Modal

In this short talk I (P. Pablo Rivas Robledo) will try to provide a concise and self-contained introduction to Non-deterministic semantics for modal logic, which are sometimes dubbed as ‘semantics for modal logic without possible worlds’ or ’truth-tables for modal logic’. We will start from the weakest (non-normal) modal logic and recover significant fragments of the modal cube. To do so, we will use standard tools of the literature, namely Coniglio’s snapshots and Gräz’s algorithm to recover necessitation. I will finish by mentioning some recent applications of the semantic framework to non-classical modal logic and intuitionistic logic. Please bring something to write and where to do it, so that you can see the magic for yourselves.

Note the reschedule!

For this session we will look at Modality and Structuralism, section 15 of: Charles Parsons, Mathematical Thought and Its Objects, 1st ed. (Cambridge University Press, 2007). As the title suggests, Parson explores an acocunt of structuralism trough modal notions; the role of necessity and possibility in mathematical discourse, and the ontology of mathematical objects.

Does mathematics require an ontological commitment to abstract objects or can modal formulations capture mathematical truth without such commitments? Can modal logic provide a foundation for mathematical necessity? How does this modal structuralism compare to eliminative forms of structuralism?

For our final meeting on structuralism, we cover the modal set-theoretic structuralism from Parsons and Linnebø. This rounds out the book we have been following and also our time with structuralism (for now).

For our next meeting on structuralism, we will be covering the two views championed by the authors of the textboock: Shapiro’s ante rem, or Sui Generis, structuralism and Hellman’s Modal Structuralism.

Main Readings

Chapter 5 and 6 in Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism (Elements in the Philosophy of Mathematics). Cambridge: Cambridge University Press. doi:10.1017/9781108582933

For this special Christmas meeting, Rodrigo will be introducing Kurt Gödel’s argument about the existence of God, a classic medieval argument formalized in modal logic.

No reading preparation is required.

During the session, Rodrigo will present the argument, followed by a discussion.

Slides used during the presentation are available here. See also a handout with the proof.