First-Order Logic

• Wolfgang Rautenberg
Chapter
Part of the Universitext book series (UTX)

Abstract

Mathematics and some other disciplines such as computer science often consider domains of individuals in which certain relations and operations are singled out. When using the language of propositional logic, our ability to talk about the properties of such relations and operations is very limited. Thus, it is necessary to refine our linguistic means of expression, in order to procure new possibilities of description. To this end, one needs not only logical symbols but also variables for the individuals of the domain being considered, as well as a symbol for equality and symbols for the relations and operations in question. First-order logic, sometimes called also predicate logic, is the part of logic that subjects properties of such relations and operations to logical analysis.

Linguistic particles such as “for all” and “there exists” (called quantifiers) play a central role here, whose analysis should be based on a well prepared semantic background. Hence, we first consider mathematical structures and classes of structures. Some of these are relevant both to logic (in particular model theory) and to computer science. Neither the newcomer nor the advanced student needs to read all of 2.1, with its mathematical flavor, at once. The first five pages should suffice. The reader may continue with 2.2 and later return to what is needed.

Keywords

Free Variable Propositional Logic Axiom System Explicit Definition Constant Symbol
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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D. Hilbert, P. Bernays, Grundlagen der Mathematik, I, II, Berlin 1934, 1939, 2⟨{ nd}⟩ ed. Springer, Vol. I 1968, Vol. II 1970.Google Scholar