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### Formalizing Cut Elimination of Coalgebraic Logics in Coq [chapter]

Hendrik Tews
2013 Lecture Notes in Computer Science
The work presented here can be used to verify cut elimination theorems for different modal logics with considerably less effort in the future.  ...  The present paper reports on a formalization of Pattinson's and Schröder's work in the proof assistant Coq that provides machine-checked proofs for soundness, completeness and cut elimination of their  ...  Actually, all proofs in the formalization only require finitely many distinct variables. The number of variables needed depends on the syntactic structure of the sequent s.  ...

### Page 1477 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews
(I-MILAN-I; Milan) Cut-free tableau calculi for some intuitionistic modal logics. (English summary) Studia Logica 59 (1997), no. 3, 303-330.  ...  Also, by introducing an- other infinite sequence of variables, formula variables, and calling the old variables concrete, the authors obtain a finitely axioma- tized logic in terms of formulas, not schemata  ...

### Page 755 of Mathematical Reviews Vol. 34, Issue 4 [page]

1967 Mathematical Reviews
It is proved that the intuitionist logic K admits a Shefferian with three variables, and that it admits a Shefferian B with five variables each of which occurs in B only once.  ...  MR 26 #6036], the author’s counterexample would have a simple cut-free proof.}  ...

### Proof Theory: Some Applications of Cut-Elimination [chapter]

Helmut Schwichtenberg
1977 Studies in Logic and the Foundations of Mathematics
We restrict ourselves throughout to derivations with only finitely many free and bound variables. The set of variables free in a derivation d is denoted by Var(d). 3.4.2. EMBEDDING LEMMA.  ...  have finitely many [CH.  ...

### Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

A. Carbone, S. Semmes
1997 Bulletin of the American Mathematical Society
This is a basic rule that we take for granted and use repeatedly, but there is a gem of a theorem in logic by Gentzen to the effect that it is not needed in some logical systems.  ...  It is fun to say, "You can make proofs without lemmas" to mathematicians and watch how they react, but our true intention here is to let go of logic as a reflection of reasoning and move towards combinatorial  ...  Remember from Section 3 that the cut-free proofs of the finite versions of the pigeon-hole principle in propositional logic must have exponential size.  ...

### Existence and Consistence

Yuzuru KAKUDA
1997 Annals of the Japan Association for Philosophy of Science
We introduce a symbol S for every set S of formulas of _??_ with finite free variables. A variable occurring free in S is said to be a free variable in S. Definition 1.  ...  Let "C and ‡™ be finite sequences of hyperformulas with at most x1,..., xn free variables. We define "C_??_ ‡™ as follows: For all structure _??_ of _??  ...

### System Description: aRa – An Automatic Theorem Prover for Relation Algebras [chapter]

Carsten Sinz
2000 Lecture Notes in Computer Science
Gordeev developed a cut free formalization of predicate logic without equality using only finitely many distinct This work was partially supported by DFG under grant Ku 966/4-1. 1 This "simple" variant  ...  It is based on Gordeev's Reduction Predicate Calculi for n-variable logic (RPCn) which allow first-order finite variable proofs.  ...  Note, that the variable elimination operators are -just like substitution -metaoperators on formulae, and not part of the formal language of the logic itself.  ...

### Page 2238 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews
The authors give a cut-free Gentzen-type formalization for this logic. The formalization consists of 23 inference rules, defined as the decomposition rules.  ...  Jones, John Gordon] (4-EDIN-A) Formalisations of many-valued propositional calculi with variable functors.  ...

### Page 14 of Mathematical Reviews Vol. , Issue 86a [page]

1986 Mathematical Reviews
This paper establishes an extended joint-consistency theorem and an interpolation lemma for a family of (presupposition) free modal logics with equality which includes the (presupposition) free ver- sions  ...  Bloom (Hoboken, N.J.) 86a:03013 Bellissima, Fabio (I-SIN); Mirolli, Massimo (I-SIN) On the axiomatization of finite K-frames. Studia Logica 42 (1983), no. 4, 383-388 (1984).  ...

### Page 8167 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews
63-91); Anuj Dawar, Finite models and finitely many variables (93-117); Jorg Flum, On the existence of prime ideals in Boolean algebras (119-123); Pawet M.  ...  Khorokhorin, Dyadic semantics for the systems of formal syllogistics Cl and C3 (Russian) (134— 136); V. I. Markin, Aristotle’s singular negative syllogistics, and free logic (Russian) (137-153); A.  ...

### On intuitionistic many-valued logics

Masazumi HANAZAWA, Mitio TAKANO
1986 Journal of the Mathematical Society of Japan
If U=T, then the intuitionistic many-valued logic is of course identical with the usual many-valued logic (cf. 3.43) ; if T = {t, f } and U= {f}, then the logic is identical with the intuitionistic logic  ...  We put Kti =K `J { <2i, A>} for i=1, 2. For the cut-free provable matrix K1 (i=1, 2), one of the following five cases occurs : I. Ki is basic. ii.  ...

### Structural Extensions of Display Calculi: A General Recipe [chapter]

Agata Ciabattoni, Revantha Ramanayake
2013 Lecture Notes in Computer Science
As a case study, we present cut-free calculi for extensions of well-known logics including Bi-intuitionistic and tense logic.  ...  We present a systematic procedure for constructing cut-free display calculi for axiomatic extensions of a logic via structural rule extensions.  ...  Here we show how to construct cut-free display calculi for infinitely many axiomatic extensions of this logic in a uniform way.  ...

### Substructural Logics and Residuated Lattices — an Introduction [chapter]

Hiroakira Ono
2003 Trends in Logic
Based on these facts, we conclude at the end that substructural logics are logics of residuated structures, and in this way explain why sequent systems are suitable for formalizing substructural logics  ...  This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics.  ...  Finite model property and finite embeddability property A substructural logic L has the finite model propery if there exists a set {P i : iI} of finite FL-algebras such that L = {L(P i ) : iI} holds  ...

### Sequent of Relations Calculi: A Framework for Analytic Deduction in Many-valued Logics [chapter]

Matthias Baaz, Agata Ciabattoni, Christian G. Fermüller
2003 Studies in Fuzziness and Soft Computing
All finite-valued logics as well as infinite-valued Gödel logic are projective. As a case-study, sequent of relations calculi for Gödel logics are derived.  ...  We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) many-valued logics -called projective logics -characterized by a special format  ...  Any simple formula Des(x) of T with exactly one free variable x may be chosen for this purpose. 1 In general, for a propositional many-valued logic, an interpretation I is a mapping from the set of propositional  ...

### Extending the Family of Intuitionistic Many-Valued Logics Introduced by Rousseau

Mitio TAKANO
1986 Annals of the Japan Association for Philosophy of Science
of K being cut-free provable with rank n (_??  ...  This completes the proof of (2) and hence of Theorem 7. • •˜4. A formal system for the intuitionistic many-valued logic based on a partial order. 4.1.  ...
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