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explain why the negation of an if then statement can not be an if then statement

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3 Answers

Perhaps a Truth Table might shed some light on this.  Below is a TT for "if p, then q."

p. q. if p, then q.

T.  T.  T.

T.  F.  F.  [note this case.  "if T, then F" = F.]

F.  T.  T.

F.  F.  T.

Notice that an implication "if p, then q" is only F when then premise, p, is T and the conclusion, q, is F.

This is also the only case the negation of an implication is T.

So considering this, we see that a negation of an "if-then", being true in only one case, cannot also be an "if-then", which is T in three cases.

Incidently, the negation of "if p, then q" is "p and (not q)."

Hope that helps.

Comments

Your welcome!

Logic & Proof was one of my favorite courses.

Learning that "If [falsehood], then [anything]." was a logically true statement showed me how strangely entertaining math can be.

I also liked that the word "truer" has no place in logic.  I like words, with a few exceptions.  That one has always bothered me though. :)

 

Robert,this is just to let you know that your spelling of,"Your" should be,"You're welcome!"  A lot of people get this wrong.  Examples: Is that your car? and You're late! You're means you are.

The "if" clause of a conditional provides a hypothetical situation, this is true, but not in all cases. A quick review:

1) If I run, I will cough.
2) If I ran, I would cough.
3) If I had run, I would have cough.

Let's take a look at the third conditional and modify to make it negative and real.

If I hadn't run then I wouldn't be sweating.

Also, when things are habitual, the second conditional becomes real in the "if" clause.

If I didn't used to play soccer as a kid, I wouldn't be as good as I am now.

The negation of an if/then statement cannot be an if/then statement because the negation is a selection, or choice, whereas an if/then statement is conditional. The if/then statement merely states a causal relation between two possibilities but it does not guarantee the outcome or actuality of these possibilities. For example, "if A, then B" means that if it is true that A is, then B is too; however, it does not establish that A is or B is. The negation of this if/then statement, "A, but not B," is a negation in the sense that it is an outcome which establishes that B does not necessarily follow from A, or, in other words, it proves that the relation stated in the if/then statement is false.