Are Uniqueness and Deducibility of Identicals the Same?

Abstract

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of analysis is driven by exploiting the Curry-Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity.1 Note also that Belnap requires the deducibility relation to satisfy a set of structural properties that contains the cut rule and the weakening rule(s); however, unlike Hacking, he does not explicitly ask for the admissibility of these rules (cf. Belnap, 1962, p. 132).3 The difference between uniqueness and strict uniqueness (Došen and Schroeder-Heister, 1985, p. 166) is not taken into account here: in a sequent calculus setting such a difference is difficult to identify in a very neat way. 4 Belnap's original definition of UNI is slightly different from the one we presented here. However, in presence of transitivity of the deducibility relation -i.e., in presence of the cut rule -our own definition entails Belnap's one. The latter will be discussed in more details in section 5.2.

Dialogue Games for Minimal Logic

“…Any attack may be defended at most once. 15 Attack-rule. O can attack P 's one and the same formula only once, whereas P can attack O's formula at most twice.…”

“…However, for LM 9 to be equal to NM, we should add there the bottom (⊥) as a constant. It is important because in minimal logic the condition of uniqueness of negation is not satisfied [15]. Following [15], we understand uniqueness as follows:…”

“…We count as a zero step the one where P proposes a formula for the dispute.14 This is a rule for minimal and intuitionistic logic only. In classical logic, any attack can be defended 15. This rule is applicable only for minimal and intuitionistic logics, but in classical logic, P can repeat his defences 16.…”

Harmonising Harmony

“…These three notions have been thoroughly investigated by Naibo & Petrolo (2015) under the names: weak deducibility of identicals, strong deducibility of identicals and uniqueness. Their primary aim was that of stressing the (mostly unnoticed) difference between the three notions.…”

Weak Disharmony: Some Lessons for Proof-Theoretic Semantics

“…From here on, the ball just rolled almost without a hitch: as far as I know, only 2014) endorses Belnap's dual criterion of harmony. Recently, uniqueness as a criterion of definitional success has come under criticism in Naibo & Petrolo (2015). Space limitations prevent from going into a discussion of their criticism here, but the reader should be warned that a separate defence of uniqueness might be called for.…”