Empty Logics

Wittgenstein and the liar

“…Haze's criticism, however, does not apply to the logical nihilist semantics proposed here. 24 An inference is valid in TS ( TS A) if and only if there is no interpretation d such that d(B) {1} or d(B) ∅ for all B ∈ , but d(A) {1} (where d is restricted to three-valued interpretations; see, e.g., Pailos, 2022Pailos, , p. 1391). 25 While there are no valid inferences according to FDE's concept of logical consequence simpliciter, it validates some classical 'metainferences' (i.e., closure properties on the set of valid arguments) like transitivity: if A B and B C, then A C (since the 'if' part is trivially false for any sentences A, B and C).…”

“…25 While there are no valid inferences according to FDE's concept of logical consequence simpliciter, it validates some classical 'metainferences' (i.e., closure properties on the set of valid arguments) like transitivity: if A B and B C, then A C (since the 'if' part is trivially false for any sentences A, B and C). This is unproblematic since FDE' is intended to model Wittgenstein's 'logical dead ends' and not what Pailos (2022Pailos ( , p. 1390) calls a 'truly empty logic', which does not only invalidate every inference but also every metainference. Pailos' concept of a 'truly empty logic', however, equivocates between (e.g., Russell's) extensional notion of an empty logic (according to which the extension of the concept of logical consequence is empty) and an intensional one (according to which an empty logic is 'uninformative'); on this and on the topic of modelling logical nihilism with a variety of empty logics with differing degrees of 'emptiness' (in the latter sense), see Pailos (2022).…”