PTS Seminars

Base-extension semantics (B-eS) -- a strand of proof-theoretic semantics -- has been given to a number of logics within the past decade. Rooted in inferentialism and constructive spirit, B-eS is a very suitable framework for -- and can be smoothly given to - intuitionistic logics; the same cannot be said about the classical ones. B-eS for intuitionistic logics follows a certain pattern that does not diverge much from logic to logic. Meanwhile, there are various ways of tweking the B-eS machinery that allow for capturing classical behaviour. In this talk, we give an overview of such methods, and investigate whether they all rely on a specific detail that is differently disguised amongst them, or, on the contrary, whether there are multiple (unrelated) ways in which classicality can be achieved -- similar to how, syntactically, intuitionistic logic collapses to classical by adding a single, but not necessarily the same rule.

According to a well-known thesis of Quine's (1986) logics that validate different arguments use different logical vocabulary. In a certain sense, then, partisans of different logics are partisans of different languages, and they don't much disagree as talk past each others.

A common respons going back to Putnam (1957) and Morton (1973), is to claim that sides who agree on enough logical principles involving some connective or quantifier are indeed using the same connective or quantifier. The obvious difficulty with this idea is spelling out what counts as "enough" agreement. Recently, some inferentialists have tried to do just that (Restall 2002, 2014; Paoli 2013, 2014; Dicher 2016). They are often called minimalists, following Hjortland's (2014) terminology. Minimalists draw a boundary roughly along the line between the operational and structural rules of sequent calculi: operational rules confer meaning, structural rules don't. Therefore, disagreements about logic that can be seen as disagreements about structural rules need not involve a change of language.

In this talk I will argue that current minimalist proposals lead to untenable views about which connectives are identical, or have the same meaning. I will also argue that the importance of the ‘change of language’ issue has been exaggerated. In many cases, I think, there is likely no fact of the matter as to whether different logics share their stock of operators, but there can still be substantial disagreement between partisans of each of them.

In 1980, Gibbard proved that when one extends classical logic with a symbol meant to model the indicative conditional that validates the equivalence known as Import-Export, that is supraclassical, - the is, if φ classical implies ψ, then 'if φ then ψ' is a theorem of the extended logic - and if one assumes that the indicative conditional is at least as strong as the material conditional, then both collapse into the material. Later on, Fitelson showed that it is not necessary to assume classical logic, rather intuitionistic logic suffices for the collapse to happen. These results are usually stated in terms of a dilemma: either one accepts that the indicative conditional is something in between the intuitionistic and the material conditional or one endorses a semantics for indicative conditionals that invalidates Import-Export of Supraclassicality.

In this work, I will show that the collapse can happen in even weaker logics, in particular in many non-contractive and non-monotonic logics. I will also show that it is not necessary to assume anything about the rest of the language (in particular about conjunctions), yet there are some structural assumptions needed to derive the collapse. As a consequence of this result, several non-monotonic conditionals will also collapse to the indicative. I will show it three times. First without assuming anything about the non-conditional fragment of the language. Then I will prove it twice with two different sets of more conservative assumptions.

Following Belnap's famous arguments about tonk (1962), I will reflect on how the consequence relation influences the meaning of the connectives one is defining and I will argue that if one thinks of a conditional in terms of the rules it validates, then some of these conditionals are good candidates to model indicatives.

I present a base-extension semantics (B-eS) for public announcement logic (PAL) as a first step in developing B-eS for the larger group of dynamic epistemic logics. This is then used to discuss some examples of PAL (including the famous Muddy Children Puzzle) to highlight some insights an inferential perspective can give on the role of information in the reasoning employed.

In this talk I present inference behavious semantics (IBS) for connectives in two dimensional sequent calculi. IBS is a novel approach to proof-theoretic semantics (PTS) that emphasises Wittgenstein's conception of 'meaning as use', alongside Gentzen's idea of operational rules as connective definitions. The core idea of IBS is to explore how proof rules determine the way we use connectives. 

To implement this idea, I analyse all rule parameters that affect connective usage by proving global harmony in minimal derivability relations. This method allows me to define IBS for any connective definable in two-dimensional sequent calculi. The findings offer a meaning-theoretic explanation for co-determination effects recently observed by Dicher and offer a fresh perspective on the relationships between connectives and their logics.

Ultimetaley, IBS opens a new avenue for PTS, providing a fine-grained local analysis of connective use and meaning, with potential to evolve our understandingof the connectives.