Fluted logic is the restriction of pure predicate logic to formulas in which variables play no essential role. Although fluted logic is significantly weaker than pure predicate logic, it is of interest because it seems closely to parallel natural logic, the logic that is conducted in natural language. It has been known since 1969 that if conjunction in fluted formulas is restricted to subformulas of equal arity, satisfiability is decidable. However, the decidability of sublogics lying between this restricted (homogeneous) fluted (...) logic and full predicate logic remained unknown. In 1994 it was shown that the satisfiability of fluted formulas without restriction is decidable, thus reducing the unknown region significantly. This paper further reduces the unknown region. It shows that fluted logic with the logical identity is decidable. Since the reflection functor can be defined in fluted logic with identity, it follows that fluted logic with the reflection functor also lies within the region of decidability. Relevance to natural logic is increased since the identity permits definition of singular predicates, which can represent anaphoric pronouns. (shrink)

It is well-known that first-order logic is semi-decidable. Therefore, first-order logic is less than ideal for computational purposes (computer science, knowledge engineering). Certain fragments of first-order logic are of interest because they are decidable. But decidability is gained at the cost of expressiveness. The objective of this paper is to investigate inexpressiveness of fragments that have received much attention.

The fluted fragment is a fragment of first-order logic (without equality) in which, roughly speaking, the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that this fragment has the finite model property. We consider extensions of the fluted fragment with various numbers of transitive relations, as well as the equality predicate. In the presence of one transitive relation (together with equality), the finite model property is lost; nevertheless, (...) we show that the satisfiability and finite satisfiability problems for this extension remain decidable. We also show that the corresponding problems in the presence of two transitive relations (with equality) or three transitive relations (without equality) are undecidable, even for the two-variable sub-fragment. (shrink)

By a fragment of a natural language, we understand a collection of sentences forming a naturally delineated subset of that language and equipped with a semantics commanding the general assent of its native speakers. By the semantic complexity of such a fragment, we understand the computational complexity of deciding whether any given set of sentences in that fragment represents a logically possible situation. In earlier papers by the first author, the semantic complexity of various fragments of English involving at most (...) transitive verbs was investigated. The present paper considers various fragments of English involving ditransitive verbs and determines their semantic complexity. (shrink)