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sav08:substitution_theorems_for_propositional_logic [2008/03/11 16:09] vkuncak |
sav08:substitution_theorems_for_propositional_logic [2008/03/11 16:11] vkuncak |
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- | Corollary: if $\models F$, then $\models (subst(\sigma)(F))$ for every variable substitution $\sigma$. | + | **Corollary (tautology instances):** if $\models F$, then $\models (subst(\sigma)(F))$ for every variable substitution $\sigma$. |
- | Corollary: if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then $\models (subst(\sigma)(F) \leftrightarrow subst(\sigma)(G)$. | + | We say that two formulas are equivalent if $\models (F \leftrightarrow G)$. |
- | Does the theorem hold if $\sigma$ is ++not a variable substitution?|No, because a general substitution could produce an arbitrary formula.++ | + | **Lemma:** If $\models (F \leftrightarrow G)$ then for every interpretation $I$ we have $e(F)(I) = e(G)(I)$. |
- | We say that two formulas are equivalent if $\models (F \leftrightarrow G)$. | + | From the tautology instances Corrolary we obtain. |
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+ | **Corollary:** if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then $\models (subst(\sigma)(F) \leftrightarrow subst(\sigma)(G))$. | ||
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+ | 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. | This theorem was about transforming formulas from the outside, ignoring the structure of certain subformulas. | ||
We next justify transforming formulas from inside. | We next justify transforming formulas from inside. | ||
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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 $\models (F_i \leftrightarrow G_i)$. | 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 $\models (F_i \leftrightarrow G_i)$. | ||
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- | **Lemma**: If $\models (F \leftrightarrow G)$ then for every interpretation $I$ we have $e(F)(I) = e(G)(I)$. | ||
**Theorem on Substituting Equivalent Subformulas**: if $\sigma$ is equivalence-preserving, then for every formula $F$ we have $\models (F \leftrightarrow (subst(\sigma)(F))$. | **Theorem on Substituting Equivalent Subformulas**: if $\sigma$ is equivalence-preserving, then for every formula $F$ we have $\models (F \leftrightarrow (subst(\sigma)(F))$. | ||