It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e., the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.

We show how to extend the Continuous Propositional Logic by means of an infinitary rule in order to achieve a Strong Completeness Theorem. Eventually we investigate how to recover a weak version of the Deduction Theorem.

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n -Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus δ SLTL is defined.

We present some proof-theoretic results for the normal modal logic whose characteristic axiom is \. We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.

ABSTRACT A new type of calculi is proposed for a first order linear temporal logic. Instead of induction-type postulates the introduced calculi contain a similarity saturation principle, indicating some form of regularity in the derivations of the logic. In a finitary case we obtained the finite set of saturated sequents, showing that ?nothing new? can be obtained continuing the derivation process. Instead of the ?-type rule of inference, an infinitary saturated calculus has an infinite set of saturated sequents, showing that (...) only a ?similar? sequents can be obtained continuing the dérivation process. The saturation calculi have some resemblance with resolution-like calculi. (shrink)

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used (...) for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics. (shrink)

A first-order temporal non-commutative logic TN[l], which has no structural rules and has some l-bounded linear-time temporal operators, is introduced as a Gentzen-type sequent calculus. The logic TN[l] allows us to provide not only time-dependent, resource-sensitive, ordered, but also hierarchical reasoning. Decidability, cut-elimination and completeness (w.r.t. phase semantics) theorems are shown for TN[l]. An advantage of TN[l] is its decidability, because the standard first-order linear-time temporal logic is undecidable. A correspondence theorem between TN[l] and a resource indexed non-commutative logic RN[l] (...) is also shown. This theorem is intended to state that “time” is regarded as a “resource”. (shrink)

A theorem for embedding a first-order linear- time temporal logic LTL into its intuitionistic counterpart ILTL is proved using Baratella-Masini's temporal extension of the Gödel-Gentzen negative translation of classical logic into intuitionistic logic. A substructural counterpart LLTL of ILTL is introduced, and a theorem for embedding ILTL into LLTL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic. These embedding theorems are proved syntactically based on Gentzen-type sequent calculi.

This study introduces refutation-aware Gentzen-style sequent calculi and Kripke-style semantics for propositional until-free linear-time temporal logic. The sequent calculi and semantics are constructed on the basis of the refutation-aware setting for Nelson’s paraconsistent logic. The cut-elimination and completeness theorems for the proposed sequent calculi and semantics are proven.