Dual Systems of Sequents and Tableaux for Many-Valued Logics

The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are al- ways two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one’s attention since in the classical (two-valued) case the two systems coincide. (In two-valued logic the assignment of a truth value and the exclusion of the opposite truth value describe the same situation.

Elimination of Cuts in First-order Finite-valued Logics

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information

Systematic construction of natural deduction systems for many-valued logics

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems

Labeled calculi and finite-valued logics

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finitevalued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight. Keywords: finite-valued logic, labeled calculus, signed formula, sets-as-sign

Review: Vagueness and Degrees of Truth

A Review of Nicholas J.J. Smith, Vagueness and Degrees of Truth, Oxford University Press, 2008

Review: Vagueness and Degrees of Truth

A Review of Nicholas J.J. Smith, Vagueness and Degrees of Truth, Oxford University Press, 2008

Implicational Completeness of Signed Resolution

deduction. The main reason for this is probably the conception that implicational completeness, in contrast to refutational completeness, is of no practical significance. Moreover, it fails for all important refinements of Robinson's original resolution calculus. In addition, Lee's proof [7] is presented in an unsatisfactory manner (to say the least). A fourth reason for the widespread neglect of implicational completeness might be the fact that Lee (and others at that time) did not distinguish between implication and subsumption of clauses. However, nowadays, it is well known that the first relation between clauses is undecidable [10], whereas sophisticated and efficient algorithms for testing the latter one are at the core of virtually all successful resolution theorem provers (see, e.g., [4]). With hindsight, this is decisive for the significance of Lee's Theorem. We will provide a new and independent proof of implicational complete

Hintikka Style Game Rules for Semi-Fuzzy Quantifiers

The final publication is available via https://doi.org/10.1109/ISMVL.2017.57.Extending Hintikka’s game for the evaluation of classical formulas, we explore the realm of quantifier rules that can be defined by combining several moves consisting of choices by the two strategic players, but also by a third non-strategic player ‘Nature’, representing random choices. The simple format of Hintikka-style games is compared to the seemingly much more general one of Giles’s game for Łukasiewicz logic.Austrian Science Funds (FWF