# Workshop: Second-order logic and the question of (im)predicativity

**Tuesday **

9:00 – 9:15: welcome

9:15 – 10.30: Crsipin Wright

10:45 – noon: Sam Roberts

Noon – 1:30pm: lunch and celebration of Peter Fritz’ research prize (in GM452)

1:30 – 2:45pm: Laura Crosilla

3:00 – 4:15pm: Øystein Linnebo

**Wednesday**

10:00 – 11:15am: Stewart Shapiro

11:30 – 12:45pm: Chris Scambler

12:45 – 1:45pm: lunch

1:45 – 3:00pm: Peter Fritz

3:15 – 4:30pm: Agustin Rayo

L**aura Crosilla (Oslo): "Predicativity and (intuitionistic) logic"**

Abstract: Predicativity originates within fundamental debates at the beginning of the 20th Century under the stimulus of the set-theoretic paradoxes. Poincaré, Russell and, subsequently, Weyl shaped the notion of predicativity. Interest for predicativity faded soon after Weyl’s 1918’s booklet “Das Kontinnum”, but re-appeared prominently within mathematical logic, and especially proof theory, from the 1950’s. Today a form of predicativity is characteristic of constructive foundational systems for Bishop-style mathematics, such as Martin-Löf type theory. A number of questions arise from considering today’s constructive forms of predicativity, one of which is the relation between predicativity and logic. In this talk I explore a possible route from predicativity to intuitionistic logic that makes use of Dummett’s argument from indefinite extensibility. I only aim at highlighting the main steps of such an argument and especially focus on the case of the natural numbers.

**Peter Fritz (Oslo): "Ground and Grain and Predicativity"**

Abstract: Current views of metaphysical ground suggest that a true conjunction is immediately grounded in its conjuncts, and only its conjuncts. Similar principles are suggested for disjunction and universal quantification. Here, it is shown that these principles are jointly inconsistent, and so that the notion of grounding is either not in good standing, or that widespread assumptions about it need to be revised. The inconsistency is established by showing that the relevant grounding principles entail that there is a distinct truth for any plurality of truths. Thus, the argument is similar to the Russell-Myhill argument against structured propositions. However, in contrast to the Russell-Myhill argument, the inconsistency of the relevant grounding principles can -- at least in certain cases -- be established using only predicative plural comprehension.

**Øystein Linnebo (Oslo): "Non-instance-based generality and Frege’s Theorem"**

Abstract: Shapiro and Linnebo (2015) proves Frege’s Theorem in intuitionistic logic. But certain impredicative comprehension axioms are needed, which are potentially problematic from a constructivist point of view. Is the apparent problem genuine? First, I reject an argument from constructivism to the Vicious Circle Principle, which would entail that there is a problem. Then, I explore the prospects for a constructivist defense of the needed comprehension axioms, drawing on (1) an alternative conception of predicativity due to Poincaré and (2) a non-instance-based conception of generality which becomes available when the logic is intuitionistic.

**Agustin Rayo (MIT/Oslo: " A Kaplanian account of the propositional paradoxes"**

I offer a formal development of a Kaplan-style answer to the propositional paradoxes.

**Sam Roberts (Oslo): "Properties, propositions, and grounding"**

**Chris Scambler (NYU): ****"Can all things be counted?"**

Abstract: "Predicativists tend to be cagy about uncountability in mathematics; their caginess also tends to lead them to mathematically restrictive programs in foundations, including for the most part an outright rejection of standard axioms of set theory. In this talk I will present an attempt to rehabilitate standard set-theoretic mathematics in the countablist (or even predicativist) setting."

**Stewart Shapiro (Ohio State University): "Properties and Predicates; Objects and Names: Impredicativity and the Axiom of Choice"**

Abstract: Recently, Bob Hale (2010, 2013) has articulated and defended a “Fregean” theory of properties and relations. It is a natural extension of the account of objects adopted by him and Crispin Wright, as part of their abstractionist, neo-logicist philosophy of mathematics, language, and general metaphysics. The purpose of this talk is to take the measure of this perspective as an interpretation of mathematics—the mathematics that is practiced today and plays a central role in just about all scientific theories. The upshot, I think, is that Hale must either defend some prima facie implausible claims about what sorts of languages are possible, for us finite beings, or else he must reject, on purely philosophical grounds, large chunks of contemporary mathematics and, perhaps, even his own neo-logicism. Our main focus is on impredicative definitions and the axiom of choice.

**Crispin Wright (NYU and Stirling): "Is (Neo-)Logicism founded on a mistake about Logic? Wright, Hale and Heck on Higher-Order Logic”. **

The underlying logic needed for neo-logicist constructions of number theory and analysis faces familiar problems concerning the construal of its higher-order variables. Three such problems are paramount:

(i) how to conceive of the entities in the range of the higher-order quantifiers;

(ii) how to conceive of the domain comprised by such entities;

(iii) how to understand the impredicative comprehension axioms which the constructions demand.

I’ll explore answers to (i) and (ii) which draw their sting. However (iii) is another matter.