Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics for classical first-order logic admits constructive soundness and completeness proofs, setting it apart from standard model-theoretic semantics of first-order logic. Furthermore, we prove that completeness with regard to intuitionistic material dialogues fails in both constructive and classical settings. As an alternative, we propose material dialogues played over Kripke structures. These Kripke material dialogues exhibit constructive completeness when restricting to the negative fragment. The results concerning classical material dialogues have been mechanized using the Coq interactive theorem prover.

I discuss problems with Martin-Löf’s distinction between analytic and synthetic judgments in constructive type theory and propose a revision of his views. I maintain that a judgment is analytic when its correctness follows exclusively from the evaluation of the expressions occurring in it. I argue that Martin-Löf’s claim that all judgments of the forms $a : A$ and $a = b : A$ are analytic is unfounded. As I shall show, when A evaluates to a dependent function type $(x : B) \to C$, all judgments of these forms fail to be analytic and therefore end up as synthetic. Going beyond the scope of Martin-Löf’s original distinction, I also argue that all hypothetical judgments are synthetic and show how the analytic–synthetic distinction reworked here is capable of accommodating judgments of the forms $A \> \mathsf {type}$ and $A = B \> \mathsf {type}$ as well. Finally, I consider and reject an alternative account of analyticity as decidability and assess Martin-Löf’s position on the analytic grounding of synthetic judgments.

This lecture was given by Per Martin-Löf at Leiden University on August 25, 2001 at the invitation by Göran Sundholm to address the topic mentioned in the title and to reflect on Dummett’s earlier effort of almost a decade before (published in this journal). The lecture was part of a three-day conference on Gottlob Frege. Sundholm arranged for the lecture to be recorded and commissioned Bjørn Jespersen to make a transcript. The information in footnote 1, which Sundholm provided, has been independently confirmed by Thomas Ricketts in an email to the author. The present version has been edited by Ansten Klev. Following the displayed text (Int-id) there is a lacuna in the original transcript corresponding to a pause in the recording when the tape was changed. The continuous text of the present version is the result of a few additions to the original transcript suggested by Klev and agreed to by the author.

We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.