On Truth, Knowledge, Strings, and Paradoxes

One of the key issues in artiicial intelligence is the representation and processing of knowledge. The selection of an appropriate formalism for doing so is a crucial step in building a usable system. In this selection process one tries to nd an optimal balance between expressive power of the system and adequacy of the representation process on the one hand and eeciency and simplicity of the reasoning process on the other hand. Many knowledge representation formalisms are built on top of some logical system. A powerful and well-understood system is classical rst-order logic. Since it has some deecien-cies, extensions and alternatives have been studied. In particular modal logics have been studied in great detail in recent years. In recent work I advocate a system that is more powerful but simpler than modal logic. It is a simple extension of classical rst-order logic, namely rst-order logic plus strings. Although known for quite a while and in spite of its positive features, it has not attracted much attention since it is easy to formulate self-referential paradoxes within this system. I have studied the reason for self-referential paradoxes and came up with a semantical condition on the connectives such that these paradoxes can be excluded. This leads to a powerful three-valued system that makes it possible to use reasoning techniques for classical rst-order logic. First-order logic with strings is a conservative extension of rst-order logic in which strings as in short(\cat") stands for short(pair (:c; pair (:a; :t))). With such a construction it is possible to deene syntactic and semantical predicates about the language itself. In particular it is possible to deene the syntax of the language within the language itself. Details of such a construction can be found in GN87, Chapter 10]. Although it is intended that strings and the objects they stand for are closely related, we must carefully distinguish between them. There is an intuitive diierence between long(John) and long(\John") (where long is assumed to be a polymorphic predicate symbol that is true for persons over 195cm and strings consisting of at least eight characters.) In such a setting it is easy to express a problematic concept like the liar sentence: This sentence is false. If we assume a standard rst-order semantics there is no problem with the sentence above, since strings like \L" are ordinary ground terms in the language and have nothing to do with their counterparts like …