Luca Incurvati: Metalogic and the Overgeneration Argument
The so-called overgeneration argument has figured prominently in the debate over the logicality of second-order logic. However, what the argument is supposed to establish is far from clear. The first part of the talk examines the argument and locates its main source, namely the alleged entanglement of second-order logic and mathematics. Various reasons why the entanglement may be thought to be problematic are then identified. In the second part of the talk, a metatheoretic perspective on the matter is adopted. A number of results are proved which establish that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that, by moving from the standard set-theoretic semantics for second-order logic to a semantics that makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic. The talk is is based on joint work with Salvatore Florio.
Hannes Leitgeb: A Hyperintensional Logic for the Causal ‘Because’?
I am going to investigate the prospects of developing a hyperintensional semantics and logic for the causal ‘because’ operator, where by ‘hyperintensional’ I mean: logically equivalent sentences cannot always be substituted for each other salva veritate in a ‘because’-context. The starting point will be some philosophical observations that may be taken to suggest a hyperintensional account of causality and/or causal explanation. After presenting a version of truth-maker semantics in a new format that builds a bridge to a class of mathematical structures well-known from combinatorics, optimization theory, and hypergraph theory, I extend that semantics to one for the causal ‘because’ operator: this will give us a hyperintensional causal semantics in which each formula is assigned a set of minimal truth-makers composed of states of nodes in a causal network. I introduce a hyperintensional system of causal logic that is sound and complete with respect to that semantics. Finally, I discuss some possible alternatives and extensions, and I evaluate some of the positive and negative features of the resulting systems.
Johannes Korbmacher: Truthmakers and Mathematics – Applications of Truthmaker Semantics in the Philosophy of Mathematics
The aim of this talk is to highlight the applications of truthmaker semantics in the philosophy of mathematics. In particular, I will sketch how truthmaker semantics can fruitfully be applied to problems concerning:
- explanatory proofs in mathematics (e.g. Lange 2009);
- mathematical explanations in science (e.g. Baker 2005); and
- mathematical objects in metaphysics (e.g. Fine 1994).
For this purpose, I will first generalize the truthmaker semantics of Fine (forthcoming) to a first-order setting. Then, I will go on to define several useful explanatory concepts in this semantics, and finally, I will apply these concepts to the problems at hand. Based on this, I will argue that truthmaker semantics provides a fruitful framework for the philosophy of mathematics in general.
- Baker, Alan. 2005. “Are There Genuine Mathematical Explanations of Physical Phenomena?” Mind 114 (454): 223-38.
- Fine, Kit. 1994. “Essence and Modality.” Philosophical Perspectives 8:1-16.
- — Forthcoming. “Angellic Content.” Journal of Philosophical Logic: 1-28.
- Lange, Marc. 2009. “Why Proofs by Mathematical Induction Are Generally Not Explanatory.” Analysis 69 (2): 203-211.
Nina Gierasimczuk: Topologicl Epistemology through the Lens of Logic. The Case of Beliefs and Learning
Learning can be seen as a process of adjusting one’s beliefs on the basis of observations. So far combining belief revision procedures with learning-theoretic notions led to many interesting observations about reliability and rationality within different formal frameworks. However, the setting of possible worlds seems particularly well-suited for the more recent controversies surrounding the logical notions of knowledge and belief. This allows a smooth transition to general topology, where the notion of reliable, limiting learning can be elegantly characterized and generalized. The particular way of viewing learnability through the lens of topological semantics of modal logic informs the abstract axiomatizations of knowledge and belief. The main purpose of this talk is to present a topological bridge between learnability in the limit and (dynamic) epistemic logic.