Good evening!
I have got a question concerning semantics with quantifiers.
We have a Syntax:
S = {$animal$: Type}
OP = $\emptyset$
REL = {
Is_dog: $animal \to Prop$
Needs_care: $animal \to Prop$
}
Variables: $X_{animal} = \{x\}$
And an expression:
$t_{1} =_{df} \forall x ($Is_dog($x$)$\to $Needs_care($x$)$)$
X-valuation is:
$ \beta_{1}(x) = 1 $
Semantic mappings:
Is_dog $ =_{df} \left\{\begin{array}{ll} \top \ if \ x \ is \ a \ dog \\ \bot \ if \ x \ is \ no \ dog \end{array}\right. $
Needs_care $ =_{df} \left\{\begin{array}{ll} \top \ if \ x \ needs \ care \\ \bot \ if \ x \ doesn't \ need \ care \end{array}\right. $
Now I wonder how to determine the value of $ [[t_{1}]]_{\beta_{1}} $.
Has anyone an idea?
Thanks