6.00 Thursdays at The Ertegun House
The aim of this reading group is to familiarise participants with the technical apparatus of higher-order logic, and discuss recent philosophical research which applies higher-order logic to issues in metaphysics and philosophical logic. Some of the issues we will discuss include: the ‘primitivist’ interpretation of higher-order logic, absolute generality, the individuation of propositions, properties and relations, Frege’s Puzzle, truth, and modality. No familiarity with the issues will be presupposed: the articles to be discussed in the first two weeks are accessible introductions to the core technical material, and later articles accessibly introduce further resources when required.
[Week 1] Background: Gallin’s Type Theory & The Interpretation of Higher-Order Logic
Timothy Williamson – Modal Logic as Metaphysics, Chapter 5: From First-Order to Higher-Order Modal Logic
Abstract: First-order logic permits quantification into name position. Second-order logic permits quantification into predicate or sentence position too. Higher-order logic takes the generalization even further. The growth of higher-order modal logic is traced, starting with Lewis and Langford’s quantification into sentence position in propositional modal logic, and on to the higher-order modal logics of Barcan Marcus, Carnap, Montague, Gallin, and others. Higher-order modal logic is proposed as a suitably general setting in which to assess fundamental issues in modal metaphysics. However, there are difficulties in interpreting higher-order quantification, since it lacks adequate paraphrases in natural language. Although Boolos’s paraphrase of quantification into monadic predicate position in terms of plural quantification works well in non- modal settings, for many purposes it is unsuitable in modal settings since plurals are modally rigid. Nevertheless, we can hope to reach a suitable understanding of irreducibly higher-order quantification by the direct method, without paraphrase.
[Week 2] Type Theory & Absolute Generality
Agustin Rayo – Beyond Plurals (in Rayo & Uzquiano Absolute Generality)
Abstract: I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higher-order quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of the same logical type. I claim that this leads to a trilemma: one must choose between giving up absolutely general quantification, settling for the view that adequate semantic theorizing about certain languages is essentially beyond our reach, and countenancing an open-ended hierarchy of languages of ever ascending logical type. I conclude by suggesting that the hierarchy may be the least unattractive of the options on the table.
[Week 3] Type Theory & Set Theory
Øystein Linnebo & Agustin Rayo – Hierarchies Ontological and Ideological (in Mind (2012) 121: 269 – 308)
Abstract: Godel claimed that Zermelo-Fraenkel set theory is ‘what becomes of the theory of types if certain superfluous restrictions are removed’. The aim of this paper is to develop a clearer understanding of Godel’s remark, and of the surrounding
philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.
[Week 4] Type Confusions
Ofra Magidor – The Last Dogma of Type Confusions (in Proceedings of the Aristotelian Society 109 (1pt1): 1-29 (2009))
Abstract: In this paper I discuss a certain kind of ‘type confusion’ which involves use of expressions of the wrong grammatical category, as in the string ‘runs eats’. It is (nearly) universally accepted that such strings are meaningless. My purpose in this paper is to question this widespread assumption (or as I call it, ‘the last dogma’). I discuss a range of putative reasons for accepting the last dogma: in §II, semantic and metaphysical reasons; in §III, logical reasons; and in §IV, syntactic reasons. I argue that none of these reasons is conclusive, and that consequently we should be willing to question this last dogma of type confusions.
[Week 5] Frege’s Puzzle
Andrew Bacon & Jeffrey Sanford Russell – The Logic of Opacity (manuscript)
Abstract: We explore the view that Frege’s puzzle is a source of straightforward counterexamples to Leibniz’s law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show that these logics are not degenerate.
[Week 6] How Fine-Grained is Reality?
Jeremy Goodman – Reality is Not Structured (forthcoming in Analysis)
Abstract: The identity predicate can be defined using second-order quantification: a 1⁄4 b 1⁄4 df 8FðFa $ FbÞ. Less familiarly, a dyadic sentential operator analogous to the identity predicate can be defined using third-order quantification: ’ 1⁄4 df 8XðX’ $ X Þ, where X is a variable of the same syntactic type as a monadic sentential operator. With this notion in view, it is natural to ask after general principles governing its application. More grandiosely, how fine-grained is reality?
I will argue that reality is not structured in anything like the way that the sentences we use to talk about it are structured. I do so by formulating a higher-order analogue of Russell’s paradox of structured propositions. I then relate this argument to the Frege-Russell correspondence. When confronted with the alleged paradox, Frege agreed that reality was not structured, but maintained that propositions (i.e. thoughts) were structured all the same. Russell replied that his paradox showed Frege’s theory of structured thoughts to be inconsistent, to which Frege replied that Russell’s argument failed to heed the distinction between sense and reference. Most recent commentators have sided with Russell. In defense of Frege, I establish the consistency of one version of his rejoinder. I then consider and reject some ways of resisting the argument against a structured conception of reality. I conclude that, if propositions are structured, this is because they correspond not to distinctions in reality, but rather to ways in which those distinctions can be represented.
[Week 7] Higher-Order Logic & Modality
Andrew Bacon – The Broadest Necessity (manuscript)
Abstract: In this paper we explore the logic of broad necessity. Definitions of what it means for one modality to be broader than another are formulated, and we prove, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. We show, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. We formulate this conjecture precisely in higher-order logic, and examine concrete cases in which it fails. We end by investigating the logic of broad necessity. It is shown that consistently with higher-order logic, the logic of broad necessity can be anywhere between S4 and Ver; we give some reasons to think that it is strictly weaker than S5.
[Week 8] Propositional Quantification & Truth