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A deterministic terminating sequent calculus for Godel-Dummett logic

1999
*
Logic Journal of the IGPL
*

We give

doi:10.1093/jigpal/7.3.319
fatcat:vso2qfia7fc2thvlbcqwzjcyga
*a*short proof-theoretic treatment of*a**terminating*contraction-free*calculus*G4-LC*for*the zero-order G odel-*Dummett**logic*LC. ... In our*calculus*, all the rules of G4-LC are invertible, thus allowing*a**deterministic*proof-search procedure. ... Acknowledgments We thank both Klaus Weich and Pierangelo Miglioli (and the latter's colleagues)*for*unpublished papers ( 24] , 1], 2]) and Klaus Weich and Sara Negri*for*helpful discussions. ...##
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A Lambda Calculus for Gödel–Dummett Logic Capturing Waitfreedom
[chapter]

2012
*
Lecture Notes in Computer Science
*

We propose

doi:10.1007/978-3-642-29822-6_14
fatcat:bmt6blzkmnbx7bdsf45laba7cu
*a*typed lambda*calculus*based on Avron's hypersequent*calculus**for**Gödel*-*Dummett**logic*. This*calculus*turns out to model waitfree computation. ... The*calculus*is not only proof theoretically interesting, but also valuable as*a*basis*for*distributed programming languages. ... This work is encouraged by feedbacks from ACAN (Algebraic and Coalgebraic Approaches to Non-Classical*Logics*Workshop) and OPLSS'11 participants, supported by Grant-in-Aid*for*JSPS Fellows 23-6978, and ...##
###
Combining Proof-Search and Counter-Model Construction for Deciding Gödel-Dummett Logic
[chapter]

2002
*
Lecture Notes in Computer Science
*

We present an algorithm

doi:10.1007/3-540-45620-1_7
fatcat:oac5c2ipjzfjtnk5aml4fdx5di
*for*deciding*Gödel*-*Dummett**logic*. ... The originality of this algorithm comes from the combination of proofsearch in*sequent**calculus*, which reduces*a**sequent*to*a*set of pseudoatomic*sequents*, and counter-model construction of such pseudo-atomic ...*Gödel*-*Dummett**logic*LC In this section, we present the propositional*Gödel*-*Dummett**logic*LC, its algebraic semantics, and some admissible*sequent**calculus*rules, including the contraction-free system G4 ...##
###
Counter-Model Search in Gödel-Dummett Logics
[chapter]

2004
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/978-3-540-25984-8_19
fatcat:rjroafs3yjavtbkttzpcn2btye
*a*new method*for*deciding*Gödel*-*Dummett**logic*LC. ... In section 5, we show that proof-search can be viewed as*a*non-*deterministic*choice of arrows in*a*bi-colored graph corresponding to left or right premises of proof-rules. ... Then, we propose*a*new system*for*counter-model search in LC and LC n , mainly based on the notion of r-cycles in conditional graphs, and thus an efficient algorithm to decide these*logics*and provide ...##
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Fast decision procedure for propositional Dummett logic based on a multiple premise tableau calculus

2010
*
Information Sciences
*

We present

doi:10.1016/j.ins.2010.06.004
fatcat:jtlgbhvbrfbddhbsoysmdcqmzu
*a*procedure to decide propositional*Dummett**logic*. Such*a*procedure relies on*a*tableau*calculus*with*a*multiple premise rule and optimizations. ...*For*this reason another name*for*the*logic*under consideration is*Gödel*-*Dummett**Logic*. ... We quote [8]*for**a*throughout treatment of the subject. In the late '90 were presented tableau [1, 11] and*sequent*calculi [10]*for*propositional*Dummett**logic*. ...##
###
Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics
[chapter]

2000
*
Lecture Notes in Computer Science
*

These calculi are obtained by adding one more structural rule to the path-hypertableau

doi:10.1007/10722086_15
fatcat:fuzurxaihnedth2ib2fob2qkdq
*calculus**for*Intuitionistic*Logic*. ... Using pathhypertableaux we define analytic calculi*for*the intermediate*logics*Bd k , with k ≥ 1, which are semantically characterized by Kripke models of depth ≤ k. ...*For*instance, in [1] duplication-free tableau calculi*for*Sm, LQ and G ∞ have been defined (see also [10]*for**a**deterministic**terminating**sequent**calculus**for*G ∞ ). ...##
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Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic

2008
*
Journal of Logic and Computation
*

We present

doi:10.1093/logcom/exn067
fatcat:yxlmeg6y6zehxcxxgtxcopujza
*a*new cut-free*sequent**calculus**for*bi-intuitionistic*logic*, and prove it sound and complete with respect to its Kripke semantics. ... Bi-intuitionistic*logic*is the union of intuitionistic and dual intuitionistic*logic*, and was introduced by Rauszer as*a*Hilbert*calculus*with algebraic and Kripke semantics. ... We would like to thank Alwen Tiu*for*providing useful comments and suggestions, and pointing out errors in*a*draft of this paper. ...##
###
Proofs and Countermodels in Non-Classical Logics

2014
*
Logica Universalis
*

*A*method is presented that makes it possible to establish completeness in

*a*direct way:

*For*any given

*sequent*either

*a*proof in the given

*logical*system or

*a*countermodel in the corresponding frame class ... Finitizations

*for*intuitionistic propositional

*logic*are obtained through the search

*for*

*a*minimal derivation, through pruning of infinite branches in search trees by means of

*a*suitable syntactic counterpart ... The famous modal translation * of intuitionistic

*logic*was defined in

*a*note by

*Gödel*[

*Gödel*, 1933] together with

*a*syntactic proof that the translation is sound, namely that Int

*A*implies S4

*A** . ...

##
###
Mathematical modal logic: A view of its evolution
[chapter]

2006
*
Handbook of the History of Logic
*

This is

doi:10.1016/s1874-5857(06)80027-0
fatcat:dihvhmx5vjhdjndl7x5auvqbki
*a*survey of the origins of mathematical interpretations of modal*logics*, and their development over the last century or so. ... It reviews the ideas of*a*number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. ...*For*such assistance I thank Wim Blok, Max Cresswell, John Dawson, Allen Emerson, Saul Kripke, Neil Leslie, Ed Mares, Robin Milner, Hiroakira Ono, Lawrence Pedersen, Vaughan Pratt, Colin Stirling and Paul ...##
###
Mathematical modal logic: A view of its evolution

2003
*
Journal of Applied Logic
*

This is

doi:10.1016/s1570-8683(03)00008-9
fatcat:zhwzfvdeqjg3rgbkqstbxk5yqq
*a*survey of the origins of mathematical interpretations of modal*logics*, and their development over the last century or so. ... It reviews the ideas of*a*number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. ...*For*such assistance I thank Wim Blok, Max Cresswell, John Dawson, Allen Emerson, Saul Kripke, Neil Leslie, Ed Mares, Robin Milner, Hiroakira Ono, Lawrence Pedersen, Vaughan Pratt, Colin Stirling and Paul ...##
###
Duality between unprovability and provability in forward proof-search for Intuitionistic Propositional Logic
[article]

2018
*
arXiv
*
pre-print

To this aim we design

arXiv:1804.06689v1
fatcat:hjlq6rvi7zgild2imvnnyvzd7m
*a*forward*calculus*FRJ(G)*for*Intuitionistic unprovability which is prone to constructively ascertain the unprovability of*a*formula G by providing*a*concise countermodel*for*it; ... Here we apply this method to derive the unprovability of*a*goal formula G in Intuitionistic Propositional*Logic*. ... Finally, as*a*future work we plan to investigate the applicability of our method to other*logics*, in particular to modal*logics*such as S4 and intermediate*logics*such as*Gödel*-*Dummett**logic*characterized ...##
###
A New Unified Account of Truth and Paradox

2015
*
Mind
*

In Gentzen's original systems, one can literally perform

doi:10.1093/mind/fzu179
fatcat:dlf6xksza5dpdhqlr3vnylljre
*a*cut, by applying the Cut Rule of the*sequent**calculus*: D : l l, G : w D, G : w -or, in the case of natural deductions, by grafting*a*copy of one's ... ,*Dummett*1991 , and Tennant 1997 Here, I shall be concerned only with the reduction procedures that will be needed*for**a*discussion of certain standard paradoxes. ... to undertake whatever rational revision of one's beliefs might be called*for*. 9 ...##
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Fast Decision Procedure for Propositional Dummett Logic Based on a Multiple Premise Tableau Calculus

unpublished

We present

doi:10.29007/mbbq
fatcat:le3w6wt3wje3hivj3cj32h7dmm
*a*procedure to decide propositional*Dummett**logic*. Such*a*procedure relies on*a*tableau*calculus*with*a*multiple premise rule and optimizations. ...*For*this reason another name*for*the*logic*under consideration is*Gödel*-*Dummett**Logic*. ... We quote [8]*for**a*throughout treatment of the subject. In the late '90 were presented tableau [1, 11] and*sequent*calculi [10]*for*propositional*Dummett**logic*. ...##
###
Proof theory and proof search of bi-intuitionistic and tense logic
[article]

2018

Our

doi:10.25911/5d5fcc3a4db33
fatcat:r5767wlm7jbozcdo6dr7smkx7q
*calculus*is sound, semantically complete and allows*terminating*backward proof search, hence giving rise to*a*decision procedure*for*bi-intuitionistic*logic*. ... The interaction between these dual connectives makes it non-trivial to develop*a*cut-free*sequent**calculus**for*these*logics*. ...*logic*called*Gödel*-*Dummett**logic*. ...##
###
Proof Search in Multi-Agent Dialogues for Modal Logic

2018

These are not only used as reasoning procedures

doi:10.20378/irbo-52527
fatcat:nov4b3rthzep5iqzrysujqzqb4
*for*intuitionistic and classical*logic*, but also*for*intermediate*logics*like the*Gödel*-*Dummett**logic*. ... Multi-Proponent Dialogues (Forking and Merging) Fermüller and Ciabattoni [51] propose multi-proponent dialogues to cope with intuitionistic and the intermediate*Gödel*-*Dummett**Logic*. ... When searching*for*proofs,*a*normalization of deductions in*a**calculus*is desired.*A*new attempt to find*a*normalized*calculus*leads to dialogical*logic*,*a*game-theoretic reasoning technique. ...
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