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First-Order Logic: Theorems
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[O]rder logic (FOL) is one answer to the last question. If some theorem of a mathematical system can be proven from the axioms of that system written in first-order logic, the theorem holds. Because of this commitment to mathematical truth in general, FOL was defined so that it can be used across a broad spectrum of mathematics. It is not tailored to the study of integers, real numbers, geometry, topology, or any particular branch of mathematics.
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[O]rder logic can be useful in the creation of computer programs. It is ... of interest to researchers in artificial intelligence (AI). There are more powerful forms of logic, but first-order logic is adequate for most everyday reasoning. The Incompleteness Theorem, proven in 1930, demonstrates that first-order logic is in general undecidable. That means there exist statements in this logic form that, under certain conditions, cannot be proven either true or false.
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Theorem proving for first-order logic is one of the most mature subfields of automated theorem proving. The logic is expressive enough to allow the specification of arbitrary problems, often in a reasonably natural and intuitive way. On the other hand, it is still semidecidable, and a number of sound and complete calculi have been developed, enabling fully automated systems. In 1965 J. Alan Robinson achieved an important breakthrough with his resolution approach; to prove a theorem it tries to refute the negated theorem, in a goal-directed way, resulting in a much more efficient method to automatically prove theorems in FOL. More expressive logics, such as higher-order and modal logics, allow the convenient expression of a wider range of problems than first-order logic, but theorem proving for these logics is less well developed.
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It is important to bear in mind that the results quoted above apply only to first-order logic, and not to first-order theories in general. The latter include not only logical axioms, but ... non-logical ones, and theorems such as Gödel's completeness theorem do not apply to them. In many first-order theories, and in particular every first-order theory that includes the natural numbers, Gödel's incompleteness theorems show, for example, that there are true statements which cannot be proven.
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The purpose of this paper is to show that first order logic is adequate for formalizing functional, multivalued and mutual dependencies in relational data bases. Advantages of using logic instead of tailored formal systems are presented. This paper is decomposed into four sections. The first one presents some notions of logic and theorem proving which are relevant to this study. In the second section, a way to analyze data bases in terms of logic is briefly indicated. The third section deals with the expression of dependency statements as formulas of logic.
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The inference rule modus ponens is the only one required here for a complete formalization of first-order logic. It states that if φ and φ ψ are both theorems, then ψ is a theorem. This can be written as following;
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