Valid Argument Forms

TheAnxious0L

Can someone break down valid argument form two for me? I'm confused about the contrapositive. /B therefore /A, but isn't shouldn't "does not treat patients" be a contrapositive"?

Valid form 2 of 9
Denying the Necessary
[English] All doctors treat patients. Hercules does not treat patients. Therefore, Hercules is not a doctor.
[Lawgic]
A –> B
/B

---

/A

NotMyName

A contrapositive will always have at least 1 sufficient condition and at least 1 necessary condition. "Does not treat patients" is only 1 idea -- it is the negation of our original necessary condition "treat patients". Using that new idea (/B) we can negate our sufficient condition "Hercules isn't a doctor".

All of this does result in the contrapositive, since:

/B
__
/A

means the same thing as /B --> /A.

Was that helpful?

akistotle

[English] All doctors treat patients. Hercules does not treat patients. Therefore, Hercules is not a doctor.

[Lawgic]
All doctors (D) treat patients (TP).
D –> TP

Contrapositive: If someone does not treat patients (/TP), then that person is not a doctor (/D).
/TP –> /D

Hercules does not treat patients.
/TP

---

Hercules is not a doctor.
/D

Does this make sense?

Accounts Playable

To add a little more high level thought to the excellent responses above, think about the relationship between the original statement and the contrapositive. Namely, they are the exact same statement; no new information is provided. In fact, calling something the "contrapositive" is actually quite arbitrary. You could just as easily diagram a A--->B as ~B--->~A.

So for example, if I say that all men are mortal this can be diagrammed as:

Man--->Mortal
or
~Mortal--->~Man

The two statements mean the exact same thing. Each is the contrapositive of the other, and for the purposes of the LSAT they are logically equivalent.

Confusion comes into play linguistically; humans are fallible and easily misunderstand things when you start getting negatives and "nots" involved.
Things can start to look backwards very quickly. The LSAT writers know this and try to word conditional statements in different, confusing ways. Nevertheless, if you always remember that there is no difference between your "original" statement and "contrapositive," that can help cut through a lot of the confusion.

hihihi9993

@"Idil.Beshir" said:
Can someone break down valid argument form two for me? I'm confused about the contrapositive. /B therefore /A, but isn't shouldn't "does not treat patients" be a contrapositive"?

Valid form 2 of 9
Denying the Necessary
[English] All doctors treat patients. Hercules does not treat patients. Therefore, Hercules is not a doctor.
[Lawgic]
A –> B
/B

---

/A

Yes "does not treat patients" is the starting point of contrapositive in this scenario.
A=Doctors, B=Treat patients
Since you said the contrapositive is /B therefore /A, it would be "does not treat patients" therefore not a doctor
.
Remember, contrapositive is just another way of saying the initial statement. Initial statement and the contrapositive should be equal to each other! (Initial statement = Contrapositive) Since we know that the initial statement is a valid argument, we can say the contrapositive is also a valid argument!
.
Hope this helps!!!

TheAnxious0L

Thank you guys!! I've always been to shy to post in the discussion, but you've been so helpful

roychess

We should be clear that the word contra positive has Different CONTEXTS.
Aristotle used the word quite differently than mathematicians in the late 1850's.
Aristotelian logic uses semantics to evaluate propositions and not just swapping symbols.
Aristotle proved that semantically some propositions could no be validly and soundly contraposed: you cannot contrapose an E proposition. That is an E proposition is NO S is P.
The No quantifier cannot be contraposed. What you will find some inferences will be true and some will be clearly false. And validity expresses it is impossible to have a false conclusion from TRUE propositions.
Philosophers call this mathematical inference you call contrapositive TRANSPOSITION. All you do is swap positions and negate the variables. The original term contraposition was more involved than TRANSPOSITION. The original contraposition term had three steps you had to complete.
The purpose of mathematical logic is not the same as Aristotelian logic. Mathematical logic simulates human communication, whereas Aristotelian logic had the purpose of evaluation in SOUND arguments to prevent deception. Sound arguments are arguments that must have true premises and also must be formally valid. Valid also can refer to complete arguments and can refer to direct inferences which is seen in mathematical logic. What you call contraposition does NOT refer to valid arguments but valid propositional inferences.