Logic@GU

Team semantics is a semantic framework originally introduced by Hodges (1997) for the study of dependence and independence concepts, and later systematically developed by Väänänen (2007). It is also independently adopted in inquisitive logic by Ciardelli and Roelofsen (2011). In team semantics, formulas are evaluated with respect to sets of evaluation points, called teams, rather than single evaluation points as in standard semantics.

Logics based on team semantics are typically extensions of classical logic and thus inherit classical implication over classical formulas. However, the earliest versions of team-based logic, such as independence-friendly logic and dependence logic, do not include a conditional connective for arbitrary formulas. An adequate conditional, known as intuitionistic implication, was proposed for dependence logic by Abramsky and Väänänen (2009). This connective is also part of the syntax of inquisitive logic. Intuitionistic implication behaves well in these downward closed logics, in the sene that it preserves downward closure and satisfies both Modus Ponens and the Deduction Theorem.

In recent years, many variants of dependence logic with different closure properties have been introduced, including union closed and convex logics. In these settings, the intuitionistic implication no longer behaves well, as it either fails to preserve the relevant closure property or fails to satisfy the Deduction Theorem. In this talk, we show that this failure is unavoidable: these logics cannot be enriched with any conditional connective that simultaneously preserves the closure property and satisfies both Modus Ponens and the Deduction Theorem.

This is joint work with Fausto Barbero.