### Construction of tableaux for classical logic: Tableaux as combinations of branches, branches as chains of sets

Tomasz Jarmużek

DOI: http://dx.doi.org/10.12775/LLP.2007.004

#### Abstract

The paper is devoted to an approach to analytic tableaux for propositional logic, but can be successfully extended to other logics. The distinguishing features of the presented approach are:(i) a precise set-theoretical description of tableau method; (ii) a notion of tableau consequence relation is defined without help of a notion of tableau, in our universe of discourse the basic notion is a branch;(iii) we define a tableau as a finite set of some chosen branches which is enough to check; hence, in our approach a tableau is only a way of choosing a minimal set of closed branches;(iv) a choice of tableau can be arbitrary, it means that if one tableau starting with some set of premisses is closed in the defined sense, then every branch in the power set of the set of formulas, that starts with the same set, is closed.

#### Keywords

analytic tableaux; propositional logic; set-theoretical approach to a description of tableaux; branches as chains of sets of formulas; tableaux consequence relation; choice of branches; tableau combined with branches

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#### References

Jarmużek, T., “Tableaux for non-classical propositional logic. General pattern of completeness and correctness” (in preparation).

Fitting, M., and R. Mendelson, First-Order Modal Logic. Kluwer Academic Publishers, Dordrecht/Boston/London, 1998.

Fitting, M., Intuitionistic Logic. Model Theory and Forcing. North-Holland Publishing Company, Amsterdam/London, 1969.

Girle, R., Modal Logics and Philosophy. McGill-Queen’s University Press, Montreal & Kingston, London, Ithaca, 2000.

Priest, G., An Introduction to Non-Classical Logic. Cambridge University Press, Cambridge, 2001.

Smullyan, R.M., First-Order Logic. Dover Publications, New York, 1995.

Financed by MNiSW on the basis of agreement no. 706/P-DUN/2018 (dated 10/05/18). Project 1: “Preparation of articles in English for eight editions of the journal Logic and Logical Philosophy over the period 2018–19; Vol. 27, No. 1–4 (2018), Vol. 28, No. 1–4 (2019)”; amount from the DUN grant: 64800 zł. Project 4: “Digitalisation of eight editions of the journal Logic and Logical Philosophy over the period 2018-19; Vol. 27, No. 1–4 (2018), Vol. 28, No. 1–4 (2019)”; amount from the DUN grant: 18600 zł.

ISSN: 1425-3305 (print version)
ISSN: 2300-9802 (electronic version)