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sav08:propositional_logic_semantics [2008/03/11 14:52]
vkuncak
sav08:propositional_logic_semantics [2015/04/21 17:30] (current)
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 We next define evaluation function: We next define evaluation function:
-\[+\begin{equation*}
     e : F \to (I \to {\cal B})     e : F \to (I \to {\cal B})
-\]+\end{equation*}
 ++++by recursion on formula syntax tree:| ++++by recursion on formula syntax tree:|
-\[\begin{array}{l}+\begin{equation*}\begin{array}{l}
   e(p)(I) = I(p), \mbox{ for } p \in V \\   e(p)(I) = I(p), \mbox{ for } p \in V \\
   e({'​\lnot '​}\>​ F)(I) = \lnot (e(F)(I)) \\   e({'​\lnot '​}\>​ F)(I) = \lnot (e(F)(I)) \\
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   e(F_1\> {'​\leftrightarrow '​}\>​ F_2)(I) = (e(F_1)(I) \leftrightarrow e(F_2)(I))   e(F_1\> {'​\leftrightarrow '​}\>​ F_2)(I) = (e(F_1)(I) \leftrightarrow e(F_2)(I))
 \end{array} \end{array}
-\]+\end{equation*}
  
 We wrote symbols like $'​\land '$ on left in quotes to emphasize that those are syntactic entities, in contrast to symbols like $\land$ on right-hand side that denote propositional operations given by truth tables (stated in [[Propositional Logic Informally]]). We wrote symbols like $'​\land '$ on left in quotes to emphasize that those are syntactic entities, in contrast to symbols like $\land$ on right-hand side that denote propositional operations given by truth tables (stated in [[Propositional Logic Informally]]).
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 We denote $e(F)(I) = {\it true}$ by We denote $e(F)(I) = {\it true}$ by
-\[+\begin{equation*}
     I \models F     I \models F
-\]+\end{equation*}
 and denote $e(F)(I) = {\it false}$ by and denote $e(F)(I) = {\it false}$ by
-\[+\begin{equation*}
    I \not\models F    I \not\models F
-\]+\end{equation*}
  
 ===== Validity and Satisfiability ===== ===== Validity and Satisfiability =====
  
 Formula is valid iff $\forall I. I \models F$.  We write this simply Formula is valid iff $\forall I. I \models F$.  We write this simply
-\[+\begin{equation*}
    ​\models F    ​\models F
-\]+\end{equation*}
  
 Formula is satisfiable iff $\exists I. I \models F$ Formula is satisfiable iff $\exists I. I \models F$