From P, you may infer ~~P
From ~~P, you may infer P
So why is this seemingly arbitrary and insignificant function so important to formal proofs? Why is it that we need to state the obvious, and then negate it's own negation in order to simply restate what was stated in the first place? The answer is difficult to comprehend at first, or at the very least I found it to be, but when the rule of Modus Tollens is taken into account, it becomes rather clear. To clarify, for reference in this particular blog post, the rule of Modus Tollens is as follows:
From P -> Q and ~Q, you may infer ~P
From a conditional statement and the negation of its consequent, infer the negation of its antecedent. (Arthur 64)
Now, in a formal proof, there is a fairly good chance that you will not solely use double negation in order to solve a proof. For example, the following proof requires that before double negation can be used, Modus Tollens must first be applied.
~P -> Q, ~Q : P
- ~P -> Q Prem
- ~Q Prem
- ~~P 1, 2, MT
Modus Tollens is used in this first part of the equation. We are given the premise ~Q, which allows us to use Modus Tollens as follows; if ~P -Q and ~Q, then ~~P. This, however, does not give us the final answer of P that we are trying to get to. Instead, we must apply the rule of double negation in order to finally solve the proof.
- ~P -> Q Prem
- ~Q Prem
- ~~P 1, 2, MT
- P 3 DN QED
By applying the double negation rule in the formal proof rather than in our heads, we were able to display the very reason why this rule is so important. In a formal proof, it is imperative that each step be worked out on paper in order to show exactly how one is able to get to the answer from start to finish. In this case, if we had left the double negation step out, there would have been no answer to the proof at all. This step, simple as it may seem, is entirely necessary, and should always be written out in a formal proof if used by the logician preforming that proof.
Wish I'd read this before today's quiz! I missed the DN, twice. mmm, I wonder if my double negation of double negation means that I didn't really mess up?
ReplyDeleteDouble negation is one of those things that is so easy it's difficult. I overthink everything all the time so when i look at such a simple concept as this it automatically becomes a really difficult issue. However, it is becoming easier. Thank you for posting about this. I think you are wise in using this blog as a way to reteach hard concepts. It's definitely a lot easier to understand something and practice when you are showing others how to do it.
ReplyDeleteI completely skip over double negation most of the times we need it and then don't remember it when I can't figure out what I was doing wrong, so this reiteration was really helpful.
ReplyDeleteA swell post this is, I must say! As I see it - and as I think Prof. Silliman has pointed out - logic is to break down all point of an argument, so that we know how they moved from point to point. A lot of these operations (MP, MT, etc.) are ones that we commonly (and sometimes mistakenly) use each day. However, DN becomes so much easier to forget to put down in a proof, because it is that much easier to use in real life. As they say in a traditional mathematics classroom in the upper levels of the subject: don't forget the easy stuff!
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