Can a sequent be valid if the conclusion contains atoms that are not in the premise?

Question

Is it possible to prove the validity of the following sequent:

$p \vdash (p \to q) \to q$

Here, our premise is that $p$ is True. The conclusion references a new atom, $q$.

I would argue that this is not valid, because I cannot deduce a conclusion i.e. I cannot derive a conclusion by means of deduction. Is my question even the right question to ask?

...Or is there some unexpected conclusion like "it is valid because anything follows yadda yadda"?

Answer 1

Yes. Here is a Natural Deduction derivation :

1) $p$ --- premise

2) $p \to q$ --- assumed [a]

3) $q$ --- from 1) and 2) by $\to$-elim

4) $(p \to q) \to q$ --- from 2) and 3) by $\to$-intro, discharging [a].

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We may check the validity of the proof in the following way : we assume that the premise is True.

Two cases :

(i) if $q$ is True, then also $(p \to q) \to q$ is True.

(ii) if $q$ is False, then $(p \to q)$ is False, and thus $(p \to q) \to q$ is again True.