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Predicates

Predicates enable an abstraction over Propositions so expressions may be formed that apply to a variable, rather than a concrete statement. They behave as functions that return either $$true$$ or $$false$$, depending on their variable(s). When all variables are bound to given values, predicates become propositions.

Quantifiers

Symbol Term Description
$$\forall$$ universal $$\forall x\ P(x)$$ symbolises the predicate $$P(x)$$ is true for every value in the universe of discourse
$$\exists$$ existential $$\exists x\ P(x)$$ the predicate $$P$$ holds for at least one $$x$$
$$\exists!$$ uniqueness $$\exists! x\ P(x)$$ expresses there exists a unique value $$x$$ for which $$P(x)$$ is true

Note: the precedence of quantifiers takes higher priority than any predicate connectives.