Catálogo de publicaciones - libros

Responding to an invitation from Peter Schroeder-Heister, the paper reacts to some criticisms of ‘model theory’ voiced among proof theorists interested in proof-theoretic semantics. It argues that the criticisms are poorly targeted: they conflate model theory with model-theoretic semantics and with the model-theoretic definition of logical consequence, which are three largely unrelated areas of study. On defining the meanings of logical constants, and of natural language expressions in general, the paper lays out some methodological requirements that any satisfactory definitions would need to meet, for example about generalisability from one context of use to other contexts. On defining logical consequence, the paper argues that some points made recently by Schroeder-Heister and Kosta Došen are largely sound and probably uncontroversial if clearly stated, but their impact is blurred by some question-begging formulations.

Wilfrid Hodges’ opinion is that some ideas of Peter Schroeder-Heister and the author concerning logical consequence are largely sound and probably uncontroversial, but he criticizes some of their aspects. In this note Hodges’ critique of the author is found misplaced.

This paper deals with Michael Dummett’s attempts at a proof-theoretic justification of the laws of (intuitionistic) logic, pointing to several critical problems inherent in this approach. It discusses in particular the role played by “boundary rules” in Dummett’s semantics. For a revised approach based on schematic validity it is shown that the rules of intuitionistic logic can indeed be justified, but it is argued that a schematic conception of validity is problematic for Dummett’s philosophy of logic.

This paper concerns the characterization of paradoxical reasoning in terms of structures of proofs. The starting point is the observation that many paradoxes use self-reference to give a statement a double meaning and that this double meaning results in a contradiction. Continuing by constraining the concept of meaning by the inferences of a derivation “self-contradictory reasoning” is formalized as reasoning with statements that have a double meaning, or equivalently, cannot be given any meaning. The “meanings” derived this way are global for the argument as a whole. That is, they are not only constraints for each separate inference step of the argument. It is shown that the basic examples of paradoxes, the liar paradox and Russell’s paradox, are self-contradictory. Self-contradiction is not only a structure of paradoxes but is found also in proofs using self-reference. Self-contradiction is formalized in natural deduction systems for naïve set theory, and it is shown that self-contradiction is related to normalization. Non-normalizable deductions are self-contradictory.

We give an overview of completeness and incompleteness results within proof-theoretic semantics. Completeness of intuitionistic first-order logic for certain notions of validity in proof-theoretic semantics has been conjectured by Prawitz. For the kind of semantics proposed by him, this conjecture is still undecided. For certain variants of proof-theoretic semantics the completeness question is settled, including a positive result for classical logic. For intuitionistic logic there are positive as well as negative completeness results, depending on which variant of semantics is considered. Further results have been obtained for certain fragments of first-order languages.

I present three open problems the discussion and solution of which I consider relevant for the further development of proof-theoretic semantics: (1) The nature of hypotheses and the problem of the appropriate format of proofs, (2) the problem of a satisfactory notion of proof-theoretic harmony, and (3) the problem of extending methods of proof-theoretic semantics beyond logic.