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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] (current) |
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Substitution is a maping formulas to formulas, | Substitution is a maping formulas to formulas, | ||
- | \[ | + | \begin{equation*} |
\sigma : D \to {\cal F} | \sigma : D \to {\cal F} | ||
- | \] | + | \end{equation*} |
where $D \subseteq {\cal F}$ is the domain of substitution, usually finite. We write it | where $D \subseteq {\cal F}$ is the domain of substitution, usually finite. We write it | ||
- | \[ | + | \begin{equation*} |
\sigma = \{F_1 \mapsto G_1,\ldots, F_n \mapsto G_n\} | \sigma = \{F_1 \mapsto G_1,\ldots, F_n \mapsto G_n\} | ||
- | \] | + | \end{equation*} |
Let ${\cal S}$ be set of all substitutions. | Let ${\cal S}$ be set of all substitutions. | ||
Line 26: | Line 26: | ||
For $F \in {\cal F}$ we write $F \sigma$ instead of $subst(\sigma)(F)$, so | For $F \in {\cal F}$ we write $F \sigma$ instead of $subst(\sigma)(F)$, so | ||
- | \[ | + | \begin{equation*} |
subst : {\cal S} \to ({\cal F} \to {\cal F}) | subst : {\cal S} \to ({\cal F} \to {\cal F}) | ||
- | \] | + | \end{equation*} |
//Variable substitution// is substitution where the domain is a subset of $V$ - it only replaces variables, not complex formulas. | //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 $\sigma = \{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\}$, | **Theorem:** For formula $F$, interpretation $I$ and variable substitution $\sigma = \{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\}$, | ||
- | \[ | + | \begin{equation*} |
e(subst(\{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\})(F))(I) = | e(subst(\{p_1 \mapsto F_1,\ldots,p_n \mapsto F_n\})(F))(I) = | ||
- | \] | + | \end{equation*} |
++++| | ++++| | ||
- | \[ | + | \begin{equation*} |
e(F)(I[p_1 \mapsto e(F_1)(I),\ldots,p_n \mapsto e(F_n)(I)]) | e(F)(I[p_1 \mapsto e(F_1)(I),\ldots,p_n \mapsto e(F_n)(I)]) | ||
- | \] | + | \end{equation*} |
++++ | ++++ | ||
- | Corollary (tautology instances): 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 $\sigma$. |
We say that two formulas are equivalent if $\models (F \leftrightarrow G)$. | 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 $e(F)(I) = e(G)(I)$. | + | |
+ | **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. | From the tautology instances Corrolary we obtain. | ||
- | Corollary: if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then $\models (subst(\sigma)(F) \leftrightarrow subst(\sigma)(G))$. | + | **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.++ | Does the theorem hold if $\sigma$ is ++not a variable substitution?|No, because a general substitution could produce an arbitrary formula.++ |