Is it possible to quantify over absolutely everything? According to the extendability argument, it is not. However many objects we quantify over, it is possible to use these objects to define another object that must, on pain of paradox, lie beyond the objects over which we quantified. The topic of this mini-workshop is whether the modal operators too are subject to extendability arguments. The idea would be that however large a space of possibilities is used to interpret our modal operators, it is possible to use this space to define yet further possibilities that must lie beyond that space.
Several philosophers have recently argued that modality too is subject to a form of indefinite extendability (Justin Clarke-Doane, Kit Fine, Michael Glanzberg, Agustin Rayo, Alex Roberts, James Studd). Other philosophers reject all forms of extendability arguments (T. Williamson). An intermediate view holds that, although every extensional domain (a set or a plurality) can be extended, there are absolutely inextendable intensional domains (Florio, Linnebo).
Program:
12:30 - 13:45: Peter Fritz, Gaifman-Hales and Indefinite Extensibility
I consider a view of propositions on which they form a free Boolean algebra. This can be motivated using ideas from logical atomism, along the lines of Wittgenstein, according to which every proposition is a truth-functional combination of elementary propositions. To accommodate quantifiers, it is natural to consider only complete Boolean algebras. However, in this setting there are no infinite free algebras, due to a result of Gaifman and Hales. One way of understanding this result is as establishing a kind of indefinite extensibility of propositions, which naturally leads further to an indefinite extensibility of modalities.
14:00 - 15:15: Øystein Linnebo, “Absolute modality: a progress report”
Abstract: I first explore some parallels between modality and the debate about absolutely general quantification, paying special attention to the question of indefinite extensibility. Then I attempt to extend my conception of “absolute but indefinite generality” from the case of quantification to that of modality.
15:30 - 16:45: Agustin Rayo, A Pragmatic Account of the Liar Paradox
Abstract: Accounts of the Liar Paradox are often aimed at "taming" the natural language truth predicate, by isolating a consistent core and going on to regiment and systematize that core using a formal theory. I take a different approach. I hope to show that an untamed truth predicate can be used successfully in conversation, provided that one is careful about the sorts of contexts in which one deploys it. Using an untamed truth predicate in the wrong contexts leads to trouble, but the trouble in question is does not call for a revisionary semantics for "true". Instead, Liar phenomena are best thought of instances of a broader phenomenon of communicative failures in which the conversation's presuppositions are misaligned.