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sav08:propositional_logic_syntax [2008/03/10 16:21] vkuncak |
sav08:propositional_logic_syntax [2008/03/18 13:54] vkuncak |
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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) | 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 ${\cal F}$. | + | We denote the set of all propositional formulas given by the above grammar by ${\cal F}$. Each propositional formula is a finite sequence of symbols. 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 parantheses: | ||
- | * $\land$, $\lor$ commutative | + | * $\land$, $\lor$ associative |
* priorities, from strongest-binding: $(\lnot)\ ;\ (\land, \lor)\ ;\ (\rightarrow, \leftrightarrow)$ | * priorities, from strongest-binding: $(\lnot)\ ;\ (\land, \lor)\ ;\ (\rightarrow, \leftrightarrow)$ | ||
When in doubt, use parenthesis. | When in doubt, use parenthesis. | ||
Line 38: | Line 38: | ||
\[\begin{array}{l} | \[\begin{array}{l} | ||
FV(p) = \{ p \}, \mbox{ for } p \in V \\ | FV(p) = \{ p \}, \mbox{ for } p \in V \\ | ||
+ | FV(\lnot F) = FV(F) \\ | ||
FV(F_1 \land F_2) = FV(F_1) \cup FV(F_2) \\ | FV(F_1 \land F_2) = FV(F_1) \cup FV(F_2) \\ | ||
FV(F_1 \lor F_2) = FV(F_1) \cup FV(F_2) \\ | FV(F_1 \lor F_2) = FV(F_1) \cup FV(F_2) \\ | ||
- | FV(\lnot F) = FV(F) \\ | ||
FV(F_1 \rightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | FV(F_1 \rightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | ||
FV(F_1 \leftrightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | FV(F_1 \leftrightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | ||
\end{array}\] | \end{array}\] | ||
+ | If $FV(F) = \emptyset$, we call $F$ a //ground formula//. |