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A deterministic terminating sequent calculus for Godel-Dummett logic

R Dyckhoff
1999 Logic Journal of the IGPL  
We give 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  Acknowledgments We thank both Klaus Weich and Pierangelo Miglioli (and the latter's colleagues) for unpublished papers ( 24] , 1], 2]) and Klaus Weich and Sara Negri for helpful discussions.  ... 
doi:10.1093/jigpal/7.3.319 fatcat:vso2qfia7fc2thvlbcqwzjcyga

A Lambda Calculus for Gödel–Dummett Logic Capturing Waitfreedom [chapter]

Yoichi Hirai
2012 Lecture Notes in Computer Science  
We propose 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  ... 
doi:10.1007/978-3-642-29822-6_14 fatcat:bmt6blzkmnbx7bdsf45laba7cu

Combining Proof-Search and Counter-Model Construction for Deciding Gödel-Dummett Logic [chapter]

Dominique Larchey-Wendling
2002 Lecture Notes in Computer Science  
We present an algorithm 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  ... 
doi:10.1007/3-540-45620-1_7 fatcat:oac5c2ipjzfjtnk5aml4fdx5di

Counter-Model Search in Gödel-Dummett Logics [chapter]

Dominique Larchey-Wendling
2004 Lecture Notes in Computer Science  
We present 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  ... 
doi:10.1007/978-3-540-25984-8_19 fatcat:rjroafs3yjavtbkttzpcn2btye

Fast decision procedure for propositional Dummett logic based on a multiple premise tableau calculus

Guido Fiorino
2010 Information Sciences  
We present 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.  ... 
doi:10.1016/j.ins.2010.06.004 fatcat:jtlgbhvbrfbddhbsoysmdcqmzu

Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics [chapter]

Agata Ciabattoni, Mauro Ferrari
2000 Lecture Notes in Computer Science  
These calculi are obtained by adding one more structural rule to the path-hypertableau 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 ∞ ).  ... 
doi:10.1007/10722086_15 fatcat:fuzurxaihnedth2ib2fob2qkdq

Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic

R. Gore, L. Postniece
2008 Journal of Logic and Computation  
We present 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.  ... 
doi:10.1093/logcom/exn067 fatcat:yxlmeg6y6zehxcxxgtxcopujza

Proofs and Countermodels in Non-Classical Logics

Sara Negri
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 * .  ... 
doi:10.1007/s11787-014-0097-1 fatcat:yzxgrtllgrga5j3nsc4qm5hgoy

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

Robert Goldblatt
2006 Handbook of the History of Logic  
This is 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  ... 
doi:10.1016/s1874-5857(06)80027-0 fatcat:dihvhmx5vjhdjndl7x5auvqbki

Mathematical modal logic: A view of its evolution

Robert Goldblatt
2003 Journal of Applied Logic  
This is 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  ... 
doi:10.1016/s1570-8683(03)00008-9 fatcat:zhwzfvdeqjg3rgbkqstbxk5yqq

Duality between unprovability and provability in forward proof-search for Intuitionistic Propositional Logic [article]

Camillo Fiorentini, Mauro Ferrari
2018 arXiv   pre-print
To this aim we design 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  ... 
arXiv:1804.06689v1 fatcat:hjlq6rvi7zgild2imvnnyvzd7m

A New Unified Account of Truth and Paradox

N. Tennant
2015 Mind  
In Gentzen's original systems, one can literally perform 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  ... 
doi:10.1093/mind/fzu179 fatcat:dlf6xksza5dpdhqlr3vnylljre

Fast Decision Procedure for Propositional Dummett Logic Based on a Multiple Premise Tableau Calculus

Guido Fiorino
unpublished
We present 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.  ... 
doi:10.29007/mbbq fatcat:le3w6wt3wje3hivj3cj32h7dmm

Proof theory and proof search of bi-intuitionistic and tense logic [article]

Linda Postniece, University, The Australian National, University, The Australian National
2018
Our 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.  ... 
doi:10.25911/5d5fcc3a4db33 fatcat:r5767wlm7jbozcdo6dr7smkx7q

Proof Search in Multi-Agent Dialogues for Modal Logic

Martin Sticht
2018
These are not only used as reasoning procedures 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.  ... 
doi:10.20378/irbo-52527 fatcat:nov4b3rthzep5iqzrysujqzqb4
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