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sav08:propositional_logic_syntax [2008/03/10 10:58] vkuncak |
sav08:propositional_logic_syntax [2009/04/08 18:34] philippe.suter |
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====== Propositional Logic Syntax ====== | ====== Propositional Logic Syntax ====== | ||
- | Let $V$ be the set of propositional variables, denoted by non-terminal //V//. The context-free grammar of propositional logic formulas ${\cal F}$ is the following: | + | Let $V$ be a [[countable set]] of //propositional variables//, denoted by non-terminal //V//. The context-free grammar of propositional logic formulas ${\cal F}$ is the following: |
\[ | \[ | ||
- | {\cal F} ::= V \mid {\it false} \mid {\it true} \mid (F \land F) \mid (F \lor F) \mid (\lnot F) \mid (F \rightarrow F) \mid (F \leftrightarrow F) | + | F ::= V \mid {\it false} \mid {\it true} \mid (F \land F) \mid (F \lor F) \mid (\lnot F) \mid (F \rightarrow F) \mid (F \leftrightarrow F) |
\] | \] | ||
+ | We denote the set of all propositional formulas given by the above [[: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: $(\lnot)\ ;\ (\land, \lor)\ ;\ (\rightarrow, \leftrightarrow)$ | * 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 $p \land q \equiv q \land p$. | 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 $p \land q \equiv q \land p$. | ||
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\[\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//. |