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Author: Admin | 2025-04-28
Rule holds with ∃ in place of ∀.For example, in ∀x ∀y (P(x) → Q(x,f(x),z)), x and y occur only bound,[19] z occurs only free, and w is neither because it does not occur in the formula.Free and bound variables of a formula need not be disjoint sets: in the formula P(x) → ∀x Q(x), the first occurrence of x, as argument of P, is free while the second one, as argument of Q, is bound.A formula in first-order logic with no free variable occurrences is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil(x) is true must depend on what x represents. But the sentence ∃x Phil(x) will be either true or false in a given interpretation.Example: ordered abelian groups[edit]In mathematics, the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then:The expressions +(x, y) and +(x, +(y, −(z))) are terms. These are usually written as x + y and x + y − z.The expressions +(x, y) = 0 and ≤(+(x, +(y, −(z))), +(x, y)) are atomic formulas. These are usually written as x + y = 0 and x + y − z ≤ x + y.The expression is a formula, which is usually written as This formula has one free variable, z.The axioms for ordered abelian groups can be expressed as a set of sentences in the language. For example, the axiom stating that the group is commutative is usually written An interpretation of a first-order language assigns a denotation to each non-logical symbol (predicate symbol, function symbol, or constant symbol) in that language. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, each predicate is assigned a property of objects, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms, predicates, and formulas of the
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