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

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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.

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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 ??,?,?,?,?,?,?

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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.

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Examples of Terms

• 0
• s(s(s(0)))
• nil
• cons(1,nil)
• cons(1,cons(2,nil))
• cons(1,cons(2,cons(3,nil)))

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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.

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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.

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Example Models ?X(?Y((mother(X) ? child_of(Y,X))
? loves(X,Y))) mother(mary) child_of(tom,
mary)
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