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I've taken formal logic, but I've never heard exactly what this term is.

In my LG Bible there is a small section about them and the I understand the formula perfectly, but I don't understand what they mean by the contrapositive. Is it the exact opposite?

For example in X --> Y

I would make a truth table to see all the possibilities, at least that's what I learned from Logic.

So I know in X --> Y the truth table would like this:

X --> Y True
X --> -Y False
-X --> Y True
-X --> -Y True

However I don't think this is relating to the actual idea of a contrapositive.

Junebugman wrote:I've taken formal logic, but I've never heard exactly what this term is.

In my LG Bible there is a small section about them and the I understand the formula perfectly, but I don't understand what they mean by the contrapositive. Is it the exact opposite?

For example in X --> Y

I would make a truth table to see all the possibilities, at least that's what I learned from Logic.

So I know in X --> Y the truth table would like this:

X --> Y True
X --> -Y False
-X --> Y True
-X --> -Y True

However I don't think this is relating to the actual idea of a contrapositive.

Those are almost all wrong. You need to forget what you learned in that formal logic class.

X --> Y
~Y --> ~X (contrapositive) Valid actually. Not true, doesn't have to be true in the traditional sense.
~X --> inconclusive, can't tell anything about Y for certain.
Y --> inconclusive, can't tell anything about X for certain.

I...don't think your professor actually taught you that. You don't use a truth table for just about anything; it is only used for testing the validity of an argument, not a mere proposition. Truth tables are overrated anyways.

OT: basically contrapositive is a statement that "places/posits against/contra." simply put: just flip and negate. If A, then B. If not B, then not A (in a logic "bubble" where there are no other variables)

the positions are reversed and negated

original = A ---> B

negation in example = -

contrapositive= - B ---> - A

Mistaken Negation + Mistaken Reversal = Contrapositive (i.e. "valid reasoning").

I. Original Statement A → B

II. Mistaken Negation of (I.) the Original Statement: ~A → ~B (in other words, this does not follow from (I.))

III. Mistaken Reversal of (II.) the Mistaken Negation: ~B → ~A (in other words, this does not follow from (II.))

Therefore A → B = ~B → ~A (with ~B → ~A being the contrapositive of A → B, and vice-versa)

FantasticMrFox wrote:I...don't think your professor actually taught you that. You don't use a truth table for just about anything; it is only used for testing the validity of an argument, not a mere proposition. Truth tables are overrated anyways.

OT: basically contrapositive is a statement that "places/posits against/contra." simply put: just flip and negate. If A, then B. If not B, then not A (in a logic "bubble" where there are no other variables)

Well I only put the Truth Table up is because from what I remember about learning how the original models such as PQ,PvQ,P->Q,etc

I only showed the truth table because that's what I remember from P --> Q and when I was reading the introduction about them and they spoke about contrapositives with "X --> Y", it just made me remember my logic class.

Op, don't take the June LSAT. You're not ready.

A contrapositive is essentially just a logical way of rephrasing a conditional. Instead of saying a statement one way, a contrapositive allows you to say it another way, but they both state the same (as defined by logical equivalence) thing. Although this seems useless, it turns out that rephrasing logical statements is very useful.

Proposition: A -> B
Contrapositive: ~B -> ~A
Both statements are logically equivalent.

KevinP wrote:A contrapositive is essentially just a logical way of rephrasing a conditional. Instead of saying a statement one way, a contrapositive allows you to say it another way, but they both mean the exact same thing. Although this seems useless, it turns out that a rephrasing logical statements is very useful.

Thank you. That makes perfect sense now.

Junebugman wrote: Thank you. That makes perfect sense now.

No problem. I think you were trying to use truth tables to see how a proposition relates to its contrapositive (sounds like something I would have done). You would have saw (from the truth tables) that they are logically equivalent.