We present an axiomatisation for the first-order temporal logic with connectives Until and Since over the class of all linear flows of time. Completeness of the axiom system is proved.We also add a few axioms to find a sound and complete axiomatisation for the first order temporal logic of Until and Since over rational numbers time.

The aims of this paper are: (1) to present tense-logical versions of such classical notions as saturated and special models; (2) to establish several fundamental existence theorems about these notions; (3) to apply these powerful techniques to tense complexity.In this paper we are concerned exclusively with quantifiedK 1 (for linear time) with constant domain. Our present research owes much to Bowen [2], Fine [5] and Gabbay [6].

This paper explains how to obtain quantification over times in a tense logic in which all temporal distinctions are ultimately spelled out in terms of the two simple tense operators “it was the case that” and “it will be the case that.” The account of times defended here is similar to what is known as “linguistic ersatzism” about possible worlds, but there are noteworthy differences between these two cases. For example, while linguistic ersatzism would support actualism, the view of times (...) defended here does not support presentism. (shrink)

This article surveys some of the key issues that arise when one tries to use tense logic as a metaphysical theory of the nature of time. Topics discussed include basic tense logic, tense logic and verb tense, the structure of the time series, instants of time, quantified tense logic, and the expressive resources of tense logic.

According to Hans Kamp and Frank Vlach, the two-dimensional tense operators “now” and “then” are ineliminable in quantified tense logic. This is often adduced as an argument against tense logic, and in favor of an extensional account that makes use of explicit quantification over times. The aim of this paper is to defend tense logic against this attack. It shows that “now” and “then” are eliminable in quantified tense logic, provided we endow it with enough quantificational structure. The operators might (...) not be redundant in some other systems of tense logic, but this merely indicates a lack of quantificational resources and does not show any deep-seated inability of tense logic to express claims about time. The paper closes with a brief discussion of the modal analogue of this issue, which concerns the role of the actuality operator in quantified modal logic. (shrink)

Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us (...) to determine the congruence of the tense Nelson algebras and also to characterize the subdirectly irreducible tense Nelson algebras and particularly the simple tense Nelson algebras. Finally, we prove that there exists an equivalence between the category of IKt-algebras and the category of tense centered Nelson algebras. (shrink)

In this paper we present a category equivalent to that of tense Nelson algebras. The objects in this new category are pairs consisting of an IKt-algebra and a Boolean IKt-congruence and the morphisms are a special kind of IKt-homomorphisms. This categorical equivalence permits understanding tense Nelson algebras in terms of the better–known IKt-algebras.

We study a modal system $\overline{T}$, that extends the classical modal system T and whose language is provided with modal operators $M_{n}$ to be interpreted, in the usual kripkean semantics, as "there are more than n accessible worlds such that...". We find reasonable axioms for $\overline{T}$ and we prove for it completeness, compactness and decidability theorems.