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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.  ​
  
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 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.++
 
sav08/substitution_theorems_for_propositional_logic.txt · Last modified: 2015/04/21 17:30 (external edit)
 
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