Balthasar Grabmayr

The study of computability is traditionally taken to produce absolute truths. According to this standard view, results in computability theory do not depend on contingent cultural practices, such as denoting numbers by Arabic numerals. However, the standard view faces what I call Montague’s Problem. This problem purports to show that computability is inherently relative to notation. Montague’s Problem has led philosophers to argue that classic computability-theoretic results are relative to contingent matters, as opposed to the absolute nature of other mathematical truths. The aim of this talk is twofold. First, I will show that Montague’s Problem generalises to the case of arithmetical definability. Hence, a similar attack can be raised against the standard view that important (un-)definability results express absolute mathematical truths. Second, I will defend the standard view from Montague’s Problem. In doing so, I will introduce absolute notions of computability and definability for important classes of objects.

Sara Ayhan

An account of bilateral proof-theoretic semantics, famously proposed in [3], usually makes two assumptions. First, that the meaning of logical connectives is determined not only by some positive notion, such as assertion or proof, but also by a dual, equally primitive notion, such as denial or refutation. Second, that negation expresses this dual notion in the object language – thereby rejecting Frege’s contrary claim that denial of A is simply assertion of ¬A. This is usually taken to yield a kind of toggle negation – in the classical setting of [3] and in constructive logics capturing the ‘strong negation’ of systems like N4 [1].

I will argue that on such a negation-as-refutation-based account – first explicitly proposed in [2] – one cannot obtain that kind of toggle negation without making highly non-constructive assumptions. So the traditional bilateralist route to N4 and related logics seems blocked. The difference between bilateral representations of N4 and classical logic amounts merely to adding so-called ‘co-ordination principles’: structural rules governing the interaction between proof and refutation rules. The classicist has no problem, of course, admitting non-constructive assumptions into their reasoning. However, I will show that adding co-ordination principles without the explicit aim of recovering classical logic is not straightforwardly justifiable. The cleaner bilateralist option would be to abandon both this kind of negation (others may be available) and co-ordination principles.

References
[1] Ahmad Almukdad and David Nelson, Constructible falsity and inexact predicates, The Journal of Symbolic Logic, 49 (1984), no. 1, pp. 231–233.
[2] Edgar López-Escobar, Refutability and elementary number theory, Indagationes Mathematicae, 75 (1972), no. 4, pp. 362–374.
[3] Ian Rumfitt, ‘Yes’ and ‘No’, Mind, 109 (2000), no. 436, pp. 781–823.