## Workshop: The Logic of Totality

15–16 June 2017, University of Glasgow

Thursday 15th

[9-9.30: coffee]

9.30-11: **Tim Button (Cambridge) – – Internalism about Set Theory**

11.15-12.45: **Stephan Leuenberger (Glasgow) – The Logic of Totality**

[lunch]

2.15-3.45: **Robert Trueman (York) – Substitution in a Sense**

4-5.30: **Ian Rumfitt (Oxford) – Truth**

[7: dinner]

Friday 16th

[9-9.30: coffee]

9.30-11: **Peter Fritz (Oslo) – The Consistency of Structured Propositions**

11.15-12.45: **Michaela McSweeney (Boston) – Realism and Closure**

[lunch]

2.15-3.45: **Bruno Whittle (Glasgow) – Exceptional Logic**

4-5.30: **Gabriel Uzquiano (USC/St Andrews) – From Cantorian Propositions to Cardinal Inequalities**

[7: dinner]

All talks were in the Reid Room, of 69 Oakfield Avenue.

**Tim Button (Cambridge) – Internalism about Set Theory**

**Abstract:** Working deductively in second-order logic, we can write down some very weak principles which govern how set-membership should behave. These principles are *so* weak, that the claim "this relation behaves like set-membership" fails to entail "there is more than one set". Remarkably, though, we can prove the following *internal categoricity theorem*: "all relations which behave like set-membership are totally isomorphic". I will outline the result, and show how it serves many (though not all) of the roles of a totality fact concerning set-hood. Moreover, because the result is a deductive theorem of pure second-order logic, it can be stated without any semantic/metalinguistic ascent. However, in a tangled way, spelling out *why* it looks like a totality fact starts to force us down a path of semantic ascent. In the end, the sense that there is "nothing more to say" must be shown rather than said.

**Peter Fritz (Oslo) – The Consistency of Structured Propositions**

**Abstract:** Some have claimed that an argument due to Russell and Myhill shows theories of structured propositions to be inconsistent. I argue that no progress can be made on the matter without narrowing down the notion of proposition at issue, building on the common observation that the various theoretical roles of propositions need not be jointly satisfiable. I illustrate this using the role of compositional semantic values of sentences and the role of states of affairs, and by showing how a kind of structured propositions can be constructed within a metaphysics of unstructured propositions and relations.

**Stephan Leuenberger (Glasgow) – The Logic of Totality**

**Abstract:** This talk will examine totality operators, expressed in English by locutions such as "p, and that's it" or "it is the whole truth that p". I argue that they can be analysed conjunctively, as "p and B¬p", where B is a box-like operator. This allows us to apply familiar concepts from modal logic to the study of totality operators. I will pay particular attention to the interaction of different totality operators, generated using different accessibility relations for the box-like operator.

**Michaela McSweeney (Boston) – Realism and Closure**

**Abstract: **I will explore what (scientific and metaphysical) realists should say about whether (scientific and metaphysical) theories should be thought of as closed under logical consequence. I will argue that characterising theories as closed under logical consequence is inconsistent with the spirit, if not the letter, of realism.

**Ian Rumfitt (Oxford) – Truth**

**Abstract: **P.F. Strawson explained truth, as it applies to statements, by saying: 'one who makes a statement or assertion makes a true statement if and only if things are as, in making the statement, he states them to be'. This explanation differs from others in taking a statement's having a content (its saying that things are thus-and-so) to be a presupposition of an attribution of truth to that statement. I show how this feature opens the way to a distinctive solution to the Liar Paradox and to an underpinning of the axiomatic theories of truth now favoured by many logicians.

**Robert Trueman (York) – Substitution in a Sense**

**Abstract:** Can we refer to properties with singular terms? Here is one argument that we can’t: properties are referred to by predicates, like '___ is a horse', and singular terms cannot co-refer with predicates, because singular terms and predicates are not grammatically intersubstitutable. This argument relies on the Reference Principle: If two expressions co-refer, then they are everywhere intersubstitutable *salva congruitate*. Unfortunately, there are a number of counter-examples to this principle: 'I' and 'me', for example, co-refer (in any given context), and yet are not grammatically intersubstitutable. In this talk I will patch up the Reference Principle, and thus hopefully save the above argument: we cannot refer to properties with singular terms.

**Gabriel Uzquiano (USC/St Andrews) – From Cantorian Propositions to Cardinal Inequalities**

**Abstract:** There are at least two different generalizations of Cantor’s theorem designed to support the claim that a given class has fewer members than subclasses. But unlike their set-theoretic counterparts, they are not equivalent to each other. Instead, one is weaker than the other and it is generally cited in support of substantive theses in logic and metaphysics, e.g., the thesis that there are more propositions than possible worlds or the thesis that a satisfactory account of logical consequence cannot identify interpretations with objects. The purpose of this talk is to assess the concern that there is a significant gap between the truth of the weaker Cantorian proposition and the cardinal inequalities it is supposed to establish.

**Bruno Whittle (Glasgow) – Exceptional Logic**

Abstract: The aim of the paper is to argue that all—or *almost* all—logical rules have exceptions. In particular, it is argued that this is a moral that we should draw from the semantic paradoxes. The idea that we should respond to the paradoxes by revising logic in some way is familiar. But previous proposals advocate the replacement of classical logic with some alternative logic. That is, some alternative system of rules, where it is taken for granted that these hold without exception. The present proposal is quite different. According to this, there is no such alternative logic. Rather, classical logic retains the status of the 'one true logic', but this status must be reconceived so as to be compatible with (almost) all of its rules admitting of exceptions. This would seem to have significant repercussions for a range of widely held views about logic: e.g. that it is a priori, or that it is necessary. The arguments of the paper would seem to overturn such views.