Øystein Linnebo (University of Oslo) Non-instantial generality: What it is and why we need it

What features of reality are responsible for the truth of a universal generalization? The orthodox answer proceeds via the instances of the generalization. Everything is F because a is F, b is F, and so on, plus (perhaps) the fact that these are all the objects. I show that the orthodoxy needs to be supplemented with (wholly or partially) non-instantial explanations. E.g., we can explain why everything crimson is red or why every object has a singleton set without invoking any instances of these generalizations. Although non-instantial generality is familiar from mathematical intuitionism, I divorce the idea from the intuitionistic philosophy and show how it can be put on a robustly realist footing (say, in terms of Finean essences). With non-instantial generality on board, all the truths of intuitionistic (but not classical) first-order logic turn out to have a trivial truthmaker.

The talk will describe the truthmaker analysis of non-instantial generality developed in my “Generality explained”. Some more recent developments will also be discussed, especially an extension of my analysis of non-instantial generality to the framework of metaphysical grounding, as well as some applications of the analysis in philosophy and the foundations of mathematics.

Matteo Plebani  (University of Turin) Variations on a theme from Linnebo

I will use Linnebo’s notion of non-instantial generality to shed some light on Wittgenstein’s tantalizing claim that “the generality required in mathematics is not an accidental generality” [T 6.031]. I will also compare the type of truthmaker semantics presented in Linnebo’s “Generality explained” with Kleene realizability semantics.

Francesca Boccuni (Vita-Salute San Raffaele University, Milan) Frege’s proof of referentiality 

In this talk, I will investigate a novel reading of Frege’s proof of referentiality (and its failure) based on so-called generic generality. First, I will explain what the latter amounts to, and, secondly, I will motivate why it is apt to capture Frege’s conception of quantification. I will then compare the reading of Frege’s proof of referentiality in terms of generic generality to its traditional readings. Finally, I will comment on a few consequences of applying generic generality to Frege’s conception of quantification.

Alice van't Hoff (University of Vienna) Against higher-order unrestrictedness

Generality absolutists argue that it is possible to quantify absolutely unrestrictedly over everything that there is. A challenge to their view arises if we take seriously the possibility of genuinely higher-order quantification. An influential proposal, first put forward by Timothy Williamson, however, suggests that to conclude on this basis against generality absolutism would be premature. Williamson's claim is that higher-order quantification in our object language is only a threat to generality absolutism if we, misleadingly, adopt a first-order metalanguage. I argue, though, that this approach is subject to a counter-example: intuitively, quantification in many-sorted languages need not be absolutely general. Yet both higher-order and many-sorted systems may involve multiple distinct quantifiers and there are, I claim, no differences of the kind relevant to a quantifier's unrestrictedness that distinguish these two versions of quantifier pluralism. This suggests, contra Williamson and others, that higher-order quantification is a threat to generality absolutism, at least insofar as we take quantification of this kind ontologically seriously.

Bruno Iacinto (University of Lisbon, CFUL) Ordinals, Cardinals and Logicism

According to abstractionist accounts of ordinal and cardinal numbers, facts involving them are grounded in facts concerning equivalence relations between entities of a more concrete kind. In this talk I argue that these accounts suffer from important problems vis-à-vis the grounding of modal facts involving ordinals and cardinals. In particular, I show that Linnebo’s reply to Donaldson’s (2023) metaphysical problem for abstractionist accounts is ultimately unsuccessful.

In addition, I propose an alternative account of both ordinals and cardinals. The proposed account is broadly neoRussellian in that it takes ordinals and cardinals to be higher-order properties characterizable in purely logical terms. Furthermore, it is not abstractionist, as ordinals and cardinals are characterized in terms of relations between equally abstract entities. I conclude by showing how these characterizations pave the way for an Upper logicist theory (in the sense of (Jacinto 2024)) of ordinals and cardinals.