How many times a premise can be use while making deductions with arguments? Exercise 3.9.9 In DMFP Book.

Question

I've already mentioned in a previous question I am going through the book Discrete Mathematics and Functional Programming by Thomas VanDrunen.

In exercise 3.9.9 from the aforementioned book I've used a premise two times, is it allowed to do so? or is it the case that a given premise should be used only once?

Here's what I am talking about:

(a) $p \implies u$
(b) $x$
(c) $\lnot(p \lor w) \implies r$
(d) $q \implies w$
(e) $x \implies t$
(f) $w \implies u$
(g) $r \lor s$
(h) $r \implies F$
(i) $\therefore t \land s \land u$

This is the way I have proceeded.

(i) $p \implies u$
Using (a), (d), and the transitivity law.

(ii) $t$
By (b), (e), and Modus Ponens.

(iii) $\lnot r$
By (h) and the contradiction law.

(iv) $p \lor w$
Using (c), (iii), Modus Tollens, and Double Negative Law.

(v) $u$
By (iv), (i), (f), and Division Into Cases.

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(vi) $s$
Given (iii), (g), and Elimination. And here's exactly where my question arises, since I already used (iii) to deduce (iv), can I use it again over here? Or is it the case that the premises should be used only once? I mean, (iii) remains true the whole time, I don't know why don't use it again.

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(vii) $\therefore t \land s \land u$
By (ii), (vi), (v), and Conjunction.

Thanks a lot for your insights.

--
Caleb

Answer 1

Yes, in the logic you are working with you can use a premise as many times as you like. If this were not allowed, tautologies such as $q \Rightarrow q \land q$ would be unprovable. This is closely analogous (via the Curry-Howard correspondence) to the way that in most programming languages, you can use a parameter as many times as you like.

There are logical systems which place restrictions on how many times you can use a premise. E.g., in affine logic, a premise can be used exactly once and $q \Rightarrow q \land q$ is unprovable. These logics correspond to programming languages where parameters represent resources that are consumed when they are used.