On Intermediate Predicate Logics of some Finite Kripke Frames,I. Levelwise Uniform Trees

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.     

Kripke Incompleteness of Predicate Extensions of the Modal Logics Axiomatized by a Canonical Formula for a Frame with a Nontrivial Cluster

We generalize the incompleteness proof of the modal predicate logic Q-S4+ p p + BF described in Hughes-Cresswell [6]. As a corollary, we show that, for every subframe logic Lcontaining S4, Kripke completeness of Q-L+ BF implies the finite embedding property of L.     

Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.     

Predicate Logical Extensions of some Subintuitionistic Logics

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier. Presented by Melvin Fitting     

Kripke Bundle Semantics and C-set Semantics

Kripke bundle [3] and C-set semantics [1] [2] are known as semantics which generalize standard Kripke semantics. In [3] and in [1], [2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics [5].In this paper, we show that Q-S4.1 is not Kripke bundle complete via C-set models. As a corollary we can give a simple proof showing that C-set semantics for modal logics are stronger than Kripke bundle semantics.     

On the Existence of Continua of Logics Between Some Intermediate Predicate Logics

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.     

A Kripke Semantics for the Logic of Gelfand Quantales

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic.     

Gentzen Calculi for the Existence Predicate

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.     

A Predicate Logical Extension of a Subintuitionistic Propositional Logic

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke's semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong completeness is proved making use of predicate logical principles, which are only classically acceptable.     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and s, it follows constructively that the K-completeness of s implies MP(S), a form of Markov's Principle. If s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when s is many-one complete, MP(S) implies the usual Markov's Principle MP.An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implics MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP.These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Gödel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

Combinatory Logic and the Semantics of Substructural Logics

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself. Presented by Rob Goldblatt     

Combinations of Tense and Modality for Predicate Logic

In recent years combinations of tense and modality have moved into the focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate logic are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T×W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T×W-semantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed.     

A Nonmonotonic Modal Formalization of the Logic of Acceptance and Rejection

The problems we deal with concern reasoning about incomplete knowledge. Knowledge is understood as ability of an ideal rational agent to make decisions about pieces of information. The formalisms we are particularly interested in are Moore's autoepistemic logic (AEL) and its variant, the logic of acceptance and rejection (AEL2). It is well-known that AEL may be seen as the nonmonotonic KD45 modal logic. The aim is to give an appropriate modal formalization for AEL2.     

The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Logic of Nonmonotonic Conditionals and Conditional Probabilities: Predicate Logic

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field's probabilistic semantics. Along the way I will show how Field's semantics differs from a substitutional interpretation of quantifiers in crucial ways, and show that Field's approach is closely related to the usual objectual semantics. One of Field's quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics.     

An Axiomatisation for the Multi-modal Logic of Knowledge and Linear Time LTK

The paper aims at providing the multi-modal propositional logicLTK with a sound and complete axiomatisation. This logic combinestemporal and epistemic operators and focuses on m odeling thebehaviour of a set of agents operating in a system on the backgroundof a temporal framework. Time is represented as linear and discrete,whereas knowledge is modeled as an S₅-like modality. A furthermodal operator intended to represent environment knowledge isadded to the system in order to achieve the expressive powersufficient to describe the piece of information available tothe agents at each moment in the flow of time.