Labeled Calculi and Finite-Valued Logics

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.     

Embedding Logics into Product Logic

We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).     

Elementary Extensions of External Classes in a Nonstandard Universe

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements.     

The place of human values in the language of science: Kuhn,Saussure, and structuralism

The current paradigm in medicine generally distinguishes between genetic and environmental causes of disease. Although the word paradigm has become a commonplace, the theories of Thomas Kuhn have not received much attention in the journals of medicine. Kuhn's structuralist method differs radically from the daily activities of the scientific method itself. Using linguistic theory, this essay offers a structuralist reading of Thomas Kuhn's The Structure of Scientific Revolutions. Our purpose is to highlight the similarities between these structuralist models of science and language. In part, we focus on the logic that enables Kuhn to assert the priority of perception over interpretation in the history of science. To illustrate some of these issues, we refer to the distinction between environmental and genetic causes of disease. While the activity of scientific research results in the revision of concepts in science, the production of significant differences that shape our knowledge is in part a social and linguistic process.     

A Note on Graded Modal Logic

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.     

On the Consistency of the Δ1 1-CA Fragment of Frege's Grundgesetze

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing ¹ ₁-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case.     

Classical Non-Associative Lambek Calculus

We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.