Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:herbrand_universe_for_equality [2008/04/02 20:40] vkuncak |
sav08:herbrand_universe_for_equality [2008/04/02 20:45] vkuncak |
||
---|---|---|---|
Line 15: | Line 15: | ||
===== Constructing Model for Formulas with Equality ===== | ===== Constructing Model for Formulas with Equality ===== | ||
- | Let $S$ be a set of formulas in first-order logic with equality and $S'$ result of replacing '=' with 'eq' in $S$. Suppose that $S \cup AxEq$ is satisfiable. Let $(GT,I_H)$ be Herbrand model for $S \cup AxEq$. We construct a new model using //quotient// construction, described as follows. | + | Let $S$ be a set of formulas in first-order logic with equality and $S'$ result of replacing '=' with 'eq' in $S$. Suppose that $S \cup AxEq$ is satisfiable. Let $(GT,I_H)$ be Herbrand model for $S \cup AxEq$. We construct a new model using //quotient// construction, described as follows (recall notation in [[First-Order Logic Semantics]]). |
For each element $t \in GT$, define | For each element $t \in GT$, define | ||
Line 35: | Line 35: | ||
In particular, when $R$ is $eq$ we have | In particular, when $R$ is $eq$ we have | ||
- | $I_Q(eq) = $ ++| \{ ([t_1],[t_2]) \mid (t_1,t_2) \in I_H(eq) \} = \{ (a,a) \mid a \in [GT] \}$ | + | $I_Q(eq) = $ ++| $\{ ([t_1],[t_2]) \mid (t_1,t_2) \in I_H(eq) \} = \{ (a,a) \mid a \in [GT] \}$ |
that is, the interpretation of eq in $([GT],I_Q)$ is equality. | that is, the interpretation of eq in $([GT],I_Q)$ is equality. |