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¬P(X) in the first clause, and in non-negated form P(a) in the second clause. X is an unbound variable, while a is a bound value (term). Unifying the two produces the substitution X ↦ a. Discarding the unified predicates, and applying this substitution to the remaining predicates (just Q(X), in this case), produces the conclusion: Q(a)
In predicate logic, generalization (also universal generalization, universal introduction, [1] [2] [3] GEN, UG) is a valid inference rule. It states that if ⊢ P ( x ) {\displaystyle \vdash \!P(x)} has been derived, then ⊢ ∀ x P ( x ) {\displaystyle \vdash \!\forall x\,P(x)} can be derived.
A statement can be called valid, i.e. logical truth, in some systems of logic like in Modal logic if the statement is true in all interpretations. In Aristotelian logic statements are not valid per se. Validity refers to entire arguments. The same is true in propositional logic (statements can be true or false but not called valid or invalid).
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.
More recent work on automated theorem proving has had a stronger basis in formal logic. An inference system's job is to extend a knowledge base automatically. The knowledge base (KB) is a set of propositions that represent what the system knows about the world. Several techniques can be used by that system to extend KB by means of valid inferences.
Logic studies valid forms of inference like modus ponens. Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and ...
Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. 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.
Constructive dilemma [1] [2] [3] is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too.