In this paper we investigate computable models of $\aleph_1$-categorical theories and Ehrenfeucht theories. For instance, we give an example of an $\aleph_1$-categorical but not $\aleph_0$-categorical theory $T$ such that all the countable models of $T$ except its prime model have computable presentations. We also show that there exists an $\aleph_1$-categorical but not $\aleph_0$-categorical theory $T$ such that all the countable models of $T$ except the saturated model, have computable presentations.

We investigate the higher-order modal logic $S_{\omega}I$, which is a variant of the system $S_{\omega}$ presented in our previous work. A semantics for that system, founded on the theory of quasi sets, is outlined. We show how such a semantics, motivated by the very intuitive base of Schrödinger logics, provides an alternative way to formalize some intensional concepts and features which have been used in recent discussions on the logical foundations of quantum mechanics; for example, that some terms like 'electron' have no precise reference and that 'identical' particles cannot be named unambiguously. In the last section, we sketch a classical semantics for quasi set theory.

From 1929 through 1944, Wittgenstein endeavors to clarify mathematical meaningfulness by showing how (algorithmically decidable) mathematical propositions, which lack contingent "sense," have mathematical sense in contrast to all infinitistic "mathematical" expressions. In the middle period (1929-34), Wittgenstein adopts strong formalism and argues that mathematical calculi are formal inventions in which meaningfulness and "truth" are entirely intrasystemic and epistemological affairs. In his later period (1937-44), Wittgenstein resolves the conflict between his intermediate strong formalism and his criticism of set theory by requiring that a mathematical calculus (vs. a "sign-game") must have an extrasystemic, real world application, thereby returning to the weak formalism of the Tractatus.

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for practical mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace.

The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that $\vee$-$\wedge$-typed combinatory logic terms are proved to completely codify the associated logical proofs.

A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. Barendregt has shown that if we can represent, for a numeral system, the functions Successor, Predecessor, and Zero Test, then all total recursive functions can be represented. In this paper we prove the independancy of these three particular functions. We give at the end a conjecture on the number of unary functions necessary to represent all total recursive functions.

This paper examines and discusses an argument for the collapse of modal distincions offered by Quine in "Reference and Modality" and in Word and Object that relies exclusively on a version of the Principle of Substitution. It is argued that the argument does not affect its historical targets: Carnap's treatment of modality, presented in Meaning and Necessity, and Church's Logic of Sense and Denotation, developed by Kaplan; nor does it affect a treatment of modality inspired in Frege's treatment of oblique contexts. It is argued, nevertheless, that the immunity of those systems to Quine's argument depends on the success of their rejection of the Principle of Substitution presupposed by Quine.

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems ${\bf S5 \pi +}$, ${\bf S4 \pi +}$, ${\bf S4.2 \pi +}$: given a Kripke frame, the quantifiers range over all the sets of possible worlds. ${\bf S5 \pi +}$ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub ${\bf S4.2 \pi t}$, is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in ${\bf S4.2 \pi t}$. In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system ${\bf S4 \pi +}$ is recursively isomorphic to second-order logic.