generic question about contrapositives Forum

[jjinva]

I understand what they are, obviously. To a degree I guess.

I get what I think is the basics. Here are examples. Am I right? Rule then contrapoz below:

If L then T
If -T then -L

If L then not T
If T then -L

If L then R and T
If -R OR -T then -L

If L then not R or T
If R AND T then -L

Do those look right so far?

Where I am missing something key (or have forgotten it since I last studied a few months ago) is the meanings of things like "neither R nor T" and its contrapoz and the contrapoz of the IF and ONLY IF stuff.

Is there a place online that has that stuff written out somewhere. I am using PS LG Bible and going through the games, and I am finding when I get things wrong its because I am not writing rules/contras correctly, especially on the AND/OR things and NEITHER/NOR things.

So I am missing an obvious "a-ha!" moment that is probably something a 3rd grader would get! Thanks for your help.

[pattymac]

For this type of stuff, I don't think it matters what the arguments say because you have to accept it as true in the LSAT world. What helped me was looking past what they said and knowing how to diagram them. At first I was wasting my time worrying about whether or not it made sense, but thats how you get tripped up.

Heres a sweet example I found that helped me.

if I live in Cleveland, I live in the USA.

CONTRAPOSITIVE: I don't live in the USA, so I don't live in Cleveland.

MISTAKEN NEGATION: I don't live in Cleveland, so I don't live in the USA.

MISTAKEN REVERSAL: I live in the USA so I live in Cleveland.

Just focus on diagramming these things quickly and not reading too far into them.

[jjinva]

For this type of stuff, I don't think it matters what the arguments say because you have to accept it as true in the LSAT world. What helped me was looking past what they said and knowing how to diagram them. At first I was wasting my time worrying about whether or not it made sense, but thats how you get tripped up.

Heres a sweet example I found that helped me.

if I live in Cleveland, I live in the USA.

CONTRAPOSITIVE: I don't live in the USA, so I don't live in Cleveland.

MISTAKEN NEGATION: I don't live in Cleveland, so I don't live in the USA.

MISTAKEN REVERSAL: I live in the USA so I live in Cleveland.

Just focus on diagramming these things quickly and not reading too far into them.

Thanks. What about NEITHER NOR and IF AND ONLY IF contrapositives?

[scionb4]

According to the label, contrapositives are at best 98% effective, and that's only when used properly according to the numerous guidelenes.

[The Pen Is Mightier]

The only contrapositive is abstinence!

[scionb4]

The only contrapositive is abstinence!

Well, at least it's the only 100% fool-proof contrapositive.

[The Pen Is Mightier]

The only contrapositive is abstinence!

Well, at least it's the only 100% fool-proof contrapositive.

Well saying positive contrapositive would be redundant.

[jjinva]

So, super-comic geniuses....what is the contrapoz of a NEITHER NOR rule? Or an EITHER OR rule.

[The Pen Is Mightier]

Oh how quickly lsat material leaves us!

[aristotle1776]

So, super-comic geniuses....what is the contrapoz of a NEITHER NOR rule? Or an EITHER OR rule.

"Neither X Nor Y" is the same thing as saying 'Not X and not Y"

p -> ~y nor ~x = p -> ~y and ~x

This is a case where the English language masks the logical operation that truly connects y and x. Negated disjunctions are logically equivalent to affirmative conjunctions.

Since you change the operation when performing the contrapositive:

y or x -> ~p

With neither nors, it is probably better to just diagram the contrapositive. It is keeping in the spirit of diagramming your rules in the affirmative and is also more likely to be applicable to the game (i.e. you can quickly deduce the presence of y or x or both negates any possibility of p's presence) - useful in unbalanced games.

"Either or" depends, if the rule simply states 'either or' then we have the following rule and contrapositive:

p -> y or x
~x and ~y -> ~p

"Either or" can get tricky if they throw in an "either or, but not both" somewhere in the antecedent or consequent (usually in the consequent, if in a conditional at all). Once you practice games enough, it may become easier to keep that negation in your mind instead of taking the time to symbolize the entire thing - I found that it saves time. Eventually you will have the "situational awareness" to maybe do the same, but until then you're probably better off diagramming it.

[tomwatts]

I will say, I handle the "but not both" differently every time it shows up. I'm not totally comfortable with this (seems inconsistent), but that's what I do.

For "if and only if," you get an arrow that points both ways. So...

A if and only if B
A <-> B

For the contrapositive, you could flip and negate, but the arrow points both ways, so it doesn't matter whether you flip or not.

~B <-> ~A
~A <-> ~B

Those are the same.