Zoom Talk: Pawel Pawlowski (Gdańsk)
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Two frameworks for informal provability.
TThere is a relatively large philosophical discussion on the relation between formal derivations (or proofs) and proofs as used and formulated in mathematical practice. One of the crucial differences between these notions is the validity of the reflection schema: if a sentence is (informally) provable, it is also true. We present two frameworks that aim at capturing the inferential behavior of informal provability: BAT and T-BAT. The former framework is based on the intuition that mathematical claims are either informally provable (they have an informal proof), or informally refutable (they have an informal refutation), or they are neither. The resulting logic is three-valued. One of the drawbacks of this approach is that the semantics focuses on the provability status of sentences, and does not tell us anything about how the provability status interacts with truth or falsity of a given claim. The latter framework was developed to fill that hole. The resulting framework is four-valued. In this talk, I will present both frameworks their pros and cons and comment a bit on the possibility of extending these frameworks to model some other notions related to informal provability.