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sav08:first-order_logic_semantics [2008/04/02 20:49] vkuncak |
sav08:first-order_logic_semantics [2008/04/02 21:27] vkuncak |
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| We generalize this notion as follows: if $I$ is an interpretation and $T$ is a set of first-order formulas, we write $e_S(T)(I)={\it true}$ iff for every $F \in T$ we have $e_F(F)(I)={\it true}$ (set is treated as infinite conjunction). This is a generalization because $e_S(\{F\})(I) = e_F(F)(I)$. | We generalize this notion as follows: if $I$ is an interpretation and $T$ is a set of first-order formulas, we write $e_S(T)(I)={\it true}$ iff for every $F \in T$ we have $e_F(F)(I)={\it true}$ (set is treated as infinite conjunction). This is a generalization because $e_S(\{F\})(I) = e_F(F)(I)$. | ||
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| + | **A terminological note:** in algebra, an interpretation is often called a //structure//. Instead of using $\alpha$ mapping language ${\cal L} = \{ f_1,\ldots,f_n, R_1,\ldots,R_m\}$ to interpretation of its symbols, the structure is denoted by a tuple $(D,\alpha(f_1),\ldots,\alpha(f_n),\alpha(R_1),\ldots,\alpha(R_n))$. For example, an interpretation with domain ${\can N}$, with one binary operation whose interpretation is $+$ and one binary relation whose interpretation is $\leq$ can be written as $({\cal N},+,\leq)$. This way we avoid writing $\alpha$ all the time, but it becomes more cumbersome to describe correspondence between structures. | ||
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| ===== Examples ===== | ===== Examples ===== | ||