Why is this "proof" actually not a proof of SL's soundness?

Question

Here, I'll consider only sentential logic as : sentential language and semantics + natural deduction rules.

Alledged "proof" :

(1) each rule of natural deduction is guaranteed or justified by a tautology

(2) therefore if I assume A, B, C and derive D using any rule, then the conclusion :

                  (A & B & C)  --> D 

will be a tautology

(3) therefore, the natural deduction system cannot prove anything but tautologies, it is sound.

My " proof" simply says that SL is sound because each individual rule is guaranteed by a corresponding tautology.

My question is : can you please explain me why proving SL's soundness is not as easy as that?

I suppose that if it is not as easy as that, it is because it could be the case that (1) each rule is guaranteed by a corresponding tautology, and (2) that however the whole system is not sound. But that is precisely what I can not understand.

It is not very far from real... the only issue is that some Natural Deduction rules have a more complex structure than simple tautologies. — Mauro ALLEGRANZA, Apr 06 '19 at 18:40
That's true. I had not noticed this point (which is clearly explained in the IEP link just given). — , Apr 06 '19 at 18:58

score 1 · Accepted Answer · answered Apr 07 '19 at 10:22

The natural deduction rules aren't guaranteed by tautologies strictly speaking. The rule of conjunction introduction for example doesn't, nor does conditional introduction. You might feel tempted to believe that the tautology (p$\rightarrow$(q$\rightarrow$(p$\land$q))) justifies conditional introduction, but if you assume 'p' and 'q', and you have the above tautology, you need modus ponens to reach (p$\land$q). Thus, you haven't just used the above tautology. Conditional introduction ends up even more complicated.

Why is this "proof" actually not a proof of SL's soundness?

1 Answers1