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.
    – NNOX Apps
    Aug 14, 2016 at 22:14
  • 1
    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.
    – commando
    Aug 14, 2016 at 22:14

2 Answers 2



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.
    – NNOX Apps
    Aug 31, 2016 at 20:49
  • Would you please respond in your answer, because it is easier to read than comments?
    – NNOX Apps
    Aug 31, 2016 at 20:49

Your proof as written is wrong at step 7. Your justification does not fly. You assume Q at 6 and all you have to do is use premise 2 to get Q via disjunctive syllogism.

Validity does not mean TRUE, but validity looks at the connection between the conclusion with the premises. If the conclusion logically follows then the argument is said to be valid. Validity only looks at or expresses that if the premises were true then the conclusion must also be true as well. That is, the conclusion is impossible to be false if the premises were true.

In the real world. You can have valid arguments with false premises for certain. Because an argument is valid does not mean the person making the argument is winning a debate or winning a disagreement. I only say this because many people who argue seem to overly focus on validity when the content of the premises is not correct. One can have a valid argument with false premises still. I try to focus on true premises and then valid form. In this way I could have sound arguments. Sound arguments are valid arguments by form and have true premises.

  • 1
    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.
    – virmaior
    Oct 21, 2017 at 6:10
  • Thank you for the correction. I understand. What I was trying to relay was the difference between soundness and validity as used in real world disagreement. There are many people in the real world environment that firmly believe if their argument is valid they "must be correct and their points must be true". I also want to express that because one's argument is "valid" does not mean the person is "winning the argument". How should I have expressed this?
    – Logikal
    Oct 22, 2017 at 1:44
  • Well for one thing, you can edit your answer when you get a correction. Personally, I would suggest "validity looks only at whether the premises logically lead to the conclusion. Specifically that if you have a valid argument, then if its premises were true, its conclusion must also be true." / then in separate paragraph, "in real world arguments ..." with whatever you intend to say about the connection between the two.
    – virmaior
    Oct 22, 2017 at 1:48
  • I have edited my post. I did not want to quote you directly as you corrected me. I decided to try to put things in my own way but express the same message.
    – Logikal
    Oct 22, 2017 at 2:25

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