## Workshop 10-12 December 2018: Themes from Alan Weir

**Issued: Thu, 05 Jul 2018 14:14:00 BST**

In a number of areas, Alan Weir has elaborated strikingly original views which involve a radical departure from the main-stream. These include formalism in the philosophy of mathematics, and as well as naïve set theory, with a universal set, and a naïve theory of truth. In contrast to other contemporary defenders of the latter two theories, Weir rejects dialetheism and accepts classical rules for the logical connectives. He avoids contradictions by restricting certain structural inference rules, specifically some generalized versions of transitivity. In addition, Weir has developed radical versions of naturalism and physicalism (partly informed by his work on Quine) and perceptual realism.

The aim of the workshop is to advance research on themes from Weir’s philosophy.

**Programme:**

Venue: Reid Room, Department of Philosophy, University of Glasgow (69 Oakfield Avenue).

Monday 10^{th} December:

1-1:45: Pre-workshop Tutorial: An Introduction to the Philosophy of Alan Weir

2-3:30: Timothy Williamson (Oxford): ‘Alternative Logics and Applied Mathematics’

[coffee]

4-5:30: Stephan Krämer (Hamburg): ‘State-space semantics for state-space mereology’

Tuesday 11th December:

[9-9:30 coffee]

9:30-11: Stewart Shapiro (Ohio State): ‘Plurals, Groups, and Paradox’

11:15-12:45 Mary Leng (York): ‘Informal Proof, Formal Proof, Formalism, and Fictionalism’

2:15-3:45: Alan Weir: ‘A Mereological Theory of Properties and Relations’

4-5:30: Alex Miller (Otago, via video-link): 'What is the Sceptical Solution?'

Wednesday 12th December:

[9-9:30 coffee]

9:30-11: Elia Zardini (Lisbon): ‘Against the World’

11:15-12:45: Marianna Antonutti Marfori (Munich): tba

[lunch]

2:15-3:45: James Levine (Trinity College Dublin): ‘On Quine's Naturalism (and Russell's)’

4-5:30: John Divers (Leeds): ‘Metaphysical Modality and Objective Probability’

Available abstracts are here

We gratefully acknowledge the support of the Analysis Trust, the Aristotelian Society, the Mind Association, the Scots Philosophical Association, and the School of Humanities, University of Glasgow.

Marianna Antonutti Marfori, `Indirect Mathematical Naturalism'

Scientific naturalism traditionally struggles to give a picture of disciplines such as mathematics and logic – and particularly of those parts of mathematics and logic that are far removed from scientific applications – that is both non-revisionist on their practice and coherent with a broadly physicalistically acceptable world view.

I will present a new kind of mathematical naturalism that identifies mathematics’ rigorous methodology as its distinctive mark. I will argue that provability in some consistent, recursively axiomatisable informal theory is at the core of this methodology, and that these aspects can be formally captured by a proof theoretic interpretation along the lines of Feferman (1960). This interpretation function can be used by the naturalist to discern mathematical theories from pseudoscientific ones (although not to discern good mathematical theories from bad ones). This kind of naturalism avoids revisionism about mathematical practice while being able to commit to as much ontology as is justified by our best science, modulo a minimal commitment to the weakest theory that can prove the arithmetised completeness theorem.

John Divers: ‘Metaphysical Modality and Objective Probability’*

In recent work** I have: (a) argued that a neo-Quinean pragmatic scepticism about metaphysical modality is a perfectly reasonable position to maintain and (b) illustrated the difficulties and limitations associated with some strategies for overcoming such scepticism. One such strategy is to forge an appropriate association between metaphysical modality and objective probability. This paper is an attempt to develop an account of the difficulties and limitations associated with this particular strategy.

* Work in Progress, co-author Shyane Siriwardena (University of Cambridge)

** Divers, J. (2018) “W(h)ither Metaphysical Necessity?” (The Presidential Address)

in *Proceedings of the Aristotelian Society Supplementary Volume*, 92.1, pp.1-25.

Stephan Krämer, ‘State-space semantics for state-space mereology’*

State-space semantics, as the term is here understood, replaces the usual appeal in semantics to a space of possible worlds by an appeal to a space of (possible, perhaps also impossible) *states*, ordered by *part-whole*. I shall be interested only in so-called *exact* state-space semantics, in which the notion of a proposition’s being verified by a state is understood to impose the requirement that the state be *wholly relevant *to the truth of the proposition. State-space semantics earns its keep – if it does – to a large extent by allowing us explicate in a satisfactory way various intuitively and theoretically significant hyperintensional properties of or relations between properties. For instance, the conceptually most central relationship of *entailment* in (exact) state-space semantics – so-called *exact entailment* – has been argued to capture an important conception of *grounding*. Now, once such relationships as exact entailment have been defined, we can formulate new propositions involving them, such as the proposition that *P* exactly entails *Q*. At least in principle, it would seem, we should be able to come up with state-space semantical treatment of these *higher-degree* propositions as well, and thereby with a theory of how these propositions enter into the relevant hyperintensional relations. Since the relevant properties of and relations between propositions are defined in terms of the *mereology* of the underlying state-space, it is natural to take the verifiers of the higher-degree propositions to be states *about* the mereology of the state-space. My talk begins to explore how this idea might be implemented with respect to exact entailment.

* Work in Progress, co-author Johannes Korbmacher (University of Utrecht)

Mary Leng: ‘Informal Proof, Formal Proof, Formalism, and Fictionalism'

In Alan Weir's paper, 'Informal Proof, Formal Proof, Formalism', he considers an argument he attributes to Yehuda Rav to the effect that informal proofs cannot be mere abbreviations of formal proofs because we can prove informally many results which are not formally derivable. This raises a problem for Weir's formalism, but also, he thinks, for fictionalist accounts of mathematics to the extent that they account for fictional correctness in terms of derivability. Weir's solution to the problem involves introducing the notion of a formal idealisation of provability, such that the truth-maker for a sentence of a system F can take the form of* either* a derivation-in-F of S, *or *a metatheoretic derivation (in a suitable metatheory F*) of the existence of a F-derivation of S. Contrary to Weir's assumption that fictionalists can identify fictionality with formal derivability, I have argued that fictionalists should rather make use of a primitive modal notion of logical consequence in their account of mathematical correctness (truth-in-a-story). I will argue that the success of Weir's 'fix' for the problem also depends on accepting that our derivations are grounded in modal facts concerning logical consequence.

Alex Miller: ‘What is the Sceptical Solution?’

Alan Weir has had an interest in issues concerning the determinacy of meaning throughout his career (see for example his “Objective Content”, *Proceedings of the Aristotelian Society* Supplementary Volume (2003) and his “Indeterminacy of Translation”, in E. Lepore and B. Smith (eds.) *The Oxford Handbook of Philosophy of Language* (OUP 2006)). He is also well-known for his antipathy to Wittgenstein. So it might create an interesting frisson to discuss the determinacy of meaning in a Wittgensteinian context.

In chapter 2 of *Wittgenstein on Rules and Private Language*, Kripke's sceptic concludes that there are no facts in virtue of which ascriptions of meaning (such as "Jones means addition by '+'") are true, and concludes that the notion of meaning "vanishes into thin air". In chapter 3, with a nod to Hume, Kripke's Wittgenstein offers a "sceptical solution" to the sceptical problem of chapter 2, a solution that many commentators have taken to be broadly analogous to non-cognitivist or expressivist theories in other domains, such as metaethics. In this talk, I'll explore the nature of the sceptical solution and consider inter alia two objections raised against it in Barry Stroud's paper "Wittgenstein on Meaning, Understanding and Community". I'll also attempt to correct some misconstruals of the sceptical solution that have been promulgated by Davidson and his followers.

Stewart Shapiro: ‘Plurals, Groups, and Paradox’*

There are two prominent views regarding definite plurals like ‘the students’. According to singularism – the predominant view within linguistic semantics – ‘the students’ refers to a set-like object, e.g. a set or sum of atoms. According to pluralism – the predominant view among philosophers and logicians – ‘the students’ instead refers to a primitive multiplicity of individuals, as familiar from plural logic. The primary argument against singularism is that because set-like entities iterate, they lead to Russell’s Paradox, thus rendering singularism incoherent. And though linguists are aware of this threat, no generally agreed upon resolution exists. The purpose of this paper is to rectify this situation, by establishing two claims. First, we argue that certain natural language examples – so-called superplurals – prove empirically problematic for extant pluralist analyses. What these examples reveal, ultimately, is the need for *groups*, or collections viewed as entities in their own right. However, since these iterate like sets, they lead to Russell’s Paradox. Secondly, we claim that the resulting version of the paradox is not devastating for singularism. We sketch a *potentialist *theory of groups according to which group-formation is merely potential, as opposed to actual or complete. This is independently plausible, we argue, given the recursive nature of iterative semantic processes more generally. If so, then the primary argument against singularism is debunked, for reasons most parties to the debate should find independently plausible.

* Co-author Eric Snyder (Smith College)

Alan Weir: ‘A Mereological Theory of Properties and Relations’

Some physicalistically inclined philosophers such as Quine and Armstrong have, whilst expressing sympathy for a mereological account of properties, pointed out an apparently fatal flaw. Whilst a property such as *being red* (if there be such) can be thought of as an aggregate whose instances are its proper parts, no such idea works for *being human*: my fist is not a human being for example.

In this talk, I expound a revisionary, weaker mereology taking the notion of *being an (immediate) constituent of* as basic, defining proper part as its transitive closure and introducing a notion of fusion distinct from the two main characterisations of sum/aggregate. I argue that this enables us to explain the distinction between properties and bodily particulars. I also explore the idea that additional explanatory power results from combining this conception with a plenitude postulate roughly to the effect that the universe consists of structurally unique (under the constituent relation) miniverses, maximally many thereof. I finish by sketching how relations can brought under the theory as themselves mereological fusions.

Elia Zardini: ‘Against the World’

In previous works, I've developed a theory of transparent truth (LW) and a theory of tolerant baldness (N_{LS}) which validate the law of excluded middle (LEM) and the law of non-contradiction (LNC), and which solve the semantic paradoxes and the paradoxes of vagueness by restricting instead the structural properties of contraction and transitivity respectively. Moreover, the principle of distributivity of conjunction over disjunction (D) fails in the systems – in fact, even the weaker principles of modularity (M) and orthomodularity (O) fail. However, since neither kind of paradox seems to involve D, M or O in the first place, it might seem that the solutions I've proposed feature logics that are unnecessarily weak. I'll first argue that these appearances are deceiving: if a non-contractive or non-transitive theory of anything making certain natural assumptions (which crucially include LEM and LNC and which are shared by both LW and N_{LS}) is to work at all, D, M and O just have to fail. I'll then offer a philosophical explanation of the failures of these principles in LW and N_{LS}, which will require to bring out a common, hitherto unnoticed metaphysical consequence of these systems, and which will thus have the upshot of bringing for the first time together two systems that might up to now have seemed very remote from one another. More in detail, I'll show that both LW and N_{LS }assert the non-existence of the world, and that such assertion in turn implies the relevant instances of LEM; I'll then argue that these circumstances explain the failure of D, M and O, since these principles allow one to go from the disjunctions of incomplete ways things are licenced by LEM to a disjunction of complete ways things are, with the latter contradicting the non-existence of the world.