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sav08:first-order_logic_semantics [2008/04/02 20:46] vkuncak |
sav08:first-order_logic_semantics [2008/04/02 20:49] vkuncak |
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| ++++How do we evaluate quantifiers?| | ++++How do we evaluate quantifiers?| | ||
| \[\begin{array}{rcl} | \[\begin{array}{rcl} | ||
| - | e_F(\exists x.F)(I) &=& (\exists d \in D_I.\ e_F(F)(I[x \mapsto d])) \\ | + | e_F(\exists x.F)((D_I,\alpha_I)) &=& (\exists d \in D_I.\ e_F(F)((D_I,\alpha_I[x \mapsto d])) \\ |
| - | e_F(\forall x.F)(I) &=& (\forall d \in D_I.\ e_F(F)(I[x \mapsto d])) | + | e_F(\forall x.F)((D_I,\alpha_I)) &=& (\forall d \in D_I.\ e_F(F)((D_I,\alpha_I[x \mapsto d])) |
| \end{array} | \end{array} | ||
| \] | \] | ||
| - | where $I[x \mapsto d] = (D_I,\alpha^\prime_I)$ and | + | See [[Sets and relations#function update|function update notation]] for definition of $\alpha_I[x \mapsto d]$. |
| - | $\alpha^\prime_I(v) = \left \{ { {\alpha_I(v)~~\text{if}~ v \neq x} ~~ \atop {d ~~~~\text{if} ~ v=x}} \right. $ | + | |
| ++++ | ++++ | ||