The Logic and Metaphysics Workshop will meet on December 5th from 4:15-6:15 (NY time) via Zoom for a talk by Martin Pleitz (Muenster).

Title: Reification as identity?

Abstract: Abstract objects like properties and propositions, I believe, are the result of reification, which can intuitively be characterized as the metaphysical counterpart of nominalization (as in the shift, e.g., from ‘is a horse’ to ‘the property of being a horse’; cf. Schiffer, Moltmann), and occurs paradigmatically in the well-known bridge laws for instantiation, truth, etc. (e.g., something instantiates the property of being a horse iff it is a horse). So far, I have been working on an account of reification in terms of the technical notions of encoding & decoding, as some regulars at the L+M workshop may recall. In my upcoming talk, I wish to embed reification more clearly in higher-order metaphphysics and explore an alternative idea: Can reification be construed as identification across metaphysical categories? E.g., can the object that is the property of being a horse be identified, in some sense, with Frege’s concept horse, which is a non-objectual item because ‘is a horse’ is not a singular term? In my presentation I will argue for an affirmative answer. For this, I will sketch an ultra-generalized logic of equivalence, which has as its special cases (i) the well-known logics of first-order identity and equivalence, (ii) recent logics of generalized identities (à la Rayo, Linnebo, Dorr, Fine, Correia, Skiles, …) which connect higher-order items of the same type, and (iii) the logic of my proposed cross-level equivalences which connect items of different types. In a second step, I will re-construe reification as the cross-level equivalence that holds between higher-order items and abstract objects of the appropriate sort and argue that this account of reification as identity has certain advantages.

The Logic and Metaphysics Workshop will meet on November 28th from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by William McCarthy (Columbia).

Title: Modal pluralism and higher-order logic

Abstract: Modal pluralism is the view that there are a variety of candidate interpretations of the predicate ‘could have been the case that’ which give intuitively different answers to paradigmatic metaphysical questions (‘intuitively’ because the phrase means subtly different things on the different interpretations). It is the modal analog of set-theoretic pluralism, according to which there are a variety of candidate interpretations of ‘is a member of’.  Of course, if there were a broadest kind of counterfactual possibility, then one could define every other kind as a restriction on it, as in the set-theoretic case.  It would then be privileged in the way that a broadest kind of set would be, if there were one.  Recently, several authors have purported to prove from higher-order logical principles that there is a broadest kind of possibility. In this talk we critically assess these arguments.  We argue that they rest on an assumption which any modal pluralist should reject: namely, monism about higher-order logic. The reasons to be a modal pluralist are also reasons to be a pluralist about higher-order quantification. But from the pluralist perspective on higher-order logic, the claim that there is a broadest kind of possibility is like the Continuum Hypothesis, according to the set-theoretic pluralist.  It is true on some interpretations of the relevant terminology, and false on others.  Consequently, the significance of the ‘proof’ that there is a broadest kind of possibility is deflated.  Time permitting, we will conclude with some upshots of higher-order pluralism for the methodology of metaphysics.

Note: This is joint work with Justin Clarke-Doane.

The Logic and Metaphysics Workshop will meet on November 21st from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by Marko Malink (NYU) and Anubav Vasudevan (University of Chicago).

Title: The origins of conditional logic: Theophrastus on hypothetical syllogisms

Abstract: Łukasiewicz maintained that “the first system of propositional logic was invented about half a century after Aristotle: it was the logic of the Stoics”. In this talk, we argue that the first system of propositional logic was, in fact, developed by Aristotle’s pupil Theophrastus. Theophrastus sought to establish the priority of categorical over propositional logic by reducing various modes of propositional reasoning to categorical form. To this end, he interpreted the conditional “If φ then ψ” as a categorical proposition “A holds of all B”, in which B corresponds to the antecedent φ, and A to the consequent ψ. Under this interpretation, Aristotle’s law of subalternation (A holds of all B, therefore A holds of some B) corresponds to a version of Boethius’ Thesis (If φ then ψ, therefore not: If φ then not-ψ). Jonathan Barnes has argued that this consequence renders Theophrastus’ program of reducing propositional to categorical logic inconsistent. In this paper, we show that Barnes’s objection is inconclusive. We argue that the system developed by Theophrastus is both non-trivial and consistent, and that the propositional logic generated by Theophrastus’ system is exactly the connexive variant of the first-degree fragment of intensional linear logic.

The Logic and Metaphysics Workshop will meet on November 14th from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by Christopher Izgin (Humboldt University).

Title: A new approach to Aristotle’s definitions of truth and falsehood in Metaphysics Γ.7

Abstract: At Metaphysics Γ.7, 1011^b26–7, Aristotle defines truth and falsehood as follows: to assert of what is that it is or of what is not that it is not, is true; to assert of what is that it is not or of what is not that it is, is false. In their attempts to interpret the definitions, scholars usually distinguish between the veridical, 1-place, and 2-place uses of ‘to be’. The dominant view holds that all occurrences of ‘is’ in the definientia are interpreted veridically (Kahn 1966, Kirwan 1993, Crivelli 2004, Kimhi 2018, Szaif 2018). So the first truth condition is interpreted as follows: to assert of what is the case that it is the case, is true. I argue against this and side with those who favor a comprehensive—i.e. a jointly 1- and 2-place—interpretation (Matthen 1983, Wheeler 2011), according to which the first truth condition says: to assert of what is (F, exists) that it is (F, exists), is true. It is an open question how this interpretation makes Aristotle’s definitions sufficiently general so as to accommodate all propositional truth-value bearers. I first show that all Aristotelian propositions are reducible to propositions involving a 1- or 2-place ‘is’ and that formal properties, such as quantity and modality, merely modify the ‘is’, thus lending support to the comprehensive interpretation.

The Logic and Metaphysics Workshop will meet on November 7th from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by Victoria Gitman (CUNY).

Title: Set theory without the powerset axiom

Abstract: Many natural and useful set-theoretic structures fail to satisfy the Powerset axiom. For example, the universe of sets can be decomposed into the H_alpha-hierarchy, indexed by cardinals alpha, where each H_alpha consists of all sets whose transitive closure has size less than alpha. If alpha is a regular cardinal, then H_alpha satisfies all axioms of ZFC except, maybe, the Powerset axiom (it will only satisfy Powerset if alpha is inaccessible). Class forcing extensions of models of ZFC will often fail to satisfy ZFC, but if the class forcing is nice enough, then it will preserve all the axioms of ZFC except, maybe, the Powerset axiom. Finally, a strong second-order set theory, extending Kelley-Morse by adding a choice principle for classes (Choice Scheme), is bi-interpretable with a strong first-order set theory without the Powerset axiom. Thus working in a strong enough second-order set theory can be reinterpreted as working in a strong first-order set theory in which the Powerset axiom fails. It turns out that simply taking the axioms of ZFC and removing the Powerset axiom does not yield a robust set theory. I will discuss robust (and strong) axiomatizations of set theory without Powerset and how much of the standard set theoretic machinery is still effective even in the strongest theories in the absence of Powerset. Because of the bi-interpretability of a strong set theory without Powerset with Kelley-Morse plus Choice Scheme, these results will have consequences for which set theoretic machinery continues to work in set theories with classes. Time permitting, I will also talk about some unexpectedly strange models of set theory without Powerset.