The $$\varepsilon $$ -elimination method of Hilbert’s $$\varepsilon $$ -calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s $$\varepsilon $$ -calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses (...) of the extended first $$\varepsilon $$ -theorem, even if the formalisation incorporates so-called $$\varepsilon $$ -equality axioms. (shrink)

We present a sequent calculus with the Henkin constants in the place of the free variables. By disposing of the eigenvariable condition, we obtained a proof system with a strong locality property—the validity of each inference step depends only on its active formulas, not its context. Our major outcomes are: the cut elimination via a non-Gentzen-style algorithm without resorting to regularization and the elimination of Skolem functions with linear increase in the proof length for a subclass of derivations with cuts.

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...) with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (shrink)

We formulate Hilbert’s epsilon calculus in the context of expansion proofs. This leads to a simplified proof of the epsilon theorems by disposing of the need for prenexification, Skolemisation, and their respective inverse transformations. We observe that the natural notion of cut in the epsilon calculus is associative.

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice. We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof (...) system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions. A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals. (shrink)

Strict-Tolerant Logic ($$\textrm{ST}$$ ST ) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of $$\textrm{ST}$$ ST. Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential $$\textrm{ST}$$ ST -validity – these relations coincide only upon the addition of elimination rules and only within (...) the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order $$\textrm{ST}$$ ST with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation. (shrink)