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### 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  ...  Starting from proof-search in intuitionistic logic IL, the development of efficient proof-search procedures for intermediate logics like Gödel-Dummett logic has been the subject of recent studies [1,  ...

### 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 6, we detail some implementation techniques for a constrained r-cycle detection algorithm based on matrices of binary decision diagrams and briefly present our parallel counter-model search  ...  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  ...

### Gödel-Dummett Counter-models through Matrix Computation

Dominique Larchey-Wendling
2005 Electronical Notes in Theoretical Computer Science
We present a new method for deciding Gödel-Dummett logic. Starting from a formula, it proceeds in three steps. First build a conditional graph based on the decomposition tree of the formula.  ...  We emphasize on cycle removal through matrix computation, boolean constraint solving and counter-model extraction.  ...  We have recently proposed a combination of proof-search in sequent calculus and counter-model construction to provide a decision procedure for LC which is based on a new principle: we are able to gather  ...

### 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  Acknowledgments We thank both Klaus Weich and Pierangelo Miglioli (and the latter's colleagues) for unpublished papers ( 24] , 1], 2]) and Klaus Weich and Sara Negri for helpful discussions.  ...

### Bounding Resource Consumption with Gödel-Dummett Logics [chapter]

Dominique Larchey-Wendling
2005 Lecture Notes in Computer Science
Combining this result with our previous results on proof and counter-model construction for LCn, we conclude that bounding resource consumption is (linearly) equivalent to searching counter-models in LCn  ...  Gödel-Dummett logic LC and its finite approximations LCn are the intermediate logics complete w.r.t. linearly ordered Kripke models.  ...  to decide irreducible (hyper)-sequents and eventually build a counter-model.  ...

### Refutation in Dummett Logic Using a Sign to Express the Truth at the Next Possible World

Guido Fiorino
2011 International Joint Conference on Artificial Intelligence
The focus of the paper is on a new tableau system for Dummett logic, for which we have an implementation.  ...  In this paper we use the Kripke semantics characterization of Dummett logic to introduce a new way of handling non-forced formulas in tableau proof systems.  ...  For a survey in proof theory in Gödel-Dummett logics we quote [Baaz et al., 2003] . To perform automated deduction both tableau and sequent calculi have been proposed.  ...

### Graph-based Decision for Gödel-Dummett Logics

Dominique Larchey-Wendling
2006 Journal of automated reasoning
We present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models.  ...  From an instance graph containing no such cycle (resp. no (n + 1)-alternating chain) we extract a counter-model in LC (resp. LCn).  ...  Introduction Gödel-Dummett logic LC and its finitary versions (LC n ) n>0 are the intermediate logics (between classical and intuitionistic logics) characterized by linear Kripke models.  ...

### Deflationism and the Godel Phenomena

N. Tennant
2002 Mind
The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence.  ...  Any (-)consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems.  ...  The proof ⌫ establishes ᭙y¬Proof S ( ,y) from G; while the proof ☛ establishes the converse inference. Both ⌫ and ☛ can be constructed in accordance with the rules of intuitionistic relevant logic.  ...

### Towards a Proof Theory of Gödel Modal Logics

George Metcalfe, Nicola Olivetti, Martin Giese
2011 Logical Methods in Computer Science
Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic.  ...  The calculi are used to establish completeness and complexity results for these fragments.  ...  Hence the total amount of space needed for carrying out proof search is polynomial in |A|, and so deciding validity for the box fragment of GK is in PSPACE. derivations d 1 and d 2 end as follows with  ...

### Proof Theory and Meaning [chapter]

Göran Sundholm
2002 Handbook of Philosophical Logic
systematic search for a counter-model, one for verification and one for falsification.  ...  full predicate logic) and it is the corresponding constructive 'proof-tables' of Heyting that offer a possibility for Dummett's positive proposal.  ...

### Proof Theory and Meaning [chapter]

Göran Sundholm
1986 Handbook of Philosophical Logic
systematic search for a counter-model, one for verification and one for falsification.  ...  full predicate logic) and it is the corresponding constructive 'proof-tables' of Heyting that offer a possibility for Dummett's positive proposal.  ...

### 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.  ...

### Decidability of order-based modal logics

Xavier Caicedo, George Metcalfe, Ricardo Rodríguez, Jonas Rogger
2017 Journal of computer and system sciences (Print)
The standard infinite-valued Gödel logic (also known as Gödel-Dummett logic) interprets truth values as elements of [0, 1], conjunction and disjunction as minimum and maximum, respectively, and implication  ...  models, one for the modal operators and one for  ...  Decidability and PSPACE-completeness of the validity problem for these fragments of Gödel modal logics over [0, 1] was established in  using analytic Gentzen-style proof systems, but this methodology  ...

### On the Computational Meaning of Axioms [chapter]

Alberto Naibo, Mattia Petrolo, Thomas Seiller
2016 Logic, Epistemology, and the Unity of Science
An anti-realist theory of meaning suitable for both logical and proper axioms is investigated.  ...  As opposed to other anti-realist accounts, like Dummett-Prawitz verificationism, the standard framework of classical logic is not called into question.  ...  logical proofs, and everything can be eventually reduced to logical combinations of identity axioms.  ...

### ON ONTOLOGY AND REALISM IN MATHEMATICS

HAIM GAIFMAN
2012 The Review of Symbolic Logic
-models), and the construction yields no indication as to what happens in the standard model.  ...  The methods of constructing and extending models allowed us to switch back and forth the truth value of CH by repeated extensions of a given model.  ...
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