This presentation of a system of propositional logic is a foundational paper for systems of illocutionary logic. The language $\mathcal{L}_{.75}$ contains the illocutionary force operators '$\vdash$' for assertion and '⨽' for supposition. Sentences occurring in proofs of the deductive system $\mathcal{S}_{.75}$ must be prefixed with one of these operators, and rules of $\mathcal{S}_{.75}$ take account of the forces of the sentences. Two kinds of semantic conditions are investigated; familiar truth conditions and commitment conditions. Accepting a statement A or rejecting A commits a person to accepting some statements (the symbol '$+$' marks this value), to rejecting some statements ($-$), and will leave the person uncommitted with respect to others ($n$). Commitment valuations assign the values $+, -, n$ to sentences of $\mathcal{L}_{.75}$; such a valuation is conceived as reflecting the beliefs/knowledge of a particular person. This paper explores the relations between truth conditions and commitment conditions, and between semantic concepts defined in terms of these conditions.

We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. We also establish the consistency of this system, applying the strong normalization theorem.

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

In this paper I examine the nature of Russell's ramified type theory resolution of paradoxes. In particular, I consider the effect of construing the types in Church's cumulative sense, that is, the range of a variable of a given type includes the range of every variable of directly lower type. Contrary to what seems to be generally assumed, I show that the decision to make the levels cumulative and allow this to be reflected in the semantics is not neutral with respect to the solution of the paradoxes. I introduce a distinction between syntactical and semantical cumulativeness. It turns out that noncumulative type theories (in either sense) are equally capable of dealing with the paradoxes. Furthermore, whether cumulativeness is appropriate appears to be context dependent.

Disjunctive Syllogism, that is, the inference from 'not-A or B' and 'A', to 'B' can lead from true premises to a false conclusion if each of the sentences 'A' and 'not-A' is a statement of a partial truth such that affirming one of them amounts to denying the other, without each being the contradictory of the other. Such sentences inevitably occur whenever a situation which for its proper precise description needs the use of expressions such as 'most probably true' and so forth, is described (less precisely) by sentences not containing such expressions.

We show that for every nontrivial r.e. wtt-degree a, there are r.e. wtt-degrees b and c incomparable to a such that the infimum of a and b exists but the infimum of a and c fails to exist. This shows in particular that there are no strongly noncappable r.e. wtt-degrees, in contrast to the situation in the r.e. Turing degrees.

In this paper we shall introduce a simple temporal logic suitable for reasoning about the temporal aspects of parallel universes, parallel processes, distributed systems, or multiple agents. We will use a variant of the mosaic method to prove decidability of this logic. We also show that the logic does not have the finite model property. This shows that the mosaic method is sometimes a stronger way of establishing decidability.

Frege's intention in section 31 of Grundgesetze is to show that every well-formed expression in his formal system denotes. But it has been obscure why he wants to do this and how he intends to do it. It is argued here that, in large part, Frege's purpose is to show that the smooth breathing, from which names of value-ranges are formed, denotes; that his proof that his other primitive expressions denote is sound and anticipates Tarski's theory of truth; and that the proof that the smooth breathing denotes, while flawed, rests upon an idea now familiar from the completeness proof for first-order logic. The main work of the paper consists in defending a new understanding of the semantics Frege offers for the quantifiers: one which is objectual, but which does not make use of the notion of an assignment to a free variable.