Differences

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--- sav08:propositional_logic_syntax [2008/03/10 19:33]
vkuncak
+++ sav08:propositional_logic_syntax [2009/04/08 18:34]
philippe.suter
@@ Line 6: / Line 6: @@
    F ::= V \mid {\it false} \mid {\it true} \mid (F \land F) \mid (F \lor F) \mid (\lnot F) \mid (F \rightarrow F) \mid (F \leftrightarrow F)
 \]
-We denote the set of all propositional formulas given by the above grammar by ${\cal F}$.  This is a countable set: we can order all formulas in this set in a sequence (for example, by writing them down in binary alphabet and sorting the resulting strings alphabetically).
+We denote the set of all propositional formulas given by the above [[:Notes on Context-Free Grammars|context-free grammar]] by ${\cal F}$.  Each propositional formula is a finite sequence of symbols, given by the above context-free grammar.  The set ${\cal F}$ is a countable set: we can order all formulas in this set in a sequence (for example, by writing them down in binary alphabet and sorting the resulting strings alphabetically).
-Omitting parantheses:
+Omitting parentheses:
-  * $\land$, $\lor$ commutative
+  * $\land$, $\lor$ associative
   * priorities, from strongest-binding: $(\lnot)\ ;\ (\land, \lor)\ ;\ (\rightarrow, \leftrightarrow)$
-When in doubt, use parenthesis.
+When in doubt, use parentheses.
 Notation: when we write $F_1 \equiv F_2$ this means that $F_1$ and $F_2$ are identical formulas (with identical syntax trees).  For example, $p \land q \equiv p \land q$, but it is not the case that $p \land q \equiv q \land p$.
@@ Line 38: / Line 38: @@
 \[\begin{array}{l}
   FV(p) = \{ p \}, \mbox{ for } p \in V \\
+  FV(\lnot F) = FV(F) \\
   FV(F_1 \land F_2) = FV(F_1) \cup FV(F_2) \\
   FV(F_1 \lor F_2) = FV(F_1) \cup FV(F_2) \\
-  FV(\lnot F) = FV(F) \\
   FV(F_1 \rightarrow F_2) = FV(F_1) \cup FV(F_2) \\
   FV(F_1 \leftrightarrow F_2) = FV(F_1) \cup FV(F_2) \\