Are CP and RAA expanded versions of MP and MT?

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Reply Wed 26 Nov, 2008 06:12 pm
Are CP and RAA expanded versions of MP and MT?

Is this a correct proof of MI:

1. A -> B /~A v B
-------------------------------
2. ~(~A v B) RAA
3. A + ~B 2 DeM (correct?)
4. A 3 Simp.
5. B 1,4 MP
6. A + B 4,5 Conj.
7. ~B 3 Simp.
8. B + ~B 5,7 Conj.
9. ~A v B 2-8 RA

Thanks!
 
VideCorSpoon
 
Reply Fri 28 Nov, 2008 12:30 pm
@Protoman2050,
So you are asking whether conditional proofs and RAA proofs (i.e. Reductio ad absurdum -->contradictory proof-->indirect proof) are elaborations of Modus ponens and Modus Tollens?

Honestly, they may look the same in format but they are nothing alike in structure (in my opinion). Modus ponens for example infers Q from P and P-->Q and thus only infers the consequent. A conditional, which requires the formation of a conditional, infers the whole thing, antecedent and consequent. Indirect proof wise, modus Tollens would have you conclude in the negated antecedent from a negated consequent. But in the case of the indirect proof, you can infer anything when you derive a contradiction. The Modus Tollens inference rule will only let you infer the negated antecedent.

On a side note, you may want to spell out the inference rule or the type of logical syntactical structure because there are manyas long as you negate both variables. You need to correct #3. This needs to corrected because you entire proof rests on the A wrongly inferred from line 3.
 
Protoman2050
 
Reply Fri 28 Nov, 2008 02:31 pm
@VideCorSpoon,
VideCorSpoon wrote:
So you are asking whether conditional proofs and RAA proofs (i.e. Reductio ad absurdum -->contradictory proof-->indirect proof) are elaborations of Modus ponens and Modus Tollens?

Honestly, they may look the same in format but they are nothing alike in structure (in my opinion). Modus ponens for example infers Q from P and P-->Q and thus only infers the consequent. A conditional, which requires the formation of a conditional, infers the whole thing, antecedent and consequent. Indirect proof wise, modus Tollens would have you conclude in the negated antecedent from a negated consequent. But in the case of the indirect proof, you can infer anything when you derive a contradiction. The Modus Tollens inference rule will only let you infer the negated antecedent.

On a side note, you may want to spell out the inference rule or the type of logical syntactical structure because there are manyas long as you negate both variables. You need to correct #3. This needs to corrected because you entire proof rests on the A wrongly inferred from line 3.


But line 3 would be ~~A v ~B. That's the same as A v ~B.
 
VideCorSpoon
 
Reply Fri 28 Nov, 2008 05:30 pm
@Protoman2050,
First, I think you probably mean (&) rather than (v) in you previous post. If line 2 has an (v), then line 3 will have a (&) or vice versa.

Though what you say is correct that ~~Av~B is the same as saying Av~B, that does not apply in this case. You are inferring Demorgans from line #2. The only way you can infer DeMorgans from line 2 is by stating ~A &~B. You are adding one negation too many to the antecedent. Also, you would have to cite double negation to get that in a previous line.

You cannot add that extra negation symbol until you have legitimately worked it into your proof through inference or replacement.
 
 

 
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