X Workshop on Philosophical Logic:
Week IV: Logical conectives
August 26, 27 2021
Online Conference via ZOOM Meetings
Sara Ayhan (Ruhr-Universität Bochum)
Bruno Da Ré (University of Buenos Aires)
Teresa Kouri Kissel (Old Dominion University)
Hitoshi Omori (Ruhr-Universität Bochum)
Dave Ripley (Monash University)
Damián Szmuc (University of Buenos Aires)
Pilar Terres Villalonga (Université Catholique de Louvain)
“Bilateralism, logical consequence, and uniqueness of connectives”
In the literature on uniqueness it has been clearly shown that the question whether a connective is uniquely characterized by its set of rules is strongly related to the logic, its specific representation, and in particular to the underlying consequence relation. I will show the problems that are encountered when dealing with uniqueness of connectives in a bilateralist setting within the larger framework of proof-theoretic semantics and suggest a solution. Therefore, I will present and argue for a specific—and so far underrated—form of bilateralism: one that is bilateral not only on the level of rules, but also on a meta-level, namely concerning inferential relations. This is realized in the logic 2Int, for which I introduce a sequent calculus system, displaying—just like the corresponding natural deduction system—a consequence relation for provability as well as one dual to provability. I will propose a modified characterization of uniqueness incorporating such a duality of consequence relations, with which we can maintain uniqueness in a bilateralist setting. Finally, I want to discuss some implications of these considerations with regard to what may be considered more or less suitable representations of logics as well as bilateral features therein.
More on conjunction and disjunction in infectious logics
In “Conjunction and Disjunction in Infectious Logics”,
Damian Szmuc and I discussed the extent to which conjunctions and
disjunctions, appearing in the context of infectious logics, can be
rightfully called conjunction and disjunction. For our answer, we
relied on the framework of plurivalent semantics due to Graham Priest
to devise a two-valued semantics for the basic three-valued infectious
logics. The aim of this talk is to revisit the topic in view of some
recent developments in the semantics for infectious logics, and
compare how they differ in discussing the two connectives.
Pilar Terres Villalonga:
“A combination of logics: recovering the material conditional within a relevance consequence relation”
It is possible to defend the material conditional as legitimately codifying the inferential role of “if… then” and still argue that whenever one uses an expression as “if p then q” one is communicating a connection between p and q. This requires a pragmatic interpretation of this connection, and a defence that the material conditional formalizes the semantic content of “if… then”. I will first argue that relevant logic LR can give a systematic diagnostic of those cases in which a conditional is pragmatically enriched and those cases in which it is not, while classical logic LK captures the literal meaning of the conditional. The result is the endorsement of two different senses of “follows from” and three different formalizations of the conditional: the classical consequence which is linked to the material conditional, and the relevant consequence, that distinguishes an additive and a multiplicative conditional. After presenting these results, I will explore the possibility of combining a non-enriched conditional with a relevant notion of consequence. In order to do so, I will focus on the logic that results from translating the multiplicative/additive conditional into a unique connective in each LR-valid sequent, while keeping the relevant interpretation of logical consequence. We can call this process the naturalization of relevant logic, and I will then argue that the resulting logic of this process has strong connections with the logic NTR presented in [Verdée et al., 2019].
Teresa Kouri Kissel:
“Proof-Theoretic Pluralism, Harmony and Identity”
Ferrari and Orlandelli (2019) presents an innovative proof-theoretic
pluralism in line with the motivations in Restall (2014), that does not have the problems proposed by Kouri (2016). This pluralism is “pluralistic” at two levels: both at the level of validity and the level of the connective meanings. Logics are admissible to this system when they meet a certain set of constraints, including that they have connectives which are conservative and unique. Thus, they must be harmonious in the sense of Belnap (1962). In this paper, I show that requiring logics be Belnap-harmonious rules out some logics we ought to admit to our logical pluralism. In particular, it rules out a system presented by French (2016). I propose three different ways of weakening the harmony constraint that Ferrari and Orlandelli suggest. This leads to a pluralism at three levels: at the level of validity and connective meanings, but also at the level of admissibility constraints.
“The list modality, the complexity condition, and elimination rules”
The list modality is a unary modal operator L with two introduction rules. According to the first, LA can be concluded from no premises, for any sentence A. According to the second, LA can be concluded from the premises A and LA, again for any sentence A. It gets its name because the canonical proofs of LA are in bijection with the finite lists of canonical proofs of A. The list modality violates a property that has been taken to be important in introduction-rule-based presentations of connectives: LA itself occurs as a premise in one of the introduction rules for LA. In Dummett’s terminology, this rule
violates the “complexity condition”. In this talk, I argue that the complexity condition is ill-motivated, and offer a diagnosis of where Dummett went wrong in proposing it. Then, I turn to the question of elimination rules for connectives like L that violate the complexity condition. I argue that elimination rules based on “general elimination” or “inversion” are not appropriate for these connectives, and offer some alternative approaches.
Damian Szmuc and Bruno Da Ré:
On three-valued presentations of classical logic
Cobreros, Égré, Ripley, and van Rooij (2012) have shown that the logic defined using the strict-tolerant consequence relation over the (three-valued) Strong Kleene schema coincides with classical logic (at least inferentially speaking). More recently, Ferguson and Szmuc (2021) have proved that the (three-valued) Weak Kleene schema serves that purpose also. The aim of this talk is twofold. First, we generalize these results and present all of the three-valued schemata such that the strict-tolerant consequence relation over them coincides with classical logic. Secondly, we generalize these results and present all of the three-valued schemata such that the tolerant-strict, strict-strict, tolerant-tolerant, and order-consequence relation over them coincides with classical logic. This is based on joint work with Emmanuel Chemla and Paul Egré.
Damián Szmuc (UBA/IIF-SADAF-CONICET)
We are thankful for the support provided by CONICET and MINCYT