# Can Deduction for a Valid Argument produce the wrong conclusion?

Source: 1. p 287, 5. p 261. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.

[1.] Predicate is the reason we started on deductions. In Sentential, remember, we can verify that an argument is valid by using truth tables or the short-cut method. With Predicate, we have no such tool. We can show that an argument is invalid if we exhibit a universe in which the premises are true and the conclusion is false, but (until now) we have had no means of showing that an argument in Predicate is valid.

1. [Write a deduction to prove the Disjunctive Syllogism, hereafter DS.]

Per 1, I already know how DS can be proven more quickly with (Indirect/Short-Cut) Truth Tables; so a deduction (which would take longer) is needless. But in general, for Valid Arguments, are Deductions fallible? For Sentential Logic, are (Indirect/Short-Cut) Truth Tables the only 100% safe method?

Consider the following. Because 4 is the the Principle of Explosion, anything can be deduced in 5: so what if you instead wrote (wrongly) ¬Q (instead of the correct Q)? Then the Deduction would continue and conclude with ¬Q, but dangerously and deceptively, would fail to reveal this error?

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• @MoziburUllah Thanks. I know that, but I meant to exclude invalid arguments. Aug 14, 2016 at 22:14
• I'm pretty sure that if you did ~Q the deduction wouldn't conclude with that, but would not be finished. You still have the second branch ending with Q, so you can't conclude ~Q. So if you put the "wrong" thing, you just don't manage to complete the deduction. Aug 14, 2016 at 22:14

No.

Sentential logic (and also first order predicate logic) with natural deduction is sound and complete. That means that any valid argument can be proved using natural deduction, and that anything that is proved using natural deduction is valid.

That makes natural deduction equivalent to truth tables in sentential logic in the sense that showing that an argument is valid using a truth table implies that it is provable using natural deduction, and showing that it is provable in natural deduction implies that its truth table would show it to be valid.

If step 5 in your deduction was ~Q, the proof would not work. The crucial move in that deduction is 8, which deduces Q from Q->Q, P->Q, PvQ. If 5 had ~Q instead, that would not work, since that would get you P->~Q instead of P->Q.

6 and 7 are needed because Q->Q is needed (for disjunction elimination in 8), and there's no other way to get it (that is, Q->Q must be proved, even if it seems trivial).

• Strictly speaking there are no deductions in classical, truth-conditional logic. You look up the appropriate line in the truth table, and it tells you what's true and what isnt. There is no deduction involved, only truth values.
– user20153
Aug 16, 2016 at 20:58
• +1. Thanks. About your last paragraph under the horizontal line, did the Deduction need Q->Q? To me, 6 and 7 appear redundant and can be skipped. Aug 31, 2016 at 20:49
• `Validity simply means the premises are consistent in the real world.` is very infelicitous wording. Validity implies consistent use of rules of inference. It doesn't say anything about the real word truth of any of the claims or even necessarily the consistency of the claims in the world. Oct 21, 2017 at 6:10