# New College 4 June 2005

The meeting will be held in the conduit room of New College.

Guests are welcome.

Maps of Oxford and further travel information can be found here.

### Schedule

 10.00-11.30 Hannes Leitgeb: A Type-Free Theory of Truth, Modality, and Satisfaction 11.30-13.00 Tim Williamson: Congruential and non-hyperintensional contexts, indicative and subjunctive conditionals lunch 14.00-15.30 Leon Horsten: Infinite Adaptive Proofs 15.30-17.00 Philip Welch: Field's theory of truth 17.00-18.00 Volker Halbach: Axiomatising Kripke's theory of truth

### Tim Williamson: Congruential and non-hyperintensional contexts, indicative and subjunctive conditionals

The talk will consider different weakenings of the notion of extensionality for sentence operators. It will be shown that the conjunction of two apparently very similar such weakenings implies extensionality in suitably rich languages (specifically, those with an "actually" operator). These notions will be used to contrast subjective and objective probability, epistemic and metaphysical modalities, and indicative and subjunctive conditionals. It is hoped that some light will be cast on the question of whether indicative conditionals in natural language are truth-functional.

### Leon Horsten: Infinite Adaptive Proofs (joint work with Philip Welch)

Adaptive logics have been proposed as systems for reasoning sensibly from inconsistent premise sets. When an inconsistent set of premises is given, the rules of adaptive logic allow one to derive sound information concerning the class of those models that are no more inconsistent than is required by the premises.

The distinguishing feature of adaptive logic is that it involves a revision rule. In general, consequences that are drawn from a premise set are provisional: it occasionally happens that the system forces the reasoner to withdraw statements that were derived earlier.By thus inferring and occasionally revising according to the rules of adaptive logic, we gradually zoom in on the structure of the minimally inconsistent models, i.e., the models that contain no more truth value gluts than are necessary to make a given inconsistent set of premises true.

We shall investigate complexity aspects of adaptive consequence relations. As our point of departure, we take a very basic inconsistency-adaptive logic, developed by Diderik Batens. Our investigation leads us to investigate derivations of transfinite length in this setting. But such a task is not completely straightforward. Although Batens admits of infinite proofs, they all have order type $\omega$, i.e., he leaves no room for transfinite proofs. But transfinite proofs are necessary to formulate a natural concept of ordinal stage at which a sentence becomes \emph{conclusively} or unrevisably established. So we will, in our proposed alternative characterization of adaptive arguments, allow proofs of transfinite length. For the purposes of this paper, we do not need to go far into the transfinite. It will turn out that all matters that are of interest to us here are settled by proofs of length $\omega + 1$.

In order to keep these questions manageable, we will work in a simplified setting. For this reason, we concentrate on the \emph{propositional} fragment of Batens' systems of adaptive logic. We will be interested in Batens' two main systems of adaptive logic, called \textbf{ACLuN1} and \textbf{ACLuN2} by him, or sometimes also the \emph{reliability reasoning strategy} and the \emph{minimal abnormality reasoning strategy}. In the first part of the paper we will be mostly concerned with the reliability strategy. At the end of the paper, some observations will be made about the minimal abnormality strategy.

We will be mainly interested in the complexity of the collection of \emph{final consequences} of a theory. Usually, propositional proof-theoretic consequence relations in logic are recursive, or at worst recursively enumerable. But it will turn out that the final consequence operation is more complicated than that. We argue that this has philosophical implications. In our view, our results cast some doubt on Batens' philosophical thesis that adaptive consequence closely reflect how people \emph{actually} reason on the basis of inconsistent theories.

### Philip Welch: Field's theory of truth

Field offers a theory of truth involving the use of an implication operator --> . (Field: "A Revenge-Immune Solution to the Semantic Paradoxes" JPL April 2003) He consistently keeps the substitutivity of Tr(<A>) everywhere with A and hence has the unrestricted truth schema "Tr(<A>) <--> A" (using the derived operator biconditional).

The price, or prices, to be paid are (i) For the operator -->, whilst fulfilling many desiderata of implication, several basic rules must be declared invalid ( --> Contraction, --> Importation for example); (ii) the complexity of the predicates for the extension of T is high: the minimal T-predicate for the "ultimately true" sentences has the same complexity as H.Herzberger's set of "stably true" sentences (indeed over a model such as that of arithmetic the two sets are recursively isomorphic, and in this context both are far beyond the reach of our best efforts of proof-theoretic analysis of, say, second order arithmetic.)

Field has further advanced the claim that this model provides some form of escape from the usual semantic paradoxes, and their revengeful strengthenings. It seems unclear (to this reader at least) how this claim can be substantiated, but partly this escape seems to be engineered by arguing that sufficiently complex "determinately true" predicates (viewed as strengthening the usual "truth"-predicate), are no longer obviously subject to the law of excluded middle.

### Volker Halbach: Axiomatising Kripke's theory of truth (joint work with Leon Horsten)

There have been several attempts to axiomatise Kripke's theory of truth. The best known attempt is Feferman's theory KF (Solomon Feferman: Reflecting on incompleteness, The Journal of Symbolic Logic, vol. 56 (1991), pp. 1–49), which is formulated in classical logic. It has been argued by Reinhardt and others that Feferman's approach is not thoroughly satisfactory, because KF tries to capture a partial notion of truth in classical logic. Reinhardt issued the challenge to provide a natural axiomatisation of Kripke's theory that does not make use of classical logic. We shall take up the challenge by providing a formal system in Strong Kleene logic. This system will be shown to be proof-theoretically much weaker than Feferman's axiomatisation in classical logic.

last change: 13 May, 2006

e-mail address (please replace "0" by the usual "@" symbol): volker.halbach at philosophy.oxford.ac.uk