# Confusion about an implication's contrapositive

I just begin learning propositional logic and find a little bit confused when trying to translate the contrapositive of an implication back to the English language.

For example, I have an implication: p implies q.

p: it is raining. q: the home team wins.

Therefore, this implication means: if it is raining, the home team wins. "it is not raining, the home team wins" is also right since this is not an "only if" case, the home team can also win when they hear cheers. However, "if it is raining, the home team loses" is false since "raining" will be a sufficient condition for q to occur.

I think my reasoning above is correct. And here is my confusion. As I know that contrapositive of an implication is equivalent to the implication, so I try to translate this contrapositive back to the English language to see whether it works:

The contrapositive of "it is not raining, the home team wins", from my perspective, should be:

"If the home team loses, it is raining". And this true contrapositive seems to be similar to that false statement "if it is raining, the home team loses", as I think that raining and losing cannot happen together. Thus this contradiction puzzles me a lot. Could anyone help me? Is there any weakness or error in my reasoning? Thanks in advance.

The contrapositive of "if p, then q" is "if not-q, then not-p".

The two are equivalent.

Thus, "if it is raining, the home team wins" is equivalent to:

"if the home team does not win, then it is not raining".

You are saying :

"However, 'if it is raining, the home team does not win' is FALSE since raining will be a sufficient condition for q to occur".

If we assume that "if it is raining, the home team wins" is TRUE, this does not implies that "if it is raining, the home team does not win" is FALSE.

If it is not raining, both conditionals have FALSE antecedent and thus they are both TRUE, because :

a conditional with FALSE antecedent is TRUE, irrespective of the truth-value of the consequent.

From "if p, then q" we cannot derive "if not-p, then q".

When p and q are both FALSE, we have that "if p, then q" is TRUE, while "if not-p, then q" is FALSE :

• Yes. But I am thinking the true statement "if not-p, then q" can also be derived from "if p, then q", and then "if not-p, then q" can also have contrapositive right? The true contrapositive of "if not-p, then q", which is "if not-q, then p", seems similar to the false statement "if p, then not-q". Here is my confusion: a true statement seems to be similar to a false statement. (I know one is right and the other is false in propositional logic, but they just seems very similar in plain English) Commented Apr 10, 2018 at 7:31
• @DannyC - your confusion is about mixing TRUE-FALSE with logical equivalent. Two formulas are logically equivalent also when they are FALSE (i.e. both FALSE). Commented Apr 10, 2018 at 12:56
• @DannyC - Please show how “if not-p, then q” is derived from “if p, then q”. One is not the corollary of the other. Commented Apr 10, 2018 at 17:46

You are confusing even the terminology. The logical operation you refer to is called material implication.

Mathematicians still use the term contrapositive for some unknown reason. The term contrapositive used to also refer to a relationship in Aristotelian logic as well.

However, there is no contrapositive for E propositions. An E proposition is a proposition of the form No S is P. It is funny people say "contrapositive" still and not know that if you understood that this process is not always valid, you would stop using that name.

Material implication also expresses this operation is not equal to conversational or slang use. The inference usually is rewritten in multiple ways to demonstrate this fact. I can use IF . . . THEN in multiple contexts --not only one. So one SYMBOL can't cover real speech.

The original claim was of the form "If S then P". Let S be the subject of your choosing. Let P be the predicate of your choosing. The contrapositive in classical logic requires three steps: obversion, conversion, and obversion again. If you have no idea what these are you are probably more confused.

Mathematicians teach this operation differently. In mathematical logic, what you do is to Invert both the subject and predicate terms and swap their positions. Invert simply means to attach a NON prefix to the subject or predicate. This "if S then P" will be allegedly contrapoposed as "if non-P then non-S". This proposition will have an identical truth table as the original claim "if S then P".

I think what you did was to negate only the subject and you kept the same predicate.

"If it is not raining, then the home team wins" is not the same as the original proposition "if it rains, then the home team wins". Follow the steps carefully.

"If the home team is non-winning then it is not raining." The term non does not express the same as NOT or Losing.