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sav08:first-order_logic_syntax [2008/03/18 12:12] vkuncak |
sav08:first-order_logic_syntax [2008/03/18 12:17] vkuncak |
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| $FV$ denotes the set of free variables in the given propositional formula and can be defined recursively as follows: | $FV$ denotes the set of free variables in the given propositional formula and can be defined recursively as follows: | ||
| - | \[\begin{array}{l} | + | \[\begin{array}{rcl} |
| - | FV(x) = \{ x \}, \mbox{ for } x \in V \\ | + | FV(x) &=& \{ x \}, \mbox{ for } x \in V \\ |
| - | FV(\lnot F) = FV(F) \\ | + | FV(f(t_1,\ldots,t_n) &=& F(t_1) \cup \ldots \cup F(t_n) \\ |
| - | FV(F_1 \land F_2) = FV(F_1) \cup FV(F_2) \\ | + | FV(R(t_1,\ldots,t_n) &=& F(t_1) \cup \ldots \cup F(t_n) \\ |
| - | FV(F_1 \lor F_2) = FV(F_1) \cup FV(F_2) \\ | + | FV(t_1 = t_2) &=& F(t_1) \cup F(t_2) \\ |
| - | FV(F_1 \rightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | + | FV(\lnot F) &=& FV(F) \\ |
| - | FV(F_1 \leftrightarrow F_2) = FV(F_1) \cup FV(F_2) \\ | + | FV(F_1 \land F_2) &=& FV(F_1) \cup FV(F_2) \\ |
| - | FV(\forall x.F) = FV(F) \setminus \{x\} \\ | + | FV(F_1 \lor F_2) &=& FV(F_1) \cup FV(F_2) \\ |
| - | FV(\exists x.F) = FV(F) \setminus \{x\} | + | FV(F_1 \rightarrow F_2) &=& FV(F_1) \cup FV(F_2) \\ |
| + | FV(F_1 \leftrightarrow F_2) &=& FV(F_1) \cup FV(F_2) \\ | ||
| + | FV(\forall x.F) &=& FV(F) \setminus \{x\} \\ | ||
| + | FV(\exists x.F) &=& FV(F) \setminus \{x\} | ||
| \end{array}\] | \end{array}\] | ||
| If $FV(F) = \emptyset$, we call $F$ a //closed formula//. | If $FV(F) = \emptyset$, we call $F$ a //closed formula//. | ||