suppose we have a model for a language in first order logic $ M=
prove or disprove the following:
a) if $ \forall x A $ is a sentence than $ M \models \forall x A $ iff for each closed noun $t$ $ M \models A\{t/x\} $ where $ A\{t/x\} $ is substitution of $t$ in place of $x$ in $A$
b) if $ \forall x A $ is a formula than $ M \models \forall x A $ iff for each closed noun $t$ $ M \models A\{t/x\} $
c) if $ \exists x A $ is a sentence than $ M \models \exists x A $ iff there exists a closed noun $t$ such that $ M \models A\{t/x\} $
d) if $ \exists x A $ is a formula than $ M \models \exists x A $ iff there exists a closed noun $t$ such that $ M \models A\{t/x\} $
i think a and c are true because in the case where x is the only free variable in A I can be thought as assignment to $ A\{t/x\} $ (where t is closed noun) but in b and d i have no idea
please help me with this exrecise