Here is a problem I am trying to do from Kenneth Rosen's
is a tautology whenever are propositions, whenever
Here is the proof I attempted.
let be arbitrary propositions.Now consider
To prove this, we assume antecedent and since the antecedent is same
as consequent, it proves the above compound proposition . So
Induction case: Let be arbitrary. Also suppose
. To prove , take arbitrary
propositions . We have to prove
is a tautology. In other words, we have to prove it. Since its an implication,
and we also suppose that
It follows that, in particular,
So by modus ponens, we have
Since n is arbitrary, result holds generally.
Is it correct reasoning ?