LYCOS RETRIEVER 
First-Order Logic: Sentences
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[O]rder logic is "classical" logic. Each sentence is true or false, and quantification is allowed so that you can say things like, "For all x, if x is a dog, then x has a tail." There is a unary negation operator and standard connectives are used: conjunction, disjunction and implication. Determining whether or not a first-order theory is consistent ("has a model", in slightly more technical terms) is in general semidecidable. This means that there are algorithms that are guaranteed to terminate with the answer "yes" if the theory is consistent, but might never terminate if the theory is inconsistent. There can be no algorithm that will always terminate in both cases.
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Herbrand logic differs from first-order logic solely in the structures it considers to be models. The semantics of a given set of sentences is defined to be the set of Herbrand models that satisfy it, for a given vocabulary.
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Second-order logic allows quantification over subsets and relations, or in other words over all predicates. For example, the axiom of extensionality can be stated in second-order logic as [X] = y ≡def P (P(x) ↔ P(y)). The strong semantics of second-order logic give such sentences a much stronger meaning than first-order semantics.
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The notion of a model for a set of sentences in first order logic has a natural definition. At the risk of some oversimplification there is a universe U, which contains all the objects referred to by the constans and on which all the functions and predicates are defined. A sentence is satisfiable if the varaibles that appear in it can be chosen from the Universe so as to make it true. This leaves only quantification. Here is a quick semantics for the quantifiers.
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The formation rules define the terms and formulas of first order logic. When terms and formulas are represented as strings of symbols, these rules can be used to write a formal grammar for terms and formulas. The concept of free variable is used to define the sentences as a subset of the formulas.
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