The finite model property for BCI and related systems

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].Presented by H. Ono     

A modern elaboration of the ramified theory of types

The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see [3].Supported by the Co-operation Centre Tilburg and Eindhoven Universities. ¹[32], Introduction, Chapter II, Section I, p. 37.Presented by Wolfgang Rautenberg     

The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms

The Lambek calculus introduced in Lambek [6] is a strengthening of the type reduction calculus of Ajdukiewicz [1]. We study Associative Lambek Calculus L in Gentzen style axiomatization enriched with a finite set Γ of nonlogical axioms, denoted by L(Γ).It is known that finite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages (see Buszkowski [2]). Then we confine nonlogical axioms to sequents of the form p → q, where p and q are atomic types. For calculus L(Γ) we prove interpolation lemma (modifying the Roorda proof for L [10]) and the binary reduction lemma (using the Pentus method [9] with modification from [3]). In consequence we obtain the weak equivalence of the Context-Free Grammars and grammars based on L(Γ).     

Choice and Logic

There is a little known paradox the solution to which is a guide to a much more thoroughgoing solution to a whole range of classic paradoxes. This is shown in this paper with respect to Berrys Paradox, Heterologicality, Russells Paradox, and the Paradox of Predication, also the Liar and the Strengthened Liar, using primarily the epsilon calculus. The solutions, however, show not only that the first-order predicate calculus derived from Frege is inadequate as a basis for a clear science, and should be replaced with Hilbert and Bernays conservative extension. Standard second-order logic, and quantified propositional logic also must be substantially modified, to incorporate, in the first place, nominalizations of predicates, and whole sentences. And further modifications must be made, so as to insist that predicates are parts of sentences rather than forms of them, and that truth is a property of propositions rather than their sentential expressions. In all, a thorough reworking of what has been called logic in recent years must be undertaken, to make it more fit for use.Portions of this paper have previously been published in Logical Studies, vol. 9, http://www.logic.ru/LogStud/09/No9-06.html, and the Australasian Journal of Logic, vol. 2, http://www.philosophy.unimelb.edu.au/ajl/2004/2004_4.pdf.     

Number of Extensions of Non-Fregean Logics

We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.     

Probability: A new logico-semantical approach

This approach does not define a probability measure by syntactical structures. It reveals a link between modal logic and mathematical probability theory. This is shown (1) by adding an operator (and two further connectives and constants) to a system of lower predicate calculus and (2) regarding the models of that extended system. These models are models of the modal systemS ₅ (without the Barcan formula), where a usual probability measure is defined on their set of possible worlds. Mathematical probability models can be seen as models ofS ₅.     

On Consistent Subsets of Large Sets of Satisfiable Sentences

We extend some results of Adam Kolany to show that large sets of satisfiable sentences generally contain equally large subsets of mutually consistent sentences. In particular, this is always true for sets of uncountable cofinality, and remains true for sets of denumerable cofinality if we put appropriate bounding conditions on the sentences. The results apply to both the propositional and the predicate calculus. To obtain these results, we use delta sets for regular cardinals, and, for singular cardinals, a generalization of delta sets. All of our results are theorems in ZFC.