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  • The set of formulas is inductively defined by the following rules:

  1. Preciate symbols : If P is an n-ary predicate symbol and t1,…,tn are terms then P(t1,…,tn) is a formula.

  2. Negation : If φ is a formula, then ¬φ is a formula

  3. Binary connectives : If φ and ψ are formulas, then (φ → ψ) is a formula. Same for other binary logical connectives.

  4. Quantifiers : If φ is a formula and x is a variable, then ∀xφ and ∃xφ are formulas.

    • Atomic formulas are formulas obtained only using the first rule

    • Example: If f is a unary function symbol, P a unary predicate symbol, and Q a ternary predicate symbol, then the following is a formula:

      ∀x∀y(P(f(x)) → ¬(P(x)) → Q(f(y), x, x)))

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