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--- sav08:substitution_theorems_for_propositional_logic [2008/03/11 16:11]
vkuncak
+++ sav08:substitution_theorems_for_propositional_logic [2015/04/21 17:30]
@@ Line 1: / Line 1: @@
-====== Substitution Theorems for Propositional Logic ======
-Relate syntax and semantics.
-Called metatheorems, because they are not formulas //within// the logic, but formulas //about// the logic.
-From semantics we can prove the following lemma by induction.
-Lemma: if $I_1(p) = I_2(p)$ for every $p \in FV(F)$, then $e(F)(I_1)=e(F)(I_2)$.
-===== Substitutions =====
-Substitution is a maping formulas to formulas,
-\[
-   \sigma : D \to {\cal F}
-\]
-where $D \subseteq {\cal F}$ is the domain of substitution, usually finite.  We write it
-\[
-   \sigma = \{F_1 \mapsto G_1,\ldots, F_n \mapsto G_n\}
-\]
-Let ${\cal S}$ be set of all substitutions.
-Substitution operator //subst// takes such a map and a formula and returns a new formula.
-Recursive definition of //subst//:
-For $F \in {\cal F}$ we write $F \sigma$ instead of $subst(\sigma)(F)$, so
-\[
-   subst : {\cal S} \to ({\cal F} \to {\cal F})
-\]
-//Variable substitution// is substitution where the domain is a subset of $V$ - it only replaces variables, not complex formulas.
-**Theorem:** For formula $F$, interpretation $I$ and variable substitution $\sigma = \{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\}$,
-\[
-   e(subst(\{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\})(F))(I) =
-\]
-++++|
-\[
-   e(F)(I[p_1 \mapsto e(F_1)(I),\ldots,p_n \mapsto e(F_n)(I)])
-\]
-++++
-Corollary (tautology instances): if $\models F$, then $\models (subst(\sigma)(F))$ for every variable substitution $\sigma$.
-We say that two formulas are equivalent if $\models (F \leftrightarrow G)$.
-**Lemma**: If $\models (F \leftrightarrow G)$ then for every interpretation $I$ we have $e(F)(I) = e(G)(I)$.
-From the tautology instances Corrolary we obtain.
-Corollary: if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then $\models (subst(\sigma)(F) \leftrightarrow subst(\sigma)(G))$.
-Does the theorem hold if $\sigma$ is ++not a variable substitution?|No, because a general substitution could produce an arbitrary formula.++
-This theorem was about transforming formulas from the outside, ignoring the structure of certain subformulas.
-We next justify transforming formulas from inside.
-We say that $\sigma = \{F_1 \mapsto G_1,\ldots,F_n \mapsto G_n\}$ is //equivalence-preserving// iff for all $i$ where $1 \le i \le n$ we have $\models (F_i \leftrightarrow G_i)$.
-**Theorem on Substituting Equivalent Subformulas**: if $\sigma$ is equivalence-preserving, then for every formula $F$ we have $\models (F \leftrightarrow (subst(\sigma)(F))$.