For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a signature, which may or may not be satisfiable with respect to a given model.
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The "'Herbrand base "'is the set of all ground atoms that can be formed from predicate symbols in the original set of clauses and terms in its Herbrand universe.
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Let a first-order language be given, with C the set of constant symbols, V the set of ( individual ) variables, F the set of functional operators, and P the set of predicate symbols.
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This means that, given an interpretation, a predicate symbol, and " n " elements of the domain of discourse, one can tell whether the predicate is true of those elements according to the given interpretation.
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For example, an interpretation " I ( P ) " of a binary predicate symbol " P " may be the set of pairs of integers such that the first one is less than the second.
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Because the first-order interpretations described here are defined in set theory, they do not associate each predicate symbol with a property ( or relation ), but rather with the extension of that property ( or relation ).
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Consider a first order language for algebraic structures with a monadic predicate symbol S . Then a " fuzzy subalgebra " is a fuzzy model of a theory containing, for any " n "-ary operation h, the axioms
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A proposition is ( i ) a predicate symbol applied to the number of terms required by its arity, ( ii ) an operator applied to the number of propositions required by its arity, or ( iii ) a quantifier applied to a proposition.
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The interpretation of such a language would define a domain ( universe of discourse ); assign an element of the domain to each individual constant; assign the donation in the domain of some property to each unary ( one-place ) predicate symbol.
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#How should we call a first / second order language, whose all symbols are logical ( like connectives quantifiers variables and brackets and identity ), i . e . when it contains neither constants nor function symbols nor predicate symbols ( but does contain the identity symbol )?
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