Inside logic # 5NegationsShe is gathering up her stuff to leave him. You are thinking that she probably is going to be better off anyway, especially if he is a criminal. The man is reading the Sports section until he notices that she is leaving. "What are you doing?" he asks. "I am leaving you," she says. "Why?" he asks. "Because you won't confess," she said. "But it's not true that I won't confess!" he protests. And then you think, He WILL confess! I wonder what it is! * Actually it is none of your business. All the same, in that very moment you have reasoned again, and your reasoning once again is just fine. The transition in thought is almost so small as to go unnoticed, but in fact it is important. Your thoughts move from It's not true that he not will confess to He will confess. To see what is going on let us examine the little word "not". We earlier let W stand for the statement "he will not confess." What is the relation between the two statements "he will confess" and "he will not confess"? Well, if one is true, the other one must be false. This suggests that a better way to represent the sentence with the "not" is to introduce a special symbol for "not". We will use ~ to represent "not". And if we let C stand for "he will confess" then ~C stands for "he will not confess". The statement ~C is the negation of C. For any statement P, we can form its negation ~P. When we represent "he will not confess" by ~C rather than the simpler W we reveal more of the logical structure of the sentence, and in general we want to be able to do this. You reasoned "if he will not confess she will leave; and yet he will not confess; so she will leave." Now that we represent "he will not confess" by ~C, we now would represent your reasoning with the two premises ~C>L and ~C, and (as before) the conclusion L. --And this is still a good example of reasoning with MP. When you hear the man say "But it's not true that I won't confess" he is saying something that can be represented with two negation symbols -- that is, as ~~C. The negation ~C is itself negated! Normally in English there are no more than two negations, but in principle there could be more, and our symbols permit as many as desired; so ~~~C also is a perfectly good statement in our symbolic language, representing "it's not true that it's not true that he will not confess." When you think He will confess! (and where your basis for thinking this is what the man said, namely, "But it's not true that I won't confess"), you are actually reasoning using a simple but important rule of logic, Double Negation. Your thoughts move from the doubly negated statement ~~C back to C. This conforms to our next rule.
DN requires only one line as a premise for its application, not two (as for MP). Your reasoning about the man can then be represented in a proof with two lines, as follows.
DN is used in the following examples, where we also use the rules A and MP that were previously introduced. Suppose we want to show that we can derive ~~Q from the two assumptions P>Q and P. We can do so as follows.
Notice that the line 4 of the proof tells us that we successfully done what we set out to do. We have shown that ~~Q can be derived from the assumptions that we made on lines 1 and 2. Notice that the assumptions of line 4 are all of the assumptions of its premise on line 3. In other words, since the premise for 4 is Q, on line 3, the assumptions of line 4 are all those of line 3, as indicated. Suppose now that we want to derive Q from P>Q and ~~P:
Line 4 tells us that given as assumptions the sentences listed on lines 1 and 2 (namely, P>Q and ~~P), we can derive Q.
|