signatures yields state-of-the-art correspondence and canonicity theory for many well known modal expansions of classical and intuitionistic logic and for substructural logics, from classical poly-modal logics to (bi-)intuitionistic modal logics to the Lambek calculus and its extensions, the Lambek-Grishin calculus, orthologic, the logic of (not necessarily distributive) De Morgan lattices, and the multiplicative-additive fragment of linear logic.     

Analytic completeness theorem for singular biprobability models

The aim of the paper is to prove tha analytic completeness theorem for a logic L(∫₁, ∫₂)_A^s with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.     

Proof complexity of intuitionistic implicational formulas

We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF) or substitution Frege (SF) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions.In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrube? (2007), generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Je?ábek (2009), can be realized by implicational tautologies.     

On the discriminant function of two commuting nonselfadjoint operators

The notion of the discriminant function of two commuting nonselfadjoint operators A₁ and A₂ in a Hilbert space H was defined by M.S. Livic in a recent paper [5].In the present paper we study the discriminant function for two special cases of commuting operators. In one case A₁ and A₂ are rational functions of one and the same operator A in H. In the second case A₁ and A₂ are Kronecker products of the form A I and I B, where I is the identity operator.As an auxiliary result a relation between the Sylvester and Bezoutian matrices of two polynomials is derived.     

积逻辑系统中的广义重言式

讨论积逻辑系统中的广义重言式理论 ,给出积逻辑系统中子代数和广义重言式的一系列性质。本文的主要结果表明 ,在几个重要的逻辑系统中 ,标准积逻辑系统具有最简单的广义重言式结构 ,而在推理过程中 ,它具有较差的真值传递性。     

扰动模糊命题逻辑的代数结构及其广义重言式性质

着眼于扰动模糊命题逻辑的代数结构,为研究二维扰动模糊命题逻辑最大子代数I2R及其广义重言式提供了一些代数理论基础,最后研究了子代数间广义重言式的关系.     

Bilinear spaces over a fixed field are simple unstable

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for ω-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are ω-categorical.     

Spectral properties of sums of certain Kronecker products

We introduce two kinds of sums of Kronecker products, and their induced operators. We study the algebraic properties of these two kinds of matrices and their associated operators; the properties include their eigenvalues, their eigenvectors, and the relationships between their spectral radii or spectral abscissae. Furthermore, two projected matrices of these Kronecker products and their induced operators are also studied.