Formal analysis of modality and pragmatics in science and mathematics

The Institut supérieur de philosophie of the UCLouvain, the research centre CEFISES, and the FNRS contact group OLOFOS ((Onto-)LOgical Frameworks Of Science) are very happy to invite everyone interested to the (due to Covid-19) online only second ordinary meeting of this year’s OLOFOS seminar with the overarching theme “Formal analysis of modality and pragmatics in science and mathematics”.

Date and time: 20/1/2021, 15:00-17:00 CET

Speaker: Peter Verdée (Université catholique de Louvain)

Commentator: Bjørn Jespersen (Universiteit Utrecht)

Title: Finer-grained mathematics without logical revision. From relevant entailment to hyperintensional content.

Zoom link: https://us02web.zoom.us/j/87878320876. To register, please contact Peter Verdée (peter.verdee@uclouvain.be)

Abstract:

In this talk I present ideas developed in close collaboration with my FNRS MIS team on “Explanatory inference” (Pilar Terres, Joao Daniel Dantas, and Pierre Saint-Germier). We start by arguing that mathematical theories, qua epistemic objects used in mathematical practice, are hyperintensional and structured by inferential relations between mathematical beliefs: the relation of (sets of) beliefs jointly supporting other beliefs. We moreover argue that the typical use of expressions like `follows from’, `can be obtained from’, and `entails’ by mathematicians refers to a relevant entailment relation, in the sense that every mentioned premise (and the conclusion) needs to contribute in some sense to the validity of the argument.   

In order to understand such a hyperintensional and relevance-sensitive view on mathematics, we need to provide an account of hyperintensional content of statements and a relevance entailment relation that can formalize the relation of support between the statements. We moreover need to do that in a non-revisionary way, so that it is compatible with the deductive logic the mathematician follows when constructing and evaluating proofs. In other words, we need to come up with an account of hyperintensional meaning and relevant entailment that is coherent with the logic under which the mathematician closes her axioms (usually classical logic).

Suppose L is the mathematician’s deductive logic. We will propose that the relevant core L^r of L (i.e. the closure of the set of non-redundantly L-valid sequents of L under Uniform Substitution) is an adequate formalisation of the relation of support. We discuss some features of this logic and  then show that this very simple relevance logic already enables the definition of an interesting notion of hyperintensional meaning. We define a notion of exact verification and exact falsification (in Kit Fine’s sense of exactness) of formulas by mereologically structured states in a space of informational states. The definition of exact falsification/verification is solely based on the L^r-consequence relation, without needing to buy other semantic assumptions. Once exact verification and falsification are defined, we can define strong verification and strong falsification. A sequent is then relevantly L-valid iff the strong verifiers of the premises exactly verify the conclusion, or, in other words, iff the conclusion is implicitly contained in all states that explicitly contain the fusion of the premises. Statements or beliefs can then hyperintensionally be individuated by stipulating that they have the same content iff they are verified by the same exact verifiers.