This two-day workshop will bring together a range of speakers who are working on the application of higher-order formal languages to various issues in metaphysics, broadly construed. In recent years, there has been a surge of interest in this research program, which has involved the application of such resources to the topics of modality, fineness of grain, propositional attitudes, quantification, essence, and many others. The aim of this workshop is to provide a venue for researchers working in this general area to engage with one another’s work.

Any updates to the schedule will be announced on this webpage.

Registration is now closed.

If you have any questions, please email Alex, Andreas, and Nick at: [email protected].

Conference Schedule

Monday 3 July

9-9.20: Welcome/registration

9.20-9.30: Introductory remarks

9.30-10.50: Øystein Linnebo, “Lifting The Veil Of Type Distinctions: A Progress Report

Chair: Andreas Ditter

10.50-11.10: Break

11.10-12.30: Maegan Fairchild, “Maximalist Design”

Chair: Harvey Lederman

12.30-1.30: Lunch

1.30-2.50: Juhani Yli-Vakkuri, “Higher-Order Anti-Metaphysics”

Chair: Annina Loets

2.50-3.10: Break

3.10-4.30: Lavinia Picollo, “A Neutralist Semantics For Quantification”

Chair: Chris Scambler

4.30-4.50: Break

4.50-6.10: Cian Dorr, "Higher-Order Quantification and Natural Language Property-Talk"

Chair: Beau Mount

6.20-7.30pm: Drinks Reception

7.30pm: Conference Dinner

Tuesday 4 July

9.30-10.50: Andrew Bacon and Peter Fritz, “The Logic of Logical Atomism”

Chair: Nicholas K Jones

10.50-11.10: Break

11.10-12.30: Jeremy Goodman, “Frege Meets Calder: Exploring A Non-Linear Higher-Order Language”

Chair: Bruno Whittle

12.30-1.30: Lunch

1.30-2.50: Tim Button, “Higher-Order Logic And Mathematical Precision”

Chair: Michael Caie

2.50-3.10: Break

3.10-4.30: Jon Litland, “Generating Propositions, Properties, and Relations”

Chair: Lukas Skiba

4.30-4.50: Break

4.50-6.10: Timothy Williamson, “Heuristics and Hyperintensional Metaphysics”

Chair: Alexander Roberts

Conference Venue

Garden Quad Auditorium @ St John's College, St Giles', Oxford

Oxford map showing St John's Colege: https://goo.gl/maps/diQ9if443ZCHU3Dz5

College map: https://www.sjc.ox.ac.uk/documents/College_Plan_2019_compressed.pdf

Abstracts

Bacon & Fritz

Title: The Logic of Logical Atomism

Abstract: At the core of logical atomism, along the lines of Wittgenstein and Russell, is the following idea about propositions:

(LA) Every proposition is a truth-functional combination of elementary propositions.

We consider and compare a number of ways of regimenting this idea in two different formal settings. The first setting, following much of the literature, takes propositions to form a Boolean algebra. In this setting we consider in particular the effect of requiring these algebras to be complete, and correspondingly taking LA to incorporate infinite truth-functional operations; such operations are natural to appeal to in order to accommodate quantified propositions. The second, following the workshop theme, uses higher-order logic to regiment talk of propositions. Corresponding to the assumption of taking propositions to form a Boolean algebra, we here assume the theory of Classicism, which requires that sentences which are provably equivalent in a classical higher-order logic express the same proposition.

Button

Title: Higher-Order Logic and Mathematical Precision

Abstract: Higher-order logic arguably fulfils a semantic need. On pain of familiar paradoxes (Russell & Cantor), it seems that some expressions cannot have (first-order) objects as their semantic values; but if we embrace higher-order logic, then higher-order entities can act as semantic values (see Williamson 2003).

In this talk, I will explain how higher-order logic fulfils a second semantic need. On pain of familiar limitative results (Compactness & Incompleteness), a (first-order) model-theoretic understanding of our mathematical concepts would leave us unable to account for their precision; but if we embrace higher-order logic, then we can explain their precision via internal categoricity.

I will explore the similarities between these two arguments from Semantic Need. In particular: neither argument supplies us with a completely sufficient reason to embrace higher-order realism, but the higher-order realist should be happy to draw upon them.

Dorr

Title: Higher-Order Quantification and Natural Language Property-Talk

Abstract: I will defend the view that the usage of words like ‘property’ in natural languages is correctly understood in higher-order terms. For example, the English sentence ’Socrates has some property’ literally means (on its most salient interpretation) exactly the same as the higher-order sentence ∃X.X(Socrates). This view implies that many natural-language expressions are highly type-ambiguous: I respond to an influential objection to such type-ambiguity by giving an account of sentences like ‘Athens and the property of being Athenian were both mentioned by Socrates’ in which a single word-occurrence seems to need to have several differently-typed interpretations simultaneously. The account is most naturally stated using the apparatus of sum-types, which I will introduce, while also explaining how they can be treated as a harmless shorthand.

Fairchild

Title: Maximalist Design

Abstract: In a slogan, Maximalism says that “anything that can exist, does exist.” Thus glossed, maximalism is the ultimate form of ontological permissivism — potentially going well beyond the bizarre fusions of mereological universalism and the modally gerrymandered coincidents of material plenitude. But, even so, maximalism is pretty attractive! Not only does it promise to secure all of the usual advantages of permissive positions, it also preserves important analogies between first- and higher-order metaphysics. Unfortunately, we don’t have particularly great resources for getting clear about what it means to say that “ontology is maximal”. This talk explores some guiding principles for maximalist design: how they help, how they get us into trouble, and where we should focus our attention instead.

Goodman

Title: Frege meets Calder: exploring a non-linear higher-order language

Abstract: What would it be like to have a formal language in which relation symbols' argument places had no intrinsic order? For example, what if we thought of "≤" and "≥"as literally the same symbol but in different orientations? The challenge for any such notation is how to talk about properties of relations, like being well-founded or being functional, which can hold in one direction but not in the other. The solution involves giving symbols internal structure. Symbols of different logical types will have different shapes, and a pair of symbols can be fitted together in more than one configuration, like LEGO bricks, yielding formulas reminiscent of Calderian mobiles.

Giving symbols this internal structure has an unexpected upshot. By liberalizing the allowable ways of attaching symbols to each other, a completely variable-free notation becomes possible. For example, the formulas corresponding to "everyone loves someone" and "someone is loved by everyone" are both comprised of the same three symbols (for love, universal generality, and existential generality), but the symbols are attached to each other in different configurations. This resulting language is inter-translatable with an expressively adequate fragment of higher-order logic, and in such a way that two sentences of ordinary higher-order logic have the same translation if and only if they are beta-eta equivalent. I'll close by explaining how this language challenges familiar ways of formulating the idea of compositionality in formal semantics.

Linnebo

Title: Lifting the veil of type distinctions: a progress report

Abstract: The use of typed languages renders inexpressible various generalizations that we might want to make. For example, we would like to ask whether all concepts, regardless of type, are individuated by necessary coextensionality or more finely. This talk explores the use of “all-purpose variables”, capable of having any of the forms of semantic value that the typed variables can have. The use of such variables restores full expressivity and is provably consistent. The bulk of the talk explores the harder question of whether this added expressivity is theoretically stable and fruitful. Drawing on a mixture of philosophical and logical considerations, an affirmative answer is tentatively defended.

Litland

Title: Generating Propositions, Properties, and Relations

Abstract: Many grounding theorists tacitly assume structured propositions holding that the proposition Pa is identical to the proposition Qb iff P=Q and a=b. However, as shown by the Russell-Myhill reasoning, this has counter-cantorian consequences. Just jettisoning structured propositions will not suffice: as shown by Fritz standard principles of immediate ground allow us to construct an injection from the properties of propositions to the propositions restoring the counter-cantorian consequence. I propose to accept these counter-cantorian consequences and will develop an iterative conception of structured propositions, properties, and relations to make sense of this.

Picollo

Title: A neutralist semantics for quantification

Abstract: The model-theoretic turn—propelled by Tarski’s work on truth and logical consequence in the middle of the 20th century—and the spread of Quinean views on quantification cast serious doubt on the legitimacy of higher-order quantification as a logical operation, once conceived as being on a par with its first-order counterpart. In “On Quantifying into Predicate Position” Crispin Wright attempts to restore the logical status of higher-order quantification by putting forward an alternative conception of quantification under which first- and higher-order operations receive a uniform treatment. Given the disparate truth-conditional semantics these operations are typically assigned, Wright opts for an inferentialist approach. He, nonetheless, provides alternative uniform truth-conditions, but he notes their inadequacy and, thus, confines them to a merely heuristic role. In this talk I put forward a novel uniform semantic treatment of first- and higher-order quantification that bypasses all the issues raised by Wright and which can be seen, even from an inferentialist perspective, as faithful to the real meaning of the quantifiers.

Williamson

Title: Heuristics and Hyperintensional Metaphysics

Abstract: I will argue that apparent cases of hyperintensionality in metaphysics are artefacts of natural but not fully reliable heuristics, in particular for assessing ‘because’ statements. This supports an intensional approach to higher-order metaphysics.

Yli-Vakkuri

Title: Higher-order anti-metaphysics

Abstract: The first time philosophers adopted the tools of logic -- or what is now known as "higher-order logic" -- it was through the influence of Frege and Russell, and especially of Whitehead and Russell's Principia Mathematica. The result was analytic philosophy, a tradition that has evolved in various ways well known to us, but that originally defined itself in part by its opposition to "metaphysics" and "traditional philosophy" (to use terms that members of the tradition themselves used to describe what they opposed). The idea was that these should be replaced by "scientific philosophy", which has one characteristic method -- namely, the method of formalization (or the "logistic method", as it was once called) -- and one characteristic program -- namely, the unity of science. I was thought to be inevitable that when that program was carried out with that method, problems of metaphysics and traditional philosophy would by and large be either solved (found to be either trivial theorems or trivially refutable in an adequate system of logic) or dissolved (found to be meaningless, i.e., not translatable into well-typed sentences). Recently, analytic philosophers have rediscovered logic under the guise of "higher-order metaphysics" -- a surprising development in view of the history. In this talk, I ask whether we have learned anything that should incline us to think that the higher-order anti-metaphysics view that was once the dominant view in analytic philosophy was wrong, and I argue that we have not.