A Sequent-Calculus Based Formulation of the Extended First Epsilon Theorem

Baaz, Matthias; Leitsch, Alexander; Lolić, Anela

doi:10.1007/978-3-319-72056-2_4

Cited by 4 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since L (U → V ) → (U → V ), and U → V and U are critical formulas, Lε V . By assumption, Lε has the extended first ε -theorem, so L proves a disjunction 5 Bell provides another proof of M in intuitionistic ε-calculus which explicitly requires, in addition to D, the assumption a = b. DeVidi [12] shows that in the intuitionistic ε -calculus, D ∧ a = b derives Lin. (Note that also LC M .)…”

Section: If a Is Quantifier-free And Lεmentioning

confidence: 99%

“…Despite this, the ability in the classical case to eliminate single critical formulas provides flexibility that can be exploited to produce smaller overall Herbrand disjunctions. As Theorem 3 of [5] shows there are sequences of -proofs where the original procedure produces Herbrand disjunctions that are non-elementarily larger than a more efficient elimination order.…”

Section: The Hilbert–bernays Elimination Proceduresmentioning

confidence: 99%

See 1 more Smart Citation

Epsilon Theorems in Intermediate Logics

Baaz¹,

Zach²

2022

J. symb. log.

Self Cite

View full text Add to dashboard Cite

Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon-and tau-operators and critical formulas. For classical logic, this results in Hilbert's ε-calculus. The first and second ε-theorems for classical logic establish conservativity of the ε-calculus over its classical base logic. It is well known that the second ε-theorem fails for the intuitionistic ε-calculus, as prenexation is impossible. The paper investigates the effect of adding critical ε-and τ -formulas and using the translation of quantifiers into ε-and τ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ετ -calculi. The "extended" first ε-theorem holds if the base logic is finite-valued Gödel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second ε-theorem also holds for finitevalued first-order Gödel logics. The methods used to prove the extended first ε-theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.

show abstract

Section: If a Is Quantifier-free And Lεmentioning

confidence: 99%

Section: The Hilbert–bernays Elimination Proceduresmentioning

confidence: 99%

Epsilon Theorems in Intermediate Logics

Baaz¹,

Zach²

2022

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…On the other hand, it is important to explore a better proof representation suitable for epsilon calculus, because a syntactic complication of ε-terms is a real practical obstacle to study epsilon calculus. Some modern formalizations of epsilon calculus are known, for example sequent calculus with function variables by Baaz, Leitsch, and Lolic [BLL18] and another formulation based on Miller's expansion tree by Aschieri, Hetzl, and Weller [AHW18].…”

Section: Future Workmentioning

confidence: 99%

“…For non-classical approaches to epsilon calculus, see the work of Bell [Bel93a,Bel93b], DeVidi [DeV95], Fitting [Fit75], Mostowski [Mos63], and Shirai [Shi71]. Our study is also motivated by the recent renewed interest in the epsilon calculus and the ε-substitution method in, e.g., the work of Arai [Ara03,Ara05], Avigad [Avi02], Baaz et al [BLL18], and Mints et al, [MT99,Min03]. The epsilon calculus also allows the incorporation of choice construction into logic [BG00].…”

Section: Introductionmentioning

confidence: 99%

The Epsilon Calculus with Equality and Herbrand Complexity

Miyamoto,

Moser

2019

Preprint

View full text Add to dashboard Cite

Hilbert's epsilon calculus is an extension of elementary or predicate calculus by a term-forming operator ε and initial formulas involving such terms. The fundamental results about the epsilon calculus are so-called epsilon theorems, which have been proven by means of the epsilon elimination method. It is a procedure of transforming a proof in epsilon calculus into a proof in elementary or predicate calculus through getting rid of those initial formulas. One remarkable consequence is a proof of Herbrand's theorem due to Bernays and Hilbert which comes as a corollary of extended first epsilon theorem. The contribution of this paper is the upper and lower bounds analysis of the length of Herbrand disjunctions in extended first epsilon theorem for epsilon calculus with equality. We also show that the complexity analysis for Herbrand's theorem with equality is a straightforward consequence of the one for extended first epsilon theorem without equality due to Moser and Zach.

show abstract

Herbrand complexity and the epsilon calculus with equality

Miyamoto

Moser

2023

Arch. Math. Logic

View full text Add to dashboard Cite

A Sequent-Calculus Based Formulation of the Extended First Epsilon Theorem

Cited by 4 publications

References 7 publications

Epsilon Theorems in Intermediate Logics

Epsilon Theorems in Intermediate Logics

The Epsilon Calculus with Equality and Herbrand Complexity

Herbrand complexity and the epsilon calculus with equality

Contact Info

Product

Resources

About