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.     

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.     

Priorean Strict Implication, Q and Related Systems

We introduce a system PSI for a strict implication operator called Priorean strict implication. The semantics for PSI is based on partial Kripke models without accessibility relations. PSI is proved sound and complete with respect to that semantics, and Prior's system Q and related systems are shown to be fragments of PSI or of a mild extension of it.     

Philosophy and language learning

In this paper, I explore different ways of picturing language learning in philosophy, all of them inspired by Wittgenstein and all of them concerned about scepticism of meaning. I start by outlining the two pictures of children and language learning that emerge from Kripke’s famous reading of Wittgenstein. Next, I explore how social-pragmatic readings, represented by Meredith Williams, attempt to answer the sceptical anxieties. Finally, drawing somewhat on Stanley Cavell, I try to resolve these issues by investigating what characteristically happens to our view of language learning when we do philosophy. The focus throughout is on the relation between the individual (the learning child) and the community (usually represented by the parents), and how that relation is deformed when we operate with a certain philosophical notion of ground.     

Kripke Models,Distributive Lattices,and Medvedev Degrees

We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice. Presented by Daniele Mundici     

Escape from Twin Earth: Putnam's 'Logic' of Natural Kind Terms

Many still seem confident that the kind of semantic theory Putnam once proposed for natural kind terms is right. This paper seeks to show that this confidence is misplaced because the general idea underlying the theory is incoherent. Consequently, the theory must be rejected prior to any consideration of its epistemological, ontological or metaphysical acceptability. Part I sets the stage by showing that falsehoods, indeed absurdities, follow from the theory when one deliberately suspends certain devices Putnam built into it , presumably in order to block such entailments. Part II then raises the decisive issue of at what cost these devices do the job they need to do. It argues that - apart from possessing no other motivation than their capacity to block the consequences derived in Part I - they only fulfil this blocking function if they render the theory unable to deal with fiction and related 'make-believe' activities. Part III indicates the affinity Putnam's account has with the classically 'denotative' view of meaning, and thus how its weaknesses may be seen as a variant of the classical weakness of 'denotative' approaches. It concludes that the theory is a conceptual muddle.     

Axiomatizing hybrid products: How can we reason many-dimensionally in hybrid logic?

The aim of this paper is to propose a two-dimensional hybrid logic in order to formalize inferences containing both spatial and temporal propositions. The semantic idea behind the proposal is to name both horizontal and vertical lines in a 2D-plane by two kinds of nominals. This is a generalization of the idea of naming a point in one-dimensional hybrid logic. I give an axiomatization of the proposed two-dimensional hybrid logic and show that it enjoys a general completeness result (called pure completeness) with respect to product Kripke frames. Moreover, in order to capture $T\times W$-frames studied by R.H. Thomason (1984), I introduce the notion of a dependent product frame, which enables us to represent the dependence of space over time. I also give a complete axiomatization of this dependent two-dimensional hybrid logic, and, as a corollary, reveal that a hybridization of $T\times W$-logic enjoys strong completeness.     

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.