Note on witnessed Gödel logics with Delta

Abstract

a b s t r a c tWitnessed Gödel logics are based on the interpretation of ∀ (∃) by minimum (maximum) instead of supremum (infimum). Witnessed Gödel logics appear for many practical purposes more suited than usual Gödel logics as the occurrence of proper infima/suprema is practically irrelevant. In this note we characterize witnessed Gödel logics with absoluteness operator w.r.t. witnessed Gödel logics using a uniform translation.

On witnessed models in fuzzy logic III - witnessed Gödel logics

“…The paper wants to be a contribution to our knowledge of the properties of Gödel logics G(V ) under their standard semantics and witnessed semantics, as developed in [1,2,3,4,6,7]. Here we have studied truth sets having finitely many left/right accumulation points (at least one) and no right/left accumulation point.…”

“…This note is a remark to the paper [4] dealing with witnessed Gödel logics G(V ). Recall the notation…”

“…Trivially, TAUT(G(V n )) = wTAUT(G(V n )) since V n is a finite set and hence each model over it is witnessed. It is known that TAUT(G(V ↑ )) = n TAUT(G(V n )) and Baaz and Fasching had proven in [4] that TAUT(G(V ↑ )) = wTAUT(G(V ↑ )). In Section 2 we are going to show that for several other V the set wTAUT(G(V )) of witnessed tautologies of G(V ) equals to TAUT(G(V ↑ )); in Section 3 we discuss witnessed Gödel logics G Δ (V ) containing the connective of (Baaz's) Delta.…”

Proof theory of witnessed Gödel logic: A negative result

“…G w was axiomatized in [20,3] by adding to the Hilbert system for Gödel logic both axioms ∃x(P (x) ⊃ ∀yP (y)) and ∃x(∃yP (y) ⊃ P (x)) expressing that each infimum is a minimum, and each supremum is a maximum, respectively.…”

“…In contrast to Gödel logic, the only existing calculus for G w is a Hilbert system [20,3], which is difficult to use even for finding simple proofs manually. The question that we address is: can we define a "well-behaved" calculus for G w using the sequent calculus or a suitable generalization?…”