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Formalizing Cut Elimination of Coalgebraic Logics in Coq
[chapter]

2013
*
Lecture Notes in Computer Science
*

The work presented here can be used to verify

doi:10.1007/978-3-642-40537-2_22
fatcat:physiyzxlzbdhp76d2ptoil4xi
*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. ...##
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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]

1977
*
Studies in Logic and the Foundations of Mathematics
*

We
restrict ourselves throughout to derivations

doi:10.1016/s0049-237x(08)71124-8
fatcat:ewrump3wpfhbjewfhzxz3j7lwi
*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

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

doi:10.1090/s0273-0979-97-00715-5
fatcat:lrej7lo6ofhj7eix4vsxxfjtnu
*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

1997
*
Annals of the Japan Association for Philosophy of Science
*

We introduce a symbol S for every set S

doi:10.4288/jafpos1956.9.69
fatcat:7or3qxowerdhvpvqsws3fn4oqe
*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*_?? ...##
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System Description: aRa – An Automatic Theorem Prover for Relation Algebras
[chapter]

2000
*
Lecture Notes in Computer Science
*

Gordeev developed a

doi:10.1007/10721959_13
fatcat:7qtxqcvyrrhrxdtlcnncllqdom
*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

1986
*
Journal of the Mathematical Society of Japan
*

If U=T, then the intuitionistic

doi:10.2969/jmsj/03830409
fatcat:2icqjupktjevhez2l4jgosj6g4
*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]

2013
*
Lecture Notes in Computer Science
*

As a case study, we present

doi:10.1007/978-3-642-39992-3_10
fatcat:nldyzckvi5a6fdxjci55qw7uy4
*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]

2003
*
Trends in Logic
*

Based on these facts, we conclude at the end that substructural

doi:10.1007/978-94-017-3598-8_8
fatcat:d7sgqudk7je3nahfs6zkynxqcm
*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*:*i*∈*I*}*of**finite*FL-algebras such that L = {L(P*i*) :*i*∈*I*} holds ...##
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Sequent of Relations Calculi: A Framework for Analytic Deduction in Many-valued Logics
[chapter]

2003
*
Studies in Fuzziness and Soft Computing
*

All

doi:10.1007/978-3-7908-1769-0_6
fatcat:xtfaqwfol5gm3pbbmex2zo5un4
*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

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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