In first-order logic, a substitution is a total mapping σ: V → T from variables to terms many, : 73 : 445 but not all : 250 authors additionally require σ( x) = x for all but finitely many variables x. This fact implies the soundness of the deduction rule described in the previous section. If Φ is a tautology, and Θ is a substitution instance of Φ, then Θ is again a tautology. If a is a closed propositional formula we count a itself as its only substitution instance.Ī propositional formula is a tautology if it is true under every valuation (or interpretation) of its predicate symbols. In first-order logic, every closed propositional formula that can be derived from an open propositional formula a by substitution is said to be a substitution instance of a. In systems that use rules of transformation, a rule may include the use of a substitution instance for the purpose of introducing certain variables into a derivation. This is how new lines are introduced in some axiomatic systems. In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instance of a previous line of the derivation (Hunter 1971, p. 118). Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for symbols in φ, replacing each occurrence of the same symbol by an occurrence of the same formula.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |