## help understanding Negation Introduction

What is the basis for reason? And mathematics?

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Averroes
Posts: 270
Joined: Thu Jul 20, 2017 8:48 pm

### Re: help understanding Negation Introduction

ProfAlexHartdegen wrote:Sorry for the crude representation but as you know the forum isn't conducive to writing symbolic logic. The subderivation is offset a bit to clarify what it is:
As has already been explained one can use the code tags of the post editor to include white spaces and format the proof accordingly. It depends on one's aesthetical preferences. There is another way also.

If you want to type your proofs in box format, you can also use the list tags of the message editor. To put an indentation/offset, you can nest a list tag within another list tag. For example, the negation introduction rule with the appropriate indentation would look like this:

• .......................................Assumption
• ...............................Assumption
• a. P...................Assumption
• .....................
• b. Q
• ....................
• c. ~Q
• ~P.............................a-c ~I
In the message editor, it looks like this:

Code: Select all

[list].......................................Assumption[/list]
[list][list]...............................Assumption[/list][/list]
[list][list] [list]a. P...................Assumption[/list][/list][/list]
[list][list][list].....................[/list][/list][/list]
[list][list][list]b. Q[/list][/list][/list]
[list][list][list]....................[/list][/list][/list]
[list][list][list]c. ~Q[/list][/list][/list]
[list][list]~P.............................a-c ~I[/list][/list]

Averroes
Posts: 270
Joined: Thu Jul 20, 2017 8:48 pm

### Re: help understanding Negation Introduction

I can't understand the use of each individual rule. I mean I know you could apply that rule but I don't see why you use the rule. The 'plan' or the 'forest from the trees' if you will.
In the example you provided, it was required to prove (U & M) ⊃ S, M & ~S ⊢ ~U

The letters are complete sentences in the English argument provided, like here M stands for "The management agrees to the term of contract." But it could be something else, it does not really matter in logic. A note though on the second premise: "The management agrees to the terms of the contract, but the contract will not be signed", here "but" has the same meaning as "and" so it is symbolised with the logical constant "&".

How to proceed from there? What particular sequence of application of rules of inference can get us from the premises to the conclusion?

We proceed by an informal analysis of the sequent. One method is that we start by analyzing the conclusion, and work backwards. So, we observe that the conclusion is a negation, ~U. So we look in the list of premises to see if the application of a rule of inference can give us directly the conclusion and we find that ~U does not even occur in the list of premises. So from this we conclude that the last step in our formal proof will require the use of the negation introduction rule. So now, as we know that for us to be able to apply the negation introduction rule here, we must first assume U and then derive a contradiction. So we decide to assume U and expecting to derive a contradiction from it and then apply a negation introduction rule to arrive at the conclusion.

How can we now derive a contradiction? We also observe that S and ~S occur in the premises. So we ask ourselves, can we derive both of them from our set of premises and assumption U? (this becomes our new goal now) It turns out to be the case (if it did not at this stage, we would have tried something else, as it is a "by trial and error" method). By applying &E to the second premise we get ~S. By applying &E to the second premise again we get M. Conjoining the M, thus obtained, with the assumption U, we get the antecedent of the first premise. Then applying the ⊃E rule we get S. So we have our contradiction and we then proceed by applying the negation introduction rule to get the conclusion. Now, you proceed to write the formal proof from this informal analysis.

Initially you will want to write down this informal proof in English so that you can keep tract of where you are in your thought processes. It does not have to be anything fancy (or lengthy as I have done above), but just write anything that makes sense to you. But as you get some practice, you will proceed directly to write in formal notation as it would have become your language!

So the proof in box format looks like this:
• 1. (U & M) ⊃ S………..................Assumption
• 2. M & ~S.....................Assumption
• 3. U...................Assumption
• 4.M....................2 &E
• 5. U & M..............3,4 &I
• 6. S....................1,5 ⊃E
• 7. ~S...................2 &E
• 8. ~U..........................3-7 ~I
Now, this argument in formal notation can be translated back into English by replacing the letters by the English meaningful sentences, and it would look or sound as something Sherlock Holmes might utter: "Suppose U, then S follows (mentally applying &E, &I,⊃E) according to our two premises, but also ~S (mentally applying &E to 2) according to our second premise. That is a contradiction. Therefore ~U!"

Try to work out some exercises on your own, and see how it works out for you now.
ProfAlexHartdegen wrote:Is the point to take any lines derived from the Assumption that contradict to show that the negative of the assumption can be derived?
Yes. It means the assumption introduces a contradiction.
ProfHartdegen wrote:Logic is the underlying basis for many forms of persuasion too.
In philosophy, mathematics and computer science, logic is indispensable. As a matter of fact, the logic that you are studying was a refinement of the more than 2300 years old Aristotelian logic by Gottlob Frege at the end of the 19th century. The latter was a mathematician and a philosopher. Frege's logic was further refined by others who came after him, and now you have modern logic that you are studying. Frege's work has inspired a lot of philosophers, namely Russell and Wittgenstein (who were also mathematicians) immediately and others later. Wittgenstein, who was by education an Engineer, was so impressed by Frege's work on logic that he became a philosopher after reading Frege.