## IRCS Technical Reports Series

#### Document Type

Technical Report

#### Date of this Version

May 1996

#### Abstract

The language we consider is that of classical first order logic augmented with the unary modal operator □. Sentences of this language are regarded as true or false in a *knowledge-base KB*, which is any finite set of □-free formulas. Truth of □α in *KB* is understood as that α is true in all classical models of *KB* and this interpretation is intended to capture the intuition "we *know* that α" behind □α.

The resulting logic is, in general, undecidable and not even semidecidable. However, there is a natural fragment of the above language, called the *constructive language*, which yields a decidable logic. The only syntactic constraint in the constructive language is that there exists x should always be followed by □. That is, we are not allowed to simply say "there is *x* such that ..." and we can only say "there is *x* for which we know that ...". Under this constraint, truth of there existsxα(x) will always imply that an object *x* for which α(x) holds not only exists, but can be effectively found. This is generally what we want of there exists in practical applications: knowing that "there exists a combination *c* that opens safe *S*" has no significance unless such a combination *c* can actually be found, which, in our semantics, will be equivalent to saying that there is *c* for which we know that *c* opens *S*. So, it is only truth of the sentence there existsc□*OPENS*(c,*S*) that really matters, and the latter, unlike there existsc□*OPENS*(c,*S*) is a perfectly legal formula of the constructive language.

I introduce a decidable sequent system *C K N* in the constructive language and prove its soundness and completeness with respect to the above semantics.

**Date Posted:** 26 August 2006

## Comments

University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-96-06.