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Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet

Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet. Countable set of predicate symbols , each with specified arity  0. Countable set of function symbols , each with specified arity  0. Function symbols with arity 0 are also called constants or individual symbols .

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Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet

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  1. Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet • Countable set of predicate symbols, each with specified arity  0. • Countable set of function symbols, each with specified arity  0. Function symbols with arity 0 are also called constants or individual symbols. • Countable set of variables.

  2. 1. Alphabet (Continued) • (Consistent with Prolog, we will begin variables with an upper-case letter and predicate/function symbols with a lower-case letter.) • Logical symbols: ,,,,,,

  3. 2. Terms • A variable is a term. • If f is a function symbol of arity n and t1,…,tn are terms then f(t1,…,tn) is a term.

  4. Examples of Terms • 0 • s(s(s(0))) • nil • cons(1,nil) • cons(1,cons(2,nil)) • cons(1,cons(2,cons(3,nil)))

  5. 3. Formulas • If p is a predicate symbol of arity n and t1,…,tn are terms, then p(t1,…,tn) is an atomic formula. • If a and b are formulas then so are a, ab, ab, ab, ab, ab. • If X is a variable and a is a formula then Xa and Xa are formulas. We say that X is quantified in the formulas Xa and Xa.

  6. Some Notes • Predicates of arity 0 are also called propositions, the only atomic formulas allowed in propositional logic. • An expression is a term or formula. A formula with no free (unquantified) variables is a sentence.

  7. Example: Models X(Y((mother(X)  child_of(Y,X))  loves(X,Y))) mother(mary) child_of(tom,mary)

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