# An Order Hyperresolution Calculus for Gödel Logic with Truth Constants

### Dušan Guller

#### Abstract

We have generalised the well-known hyperresolution principle to the first-order G¨odel logic for the general case. This paper is a continuation of our work. We propose a modification of the hyperresolution calculus suitable for automated deduction with explicit partial truth. We expand the first-order G¨odel logic by a countable set of intermediate truth constants ¯ c, c 2 (0;1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form e1 e2 where is a connective either P or . P and are interpreted by the equality and standard strict linear order on [0;1], respectively. We shall investigate the so-called canonical standard completeness, where the semantics of the first-order G¨odel logic is given by the standard G-algebra and truth constants are interpreted by themselves. The modified hyperresolution calculus is refutation sound and complete for a countable order clausal theory under certain condition for suprema and infima of sets of the truth constants occurring in the theory.

#### References

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#### Paper Citation

#### in Harvard Style

Guller D. (2014). **An Order Hyperresolution Calculus for Gödel Logic with Truth Constants** . In *Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)* ISBN 978-989-758-053-6, pages 37-52. DOI: 10.5220/0005073700370052

#### in Bibtex Style

@conference{fcta14,

author={Dušan Guller},

title={An Order Hyperresolution Calculus for Gödel Logic with Truth Constants},

booktitle={Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)},

year={2014},

pages={37-52},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005073700370052},

isbn={978-989-758-053-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)

TI - An Order Hyperresolution Calculus for Gödel Logic with Truth Constants

SN - 978-989-758-053-6

AU - Guller D.

PY - 2014

SP - 37

EP - 52

DO - 10.5220/0005073700370052