The Finite Satisfiability Problem for Two-Variable, First-Order Logic with one Transitive Relation is Decidable

We consider the two-variable fragment of first-order logic with one distinguished binary predicate constrained to be interpreted as a transitive relation. The finite satisfiability problem for this logic is shown to be decidable, in triply exponential non-deterministic time. The complexity falls to doubly exponential non-deterministic time if the distinguished binary predicate is constrained to be interpreted as a partial order

An Introduction to Ontology

Analytical philosophy of the last one hundred years has been heavily influenced by a doctrine to the effect that one can arrive at a correct ontology by paying attention to certain superficial (syntactic) features of first-order predicate logic as conceived by Frege and Russell. More specifically, it is a doctrine to the effect that the key to the ontological structure of reality is captured syntactically in the ‘Fa’ (or, in more sophisticated versions, in the ‘Rab’) of first-order logic, where ‘F’ stands for what is general in reality and ‘a’ for what is individual. Hence “f(a)ntology”. Because predicate logic has exactly two syntactically different kinds of referring expressions—‘F’, ‘G’, ‘R’, etc., and ‘a’, ‘b’, ‘c’, etc.—so reality must consist of exactly two correspondingly different kinds of entity: the general (properties, concepts) and the particular (things, objects), the relation between these two kinds of entity being revealed in the predicate-argument structure of atomic formulas in first-order logic

Temporal logic with predicate abstraction

A predicate linear temporal logic LTL_{\lambda,=} without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LTL_{\lambda,=} can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LTL_{\lambda,=} for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LTL_{\lambda,=} is not recursively axiomatizable and, therefore, fully automated verification of LTL_{\lambda,=} specifications is not, in general, possible.Comment: 14 pages, 4 figure