First-Order Modal Logic Seminar

Epsilon Modal Logics

In this talk, I present a new class of Modal logics structurally analogous to Hilbert’s Epsilon Calculus, and based on new ‘epsilon modalities’. These are connectives indexed by formulas selecting a world-witness satisfying their index (if any) through an arbitrary choice function. The obtained ‘Epsilon Modal logics’ are conservative over a language with standard modalities, and generalise many properties of Epsilon Calculus at the propositional level. Remarkably, the two systems are proven mutually embeddable. This correspondence carries over to applications. Epsilon terms have been interpreted as indefinite descriptions in linguistics,  ‘ideal objects’ of mathematical properties in Hilbert’s Program, and used to define ‘theoretical terms’ of scientific theories explicitly over their own laws by Carnap. On intensional grounds, epsilon modalities can be read as indefinite description of states, ‘ideal worlds’ abstracting over mathematical structures, and used to explicitly define ‘theoretical contexts’ of scientific theories respectively.

The Free Choice Principle and the modal logic of forcing

In this proposal, I plan to address whether the modal logic of forcing (as first developed by Hamkins and Löwe (2008)) satisfies the Free Choice Principle (Zimmermann (2000)). The Free Choice Principle (FCP) states that if a disjunction is possible, then each of the disjuncts is also possible: ♢(φ ∨ ψ) → ♢φ ∧ ♢ψ. Such a principle is usually not included as an axiom in any modal logic (Von Wright (1968)), since it would allow one to derive ♢ψ from the single assumption ♢φ. The modal logic of forcing interprets □ and ♢ in terms of set-theoretic forcing: □φ is true in M iff φ is true in all forcing extensions of M, while ♢φ is true in M iff there exists at least one extension of M in which φ is true. In this set-theoretic context, the FCP means: if there is a forcing extension of M in which φ ∨ ψ is true, then there is a forcing extensions of M in which φ is true and another one in which ψ instead is true. A natural question is to ask whether the modal logic of forcing validates the FCP. In this proposal, I argue that, in the modal logic of forcing, the FCP actually holds, but only iff φ and ψ are independent switches, i.e. iff they are always possible in any set-theoretic model M. By contrast, the FCP doesn’t hold for buttons, i.e. statements that can be forced true one time but then remain true in all further extensions. The philosophical upshot of this discussion is that it is possible to connect set-theoretic potentialism (i.e. the view that one can expand the set-theoretic universe by adding new sets) with the validity of the FCP. In particular, if the FCP holds, then the potentialist framework will be linear (i.e. new extensions includes all the previous ones), otherwise it will be branching (there are extensions with no relation to each other).

Talks are held on Zoom on the first (occasionally the second) Monday of the month at 4:30-6:00 pm CET (and CEST when Central Europe observes daylight savings time). A session consists of a 60-minute talk plus 30 minutes of questions and discussion afterwards.