Proof-Theoretic Pluralism and Harmony

Abstract: Abstract: Ferrari and Orlandelli (2019) propose that an admissibility condition on a proof-theoretic logical pluralism be that the logics in question must be harmonious, in the sense of Belnap (1962). This means that they must have connectives which are unique and conservative. This allows them to develop an innovative pluralism, which shows variance on two levels. On one level, we have a pluralism at the level of validity alone, like that in Restall (2014). But, thanks to the Ferrari and Orlandelli system, which was developed in response to some concerns of Kouri (2016), we can add a second level and admit some logics which do not share connective meanings, and hence have different operational rules. This allows for us to have a pluralism at two levels: the level of validity and the level of connective meanings.

Here, I will show that we can extend the system one step further, and induce a three-level logical pluralism. The first and second levels remain as suggested by Ferrari and Orlandelli (2019), but we can allow for multiple notions of uniqueness in the definition of Belnap-harmony, or multiple notions of harmony writ large. Either of these options generates a pluralism at the level of our admissibility conditions. This generates a pluralism at three levels: validity, connective meanings, and admissibility conditions. But it still preserves the spirit of the Ferrari and Orlandelli (2019) solution: harmony remains as the admissibility constraint across the board, and so the original worries of Kouri (2016) are still put to rest and the original Beall and Restall (2006) criteria for admissible logics are still met.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic