An argument is valid if, and only if the conclusion of the argument is true under all interpretations under which the premises are true. In contrast to most modern incarnations of this definition, I take truth (or rather satisfaction) to be a primitive notion. Interpretations are understood as substitution instances. In contrast to Tarski's typed approach, the notion of logical validity is applicable to the language in which it is formulated. In contrast to the usual model-theoretic definition of logical validity, my account features an intended interpretation. Moreover, for every model of the language, there is a substitutional interpretation, that is, a substitutional instance plus a specification of the values of the free variables of the substitution instances.

The model-theoretic definition of logical consequence presupposes a division of the expressions of a language into logical and non-logical. The idea that invariance of denotations captures an important aspect of the formality of logic has long been taken to supply at least a necessary criterion for the logicality of an expression. This idea, however, seems to rely on classical presuppositions, such as bivalence and a (classical) set-theoretic background theory. This is problematic since the model-theoretic definition finds application in a host of non-classical logics. Yet, in how far is its usage in these contexts justified if it crucially depends on classical assumptions?

In this talk, we investigate the nature, role and justification of invariance criteria in non-classical settings. We examine whether commonly used criteria continue to provide reliable classifications of logical expressions in these frameworks and study how they might be modified or supplemented in order to provide well-motivated and adequate demarcations of non-classical constants. A core contention of our proposal is that the type of invariance appropriate for a given logic is not independent of its characteristic (non-classical) features. We demonstrate how this idea can be spelled out for a selection of non-classical logics.

We argue that Gödel's Programme, understood as the search for new axioms possibly extending ZFC and ultimately leading to an exhaustive description of the set-theoretic universe, is destined to fail.

Initially motivated by the discovery of independence phenomena, Gödel's Programme seeks to identify axioms A such that ZFC+A resolves interesting independent questions and is maximal according to some canonical justification method. Various axiom justification approaches -- intrinsic and extrinsic justification, Maddy's naturalism, consistency strength, interpretability power -- have been proposed. However, they face at least two issues: uncertainty and mutual incompatibility.

Uncertainty arises when a justification method J ranks theories ZFC+A and ZFC+B equally, yet they describe vastly different set-theoretic universes. Mutual incompatibility occurs when two justification methods J and Q disagree, i.e. when J ranks ZFC+A higher than ZFC+B, while Q ranks ZFC+B higher than ZFC+A.

A possible solution is to apply multiple justification methods, but the order of application influences the outcome, still yielding incompatible theories. Another is to scrutinise the philosophical motivation for J and Q, justifying a particular order of application. We explore these possibilities and underpin them with examples. We argue that this issue is particularly challenging for the universist.

I aim to draw a connection between higher-order metaphysics and the philosophy of mathematics, in particular set theory. Higher-order metaphysics means carrying out metaphysical debates in higher-order logic, using higher-order quantifiers to regiment talk of propositions, properties, and relations. A prominent topic in this area is grain science, the investigation of individuation conditions of propositions, properties, and relations. These topics seem purely metaphysical. But I will argue that they are intimately connected to questions in (the philosophy of) mathematics. In particular, I will argue that views about grain science can resolve the continuum hypothesis. To do so, I will present an example of such a view. I won't argue for it, but I hope to motivate, first, that the view is attractive, or at least not implausible; second, that the view doesn't obviously prejudge controversial questions in (the philosophy of) set theory; and third, that the view nevertheless settles the continuum hypothesis. The view assumes that sets obey the principles of ZFC set theory, and that propositions form a structure which corresponds to a particular complete Boolean algebra. Adapting standard forcing results using Boolean-valued models, we can show that this higher-order metaphysical view entails the failure of the continuum hypothesis.

This talk aims to answer two main questions: Can foundational theories be explanatory? If yes, can explanatoriness be used as a criterion for the preference of specific foundational theories? This is investigated by bringing together two debates, the one about explanations in mathematics and the other about criteria for preference of foundational theories. To answer the first question, an account of explanatoriness of foundational theories is developed. Here foundational theories are the explanans for mathematics (the explanandum) by fulfilling at least one of the criteria of Explanatory Systematization (ES) or Explanatory Reduction (ER), therefore also allowing for degrees of explanatoriness. ES can be expressed by epistemic accounts of explanation such as the unificatory account proposed by Kitcher (1989). ER corresponds to ontic accounts and is expressed here by the account of Pincock (2015) where explanation derives from dependency relations between objects of different abstractness. With this, the second questions is answered: It is argued that set theory fulfills ES but not ER, whereas category theory fulfills ER but not ES. Therefore different preferences for foundational theories can be linked to different ways in which these theories explain.