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.