A Note on Binary Inductive Logic

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.     

Empiricism,judgment, and argument; Toward an informal logic of science

In an attempt to explore the role of argumentation in scientific inquiry, I explore the conception of argument that appears fruitful in the light of the recent trends in the philosophy of science, away from logical empiricism, and toward a greater emphasis on change, disagreement, and history. I begin by contrasting typical instances philosopers theories of both empiricism and apriorism, with typical instances of scientists uses of these two attitudes, suggesting that such practice shows a judiciousness lacking in epistemological theory. Then I examine at some length three important version of scientific judgement, namely Einstein's opportunism, Boltzmann's pluralism, and Huygens's eclecticism. I go on to discuss some recent work in the philosophy of science, exploring connections between the notion of judgment and Feyerabend's anarchism, Harold Brown's view of rationality, and Dudley Shapere's analysis of scientific change. All of this suggests a notion of argument in the sense of informallogic, and I examine some aspects of Galileo's work in its terms.     

A Propositional Dynamic Logic with Qualitative Probabilities

This paper presents an -completeness theorem for a new propositional probabilistic logic, namely, the dynamic propositional logic of qualitative probabilities (D Q P), which has been introduced by the author as a dynamic extension of the logic of qualitative probabilities (Q P) introduced by Segerberg.     

Prolegomena of a Logic of Causality and Dynamism

We present in this article a new logical system inspired from linear logic. This system is designed in order to express causality and dynamism. The cut elimination theorem holds for this logic. Examples of applications are given.     

Mathematical Fuzzy Logic – What It Can Learn from Mostowski and Rasiowa

Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.A version of this paper has been presented during the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

Definability and Interpolation in Non-Classical Logics

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

THE SUPERNATURAL AS LANGUAGE GAME

Abstract. For many in the Anglo‐American tradition of language analysis, Ludwig Wittgenstein, the great progenitor of twentieth‐century philosophy of language, showed conclusively that theological terms lack any referent in reality and therefore represent a discourse that can do no more than manifest the existential attitudes that speakers take toward reality as a whole. To think that such terms represent more is to be bewitched by the use of language. Is it possible, however, that theological language references a fundamental human drive? In this article I reexamine the dyad of nature and supernature from the perspective of Wittgenstein's philosophy. Perhaps surprisingly, Wittgenstein's thought on the subject offers much more than his famous, terse aphorism at the conclusion of his first masterwork, the Tractatus Logico‐Philosophicus ([1921] 1961, 74, §7): “What we cannot speak about we must pass over in silence.” Furthermore, the basic Tractarian drive to determine the relationship between language and reality, which is redirected but not extinguished in Wittgenstein's second, divergent, opus, the Philosophical Investigations ([1953] 1967), may be the place for a renewed examination of what the supernatural means in human discourse. Does talk of God give expression to the fundamental transcendence of human knowledge? Is it a language game we can eschew?     

From Modal Discourse to Possible Worlds

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.     

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators,and Applications to the Semantics of Non-Classical Logics I

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. This revised version was published online in June 2006 with corrections to the Cover Date.