LYCOS RETRIEVER 
First-Order Logic
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The most common convention for equality is to include the equality symbol as a primitive logical symbol, and add the axioms for equality to the axioms for first-order logic. The equality axioms are
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This is a description of a proposed extension of SWRL to handle unary/binary first-order logic. This is intended to be a minimal extension that fits well with SWRL, OWL, and RDF. Transformation-based techniques to handle functions and n-ary predicates are suggested in an appendix.
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[I]n first-order logic, regardless the vocabulary, they are always satisfiable, e.g. {{1,2}, p={<1>}, a=1}. In Herbrand logic, enlarging the vocabulary to include an extra element is sufficient for satisfiability in this example: the vocabulary {p, a, b} allows the satisfying model {p(a)}.
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This web page gives the official rigorous definition of the proof rules for first order logic. In particular it uses the syntactical substitution defined earlier to make precide the official proof rules of first order logic. It ... relies on an earlier discussion of how to write a proof - either as a tree or as a structured list of statements. This makes precise the rules given in Definition 9.8 on page 122 of The Mathematics of Logic.
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There are several different conventions for using equality (or identity) in first-order logic. This section summarizes the main ones. The various conventions all give essentially the same results with about the same amount of work, and differ mainly in terminology.
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First-Order Logic