A semantic proof of strong cut-admissibility for first-order Godel logic

Abstract

We provide a constructive direct semantic proof of the completeness of the cut-free part of the hypersequent calculus HIF for the standard first-order Gödel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). The results also apply to derivations from assumptions (or "non-logical axioms"), showing in particular that when the set of assumptions is closed under substitutions, then cuts can be confined to formulas occurring in the assumptions. The methods and results are then extended to handle the (Baaz) Delta connective as well. 2 Standard First-Order Gödel Logic Let L be a first-order language with ∧, ∨, ⊃ as binary connectives, ⊥ as a propositional constant, and ∀ and ∃ as unary quantifiers. We assume that the set of free variables and the set of bounded variables are disjoint (thus in a well-formed formula, the use of the bound variables is always in the scope of a quantification of the same variables). We use the metavariables a, b to range over the free variables, x to range over the bounded variables, p to range over the predicate symbols of L, c to range over its constant symbols, and f to range over its function symbols. The sets of L-terms and L-formulas are defined as usual, and are denoted by trm L and frm L , respectively. We mainly use t as a metavariable standing for L-terms, ϕ, ψ for L-formulas, Γ, ∆ for sets of L-formulas, and E, F for sets of L-formulas which are either singletons or empty. Given an L-term t, a free variable a, and another L-term t , we denote by t{t /a} the L-term obtained from t by replacing all occurrences of a by t. This notation is extended to formulas, set of formulas, etc. in the obvious way. Notation 1 To improve readability we use square parentheses in the metalanguage , and reserve round parentheses to the first-order language. As usual, there are two main approaches to define the standard first-order Gödel logic: Proof-theoretically The logic is defined using an Hilbert-style calculus. Such a calculus is obtained by adding the following axioms to any Hilbert-style calculus for first-order intuitionistic logic (see e.g. [7]): • The "linearity" axiom (ϕ ⊃ ψ) ∨ (ψ ⊃ ϕ). • The "quantifier-shifting" axiom (∀x(ϕ{x/a} ∨ ψ)) ⊃ (∀x(ϕ{x/a}) ∨ ψ). We shall denote such an Hilbert-style calculus by sG, and sG will stand for the consequence relation which is naturally associated with it (note that this is a relation between sets of formulas and formulas). Model-theoretically Here there are two options. First, the standard first-order Gödel logic can be defined as a first-order fuzzy logic based on the real interval [0, 1]. Second, it can be seen as an intermediate logic, defined in terms of Kripke-style semantics. In the next two subsections, we briefly review the many-valued semantics and the Kripke-style semantics of this logic. 2.1 Many-Valued Semantics Definition 2 An L-algebra is a pair D, I where D is a non-empty domain and I is an interpretation of constants and function symbols of L ...

A cut-free calculus for second-order Gödel logic

“…The properties proved in [16] for HIF are slightly stronger than those shown in this paper for HIF 2 . Obtaining these stronger results for HIF 2 seems to be straightforward.…”

“…Note that the soundness proof for HIF in [16], was with respect to Kripke-style semantics, where here we prove soundness of HIF 2 for the many-valued semantics described above. We use the following technical lemmas (full proofs are given in Appendix Appendix A): Lemma 4.4.…”

“…We begin by presenting HIF itself. We follow its formulation from [16] (adapted to our definitions, where formulas are equivalence classes of concrete formulas).…”

“…These are straightforward adaptations of the corresponding notions in [16], that were used to prove cut-free completeness of the first-order fragment. The full proofs, that are also adaptations of the corresponding proofs in [16], are given in Appendix Appendix B.…”

“…In [16] we considered derivations from (possibly) non-empty sets of hypersequents, serving as assumptions. In this case the cut rule must be added to the calculus, and obviously one cannot have full cut-admissibility.…”