Lets say I have the sentence
$$\phi=(p(x)\wedge(q(y)\supset R(z))) \supset (\forall u.\exists w.(T(u.w) \vee \neg(u=w)))$$
I am trying to find the sub formulae of this.
Is $$(\forall u.\exists w.(T(u.w) \vee \neg(u=w)))$$ a sub formula?
In the example I was given, substitutions had been made and they were also deemed to be sub formulae which is correct, but why would you make a substitution when finding the sub formulae?
The subformulae are also the sub-formulae of the subformulae ...
The set $Sub(\phi)$ of te subformulae of $\phi$ includes $\phi$ iteself.
Then the two subformuale "linked" by the main connective $\supset$, i.e. :
$(p(x)∧(q(y)⊃R(z)))$ and $∀u.∃w.(T(u,w)∨¬(u=w))$.
In turn, the first one has two subformulae :
$p(x)$ and $(q(y)⊃R(z))$
and the last one is again "decomposable" into : $q(y)$ and $R(z)$.
Regarding : $∀u.∃w.(T(u,w)∨¬(u=w))$, we have the subformula : $∃w.(T(u,w)∨¬(u=w))$ and this in turn has the subformula : $T(u,w)∨¬(u=w)$.
This one has two subformulae :
$T(u,w)$ and $¬(u=w)$
and finally we have : $(u=w)$.
