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

This is about negations of if then statements and why they can't be if then statements.

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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.