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A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ∀ {\displaystyle \forall } in the first order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P ...
An open formula can be transformed into a closed formula by applying a quantifier for each free variable. This transformation is called capture of the free variables to make them bound variables. For example, when reasoning about natural numbers, the formula "x+2 > y" is open, since it contains the free variables x and y.
Example. In a given propositional logic, a formula can be defined as follows: Every propositional variable is a formula. Given a formula X, the negation ¬X is a formula. Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), the expression (X b Y) is a formula. (Note the parentheses.)
The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ ∧ ψ is satisfied if and only if both φ and ψ are satisfied. This leaves the issue of how to interpret formulas of the form ∀ x φ(x) and ∃ x φ(x). The domain of discourse forms the range for these quantifiers.
For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers ...
Examples of sentences that are (or make) true statements: "Socrates is a man." "A triangle has three sides." "Madrid is the capital of Spain." Examples of sentences that are also statements, even though they aren't true: "All toasters are made of solid gold." "Two plus two equals five." Examples of sentences that are not (or do not make ...
It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier (" ∃x" or "∃(x)" or "(∃x)" [1]). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.