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

Question

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

Answer 1

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.

Answer 2

An if–then (“If P, then Q”) is only wrong when P happens but Q doesn’t. That means the negation isn’t another rule — it’s just pointing to one situation: “P is true AND Q is false.” That’s why the negation is an and-statement, not another if–then.

Answer 3

The way that the conditional or implication operator is taught in most logic courses is actually incorrect. Well, to be more precise, it does not correspond to reality or what we generally mean when we say "if X then Y", "X implies Y", or "X is a (sufficient) condition of Y".

This conditional/implication operator is used as a proxy for what you called an "if statement" such as "if X then Y" and is often symbolized as X ⊃ Y or X -> Y. But what this really means is "(reality is ordered such that) it is impossible for X to be true and Y to be false." That is, implication (and its cousin, disjunction) is actually a modal claim. It doesn't tell us about the actual truth of X or Y, but only what combinations are (im)possible. But very few people ever learn modal logic, so people are taught a different (and unreal) meaning for the implication operator.

Consider the claim, "If John passes the exam, then he will graduate", E > G. Consider what we REALLY mean when we say something like this. We are saying that reality is presently arranged such that it is impossible for John to pass the exam and NOT graduate. This doesn't tell us whether E or G is true or false, and we don't actually know if this compound claim is false from the truth E or G alone except in one special case: when E is true and G is false. Because the claim means that E & ~G is impossible, if it turns out that E is true and G is false, then we know that the whole claim is false. But in all other cases, we do not know from the truth of E and G whether the whole claim is true (contrary to basically every text in existence on this matter).

But wait, there's more.

The typical formal logic class will teach you that the negation of X > Y produces X & ~Y, but that is actually an error. X & ~Y doesn't follow from ~(X > Y) -- only the possibility of X & ~Y. To say that X > Y is false is to say "it is not the case that it is impossible for X to be true and Y to be false" or "it is possible for X to be true and Y to be false." This is symbolized in different ways depending on the modal system in play.

The implication operator as taught in practically all contemporary symbolic logic courses does not correspond to what we actually intend when we say "if X then Y," but is an abstract mathematical operator with its own unique quirks. This is a tremendous problem for contemporary logic and for students learning logic, as they are being taught something that doesn't really match up with how we (rightly) think about things.

Now, what I've told you here is the truth, and here is the real problem: If you apply this in your basic logic courses, your professor will probably fail you... even though you are correct and the course is wrong. However, if you apply this in your real-life reasoning, things will likely go better and make more sense in the long run.

:)

Answer 4

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.

Answer 5

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.