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In predicate logic, existential generalization [1] [2] (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition.
Examples: The column-14 operator (OR), shows Addition rule : when p =T (the hypothesis selects the first two lines of the table), we see (at column-14) that p ∨ q =T. We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.
According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x".
Axiom scheme for Existential Generalization. Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid.
In predicate logic, existential instantiation (also called existential elimination) [1] [2] is a rule of inference which says that, given a formula of the form () (), one may infer () for a new constant symbol c.
Existential generalization / instantiation; Exportation ... Example. It rains and the sun shines implies that there is a rainbow. Thus, if it rains, then the sun ...
The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions.Assume is a set of formulas, a formula, and () has been derived. The generalization rule states that () can be derived if is not mentioned in and does not occur in .
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter property is an inclusive logic.