** Title **: "Degrees of Deducibility" (slides)

** Abstract**: Not all inferences are deductively valid, but some seem to be closer to validity than others are. It is not so much a commonplace as an unstudied presumption that the appropriate way to grade the validity of an inference from a premise or assumption \(a\) to a conclusion \(c\) is by means of a logical (epistemic, judgemental) probability measure: the probability \(\mathfrak{p}(c \, |\, a)\) takes the value \(1\) when the inference is classically valid, and generally a lower non-negative value when it is invalid. It will be argued here that this presumption is contentious, since there exist several other functions, defined in terms of a probability measure \(\mathfrak{p}\), that provide tenable necessary conditions for deducibility. Indeed, there are essentially eight distinguishable pairs of truth functions \(\langle Z,X \rangle\) of \(c\) and \(a\) such that \(Z\) is deducible from \(X\) if and only if \(c\) is deducible from \(a\), and accordingly eight distinct functions \(\mathfrak{f}\) such that \(\mathfrak{f}(c \, | \, a) = \mathfrak{p}(Z \, |\, X)\) takes the value \(1\) if \(c\) is deducible from \(a\) (and, when \(\mathfrak{p}\) is a regular measure, if and only if \(c\) is deducible from \(a\)). The pair \(\langle c, a\rangle\) is obviously one such pair of truth functions, and \(\langle a′,c′ \rangle\) (where the prime represents negation) is another. Ordered by numerical dominance, these eight functions constitute a Boolean lattice whose unit and zero elements, for different reasons, fail to provide genuine generalizations of the relation of deducibility. Some of the interrelations and the differences among the other six functions will be explored.

** Title **: "Formal and informal equivalences in real analysis"

** Abstract**:
In 'Stetigkeit und irrationale Zahlen', Dedekind (1872) demonstrates the equivalence of key theorems of analysis and the principle of continuity, a basic principle about the real numbers. I will study the extent to which Dedekind’s observations are vindicated by later developments in mathematical logic, in particular the programme of reverse mathematics, and consider their ramifications for the foundations of analysis.

** Title **: "Entanglement and Formalism Freeness: Templates from Logic and Set Theory"

** Abstract**: In this talk I consider the entanglement of semantically given mathematical objects with their possible formalizations, and on the other hand their robustness with respect to these, or as I call it, their formalism freeness.

** Title **: The Liar and the Sorites: towards a uniform nonstandard treatment

** Abstract **: A long-standing ambition of philosophers (e.g. McGee 1990, Tappenden 1993, Field 2008, Priest 2010) is to formulate an account of truth and vagueness which provides a unified resolution to the Liar and Sorites paradoxes. Such accounts often depart from the observation that the conventions governing the application of the predicates "true" and (e.g.) "bald" in natural language leave it undetermined whether they apply to the Liar sentence or terms denoting borderline cases. I will argue in this talk that some observations about models of arithmetic — respectively concerning the definability of proper cuts and partial truth predicates — provide a theoretically unified account of this phenomena which does not require the adoption of non-classical apparatus.

** Title **: Exact classes and smooth approximation

** Abstract **: I will introduce the notion of an exact class, a special kind of asymptotic class whose historical origins lie with the 1992 result of Chatzidakis, van den Dries and Macintyre on definable sets in finite fields. I will then introduce the notion of smooth approximation, a definition stemming from the work of Lachlan in the 1980s on \(\aleph_0\)-categorical structures. I will then state and sketch a proof of a new result linking the two notions. Joint work with Sylvy Anscombe (UCLan), Dugald Macpherson (Leeds) and Charles Steinhorn (Vassar).

** Title **: Revisiting some conservation results on Weak König's Lemma

** Abstract **: The Weak König Lemma (WKL) states that every infinite binary tree contains an infinite path. This is known to fail in the recursive (i.e., computable) world: there exists a recursive infinite binary tree which contains no recursive infinite path. Nevertheless, as revealed by various conservation results, the non-recursive content of WKL is rather low. In the talk, I will present new proofs of these conservation results using the Arithmetized Completeness Theorem. This research is joint with Ali Enayat (Gothenburg, Sweden).

School of Mathematics

University of Birmingham

http://web.mat.bham.ac.uk/R.W.Kaye/

Dr Walter Dean

Department of Philosophy

University of Warwick

http://go.warwick.ac.uk/whdean