Is this a correct RAA proof?

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Reply Sat 19 Jul, 2008 11:58 pm
Is this a correct RAA proof:

1. T v U
2. U -> ~K
3. K /:. T
4. ~T (RAA Assumption)
5. U (1,4, Disjunctive Syllogism)
6. ~K (2,5, Modus Ponens)
Contradiction!
7. T

Line 6 contradicts line 3, so...is this valid? I know the theorem I'm trying to prove is valid, but is my proof valid?

Thanks!
 
kennethamy
 
Reply Sun 20 Jul, 2008 06:15 pm
@Protoman2050,
Protoman2050 wrote:
Is this a correct RAA proof:

1. T v U
2. U -> ~K
3. K /:. T
4. ~T (RAA Assumption)
5. U (1,4, Disjunctive Syllogism)
6. ~K (2,5, Modus Ponens)
Contradiction!
7. T

Line 6 contradicts line 3, so...is this valid? I know the theorem I'm trying to prove is valid, but is my proof valid?

Thanks!


Looks right to me, except that you should put in a step:

7. K & ~K

Then 8.
T.
 
Protoman2050
 
Reply Sun 20 Jul, 2008 06:30 pm
@kennethamy,
kennethamy wrote:
Looks right to me, except that you should put in a step:

7. K & ~K

Then 8.
T.


1. T v U
2. U -> ~K
3. K /:. T
4. ~T (RAA Assumption)
5. U (1,4, Disjunctive Syllogism)
6. ~K (2,5, Modus Ponens)
7. K & ~K (3,6 Conjunction)
Contradiction!
8. T

Any other ways to prove this? What would this look like --and would it be convincing?-- in English?

BTW: T=God is trinity, U=God is unity, K=God can know Himself.
 
VideCorSpoon
 
Reply Mon 21 Jul, 2008 11:00 am
@Protoman2050,
Yup! Looks good. The only problem I see is that #7 is derived from "conjunction" inference rather than "addition." The addition inference would require a disjunction. Not a big problem, only the citation.
 
Protoman2050
 
Reply Mon 21 Jul, 2008 03:04 pm
@VideCorSpoon,
VideCorSpoon wrote:
Yup! Looks good. The only problem I see is that #7 is derived from "conjunction" inference rather than "addition." The addition inference would require a disjunction. Not a big problem, only the citation.


Of course. Now, I was talking w/ a philosophy prof, Dr Stolze, and he said that this is valid, just not a very "strong" proof of God's triunity. What does he mean? Any other ways of proving this? Btw, I know lines 4-8 are called the "proof", but what are lines 1-3 called?
 
VideCorSpoon
 
Reply Mon 21 Jul, 2008 08:20 pm
@Protoman2050,
I don't understand why your professor would say that it is a weak argument though. You can solve that particular proof in more than three ways (at least). The main reason that I could see as to why that particular proof would be weak is because it bases its entire weight off of the fact that "because this small part of the grander argument is problematic, the whole thing is by extension wrong."

Funny that you asked about what lines 1-3 are called. The lines are officially called premises of the argument. Lines 1 - 3 can't be an argument in itself even though we typically include the conclusion at the end of the given argument in order to solve the proof. That conclusion is just the reminder of what we are supposed to conclude at. So lines 1-3 are the initial parts of a grander argument. I'm not quite sure if there is an exact word for it though. If you know of one, I'm curious as well.

I'll write up a precise proof (without nesting) for this in a little bit.
 
Protoman2050
 
Reply Mon 21 Jul, 2008 08:29 pm
@VideCorSpoon,
VideCorSpoon wrote:
I don't understand why your professor would say that it is a weak argument though. You can solve that particular proof in more than three ways (at least). The main reason that I could see as to why that particular proof would be weak is because it bases its entire weight off of the fact that "because this small part of the grander argument is problematic, the whole thing is by extension wrong."

Funny that you asked about what lines 1-3 are called. The lines are officially called premises of the argument. Lines 1 - 3 can't be an argument in itself even though we typically include the conclusion at the end of the given argument in order to solve the proof. That conclusion is just the reminder of what we are supposed to conclude at. So lines 1-3 are the initial parts of a grander argument. I'm not quite sure if there is an exact word for it though. If you know of one, I'm curious as well.

I'll write up a precise proof (without nesting) for this in a little bit.


I believe what he meant was that you can infer anything, not just T, from line 7, and that a "stronger" proof would not rely on the principle of explosion to work.
 
VideCorSpoon
 
Reply Mon 21 Jul, 2008 08:52 pm
@Protoman2050,
Fair enough. I would refer to that as a weak inference, not a weak argument because it is contingent on a single line though. Is your professor supposing that a better derivation could have been used? That is, other than the fact that a nested proof is not exactly a solid argument (i.e. principle of explosion /indirect proof).

Also, I've noticed you are learning a very exciting adjective heavy version of logic. "Explosion" and "Reductio" etc. Which method are you learning? I learned propositional with the Herrick method. Same rules (for the most part) but different names.
 
Protoman2050
 
Reply Mon 21 Jul, 2008 09:05 pm
@VideCorSpoon,
VideCorSpoon wrote:
Fair enough. I would refer to that as a weak inference, not a weak argument because it is contingent on a single line though. Is your professor supposing that a better derivation could have been used? That is, other than the fact that a nested proof is not exactly a solid argument (i.e. principle of explosion /indirect proof).

Also, I've noticed you are learning a very exciting adjective heavy version of logic. "Explosion" and "Reductio" etc. Which method are you learning? I learned propositional with the Herrick method. Same rules (for the most part) but different names.


Nested proof? What's that? Yes, he told me that he isn't qualified enough to formally judge my proof, but to pass it on to Dr Stapp, the logic prof, b/c "she'd be quite interested in it".

I've been learning from the web, b/c when I was in HS, I took a logic class, but dropped before we got to formal proofs. "Exciting"? What does that mean?

I believe this is more formally a proof by contradiction, rather then a reductio ad absurdum proof.
 
VideCorSpoon
 
Reply Mon 21 Jul, 2008 11:29 pm
@Protoman2050,
A nested proof is a proof that is indented into the proof and discharged before the rest of the proof can go on. Indirect and conditional proofs are nested proofs.

If you have learned propositional logic from the web, you have caught on really well. I have not been able to find any kind of really understandable logic tutorials as of yet on the web, so you have been able to pick up on a generally difficult subject to convey. Props.

I'm used to very boring descriptors for reference rules, so any more interesting titles are neat to encounter. Logic is typically very bland. No harm meant.

Also, I agree with you, it seems more a contradiction (indirect) proof than reductio (conditional) proof.
 
Protoman2050
 
Reply Mon 21 Jul, 2008 11:34 pm
@VideCorSpoon,
VideCorSpoon wrote:
A nested proof is a proof that is indented into the proof and discharged before the rest of the proof can go on. Indirect and conditional proofs are nested proofs.

If you have learned propositional logic from the web, you have caught on really well. I have not been able to find any kind of really understandable logic tutorials as of yet on the web, so you have been able to pick up on a generally difficult subject to convey. Props.

I'm used to very boring descriptors for reference rules, so any more interesting titles are neat to encounter. Logic is typically very bland. No harm meant.

Also, I agree with you, it seems more a contradiction (indirect) proof than reductio (conditional) proof.


Really? Cool! And, what sort of "boring descriptors"...how are mine interesting compared to what you learned? What would your method call "reductio" and "principle of explosion"? Logic is only bland when you start mixing it w/ math.
 
VideCorSpoon
 
Reply Tue 22 Jul, 2008 05:41 am
@Protoman2050,
Take the indirect proof for example.

This is the indirect proof in proof format.
http://i36.tinypic.com/2zqehyp.jpg

Here are three different different titles for that particular method of inference rule of the top of my head.

Herrick would say that is a "indirect proof."
(To prove P. Indent, assume ~P, derive a contraditction, end the indentation, assert P)
Prospesel would say that is "Dash I" (literally... -I)
(-A depends on whatever assumptions B &-B depends on (less the assumption a)
Bergmann would not even use conditional proofs, though he incorporates more than the official inference rules.

So three different people with three different takes on the exact same thing. Herrick is by far the easiest to understand from what I have seen. But they are all technical descriptors. Yours are interesting because they incorporate an abstract concept towards the technical rule. It's not interesting in a bad way, just interesting in particular because that technical title helps understand the concept in a broader way.

Reductio ad absurdum is called indirect proof and principle of explosion is called conditional proof. Same thing, different names.

Yeah, I hear what you are saying about logic when math gets involved. I had an interesting conversation with Zentetic11235 on discrete mathematics, number theory, and topology which only underlines my precarious distance from mathematical logic. Still interesting though, you should not be deterred from it, but once you learn it mathematically, its probably 99.9% certain that you would go back to the propositional method.
 
Protoman2050
 
Reply Tue 22 Jul, 2008 08:47 am
@VideCorSpoon,
VideCorSpoon wrote:
Take the indirect proof for example.

This is the indirect proof in proof format.
http://i36.tinypic.com/2zqehyp.jpg

Here are three different different titles for that particular method of inference rule of the top of my head.

Herrick would say that is a "indirect proof."
(To prove P. Indent, assume ~P, derive a contraditction, end the indentation, assert P)
Prospesel would say that is "Dash I" (literally... -I)
(-A depends on whatever assumptions B &-B depends on (less the assumption a)
Bergmann would not even use conditional proofs, though he incorporates more than the official inference rules.

So three different people with three different takes on the exact same thing. Herrick is by far the easiest to understand from what I have seen. But they are all technical descriptors. Yours are interesting because they incorporate an abstract concept towards the technical rule. It's not interesting in a bad way, just interesting in particular because that technical title helps understand the concept in a broader way.

Reductio ad absurdum is called indirect proof and principle of explosion is called conditional proof. Same thing, different names.

Yeah, I hear what you are saying about logic when math gets involved. I had an interesting conversation with Zentetic11235 on discrete mathematics, number theory, and topology which only underlines my precarious distance from mathematical logic. Still interesting though, you should not be deterred from it, but once you learn it mathematically, its probably 99.9% certain that you would go back to the propositional method.


I thought conditional proof was where you assume the antecedent of a conditional to prove the consequent, ie Conditional proof - Wikipedia, the free encyclopedia.
 
VideCorSpoon
 
Reply Tue 22 Jul, 2008 08:59 am
@Protoman2050,
No, you have it right, but Herrick uses the conditional proof as its own inference (rather than just as the name of the" if, then" statement.) Different names for different methods.

The conditional you posited is right (the one from wiki), but that is the basic conditional, as in conditional, bi-conditional, conjunction, and disjunction.

The way the conditional proof works is this. Say for your conclusion you wanted to get a conditional, i.e. P-->Q . You would have to indent, assume P, end with Q , and assert P -->Q as your conclusion. This is one of my favorite inference rules. It's basically a big all encapsulating argument, like a hypothetical syllogism, but also incorporates different inferences and replacements.
http://i36.tinypic.com/2jcwz6p.jpg
 
Protoman2050
 
Reply Tue 22 Jul, 2008 09:05 am
@VideCorSpoon,
So, what are the other methods of proving my, um, thingie?
 
VideCorSpoon
 
Reply Tue 22 Jul, 2008 09:18 am
@Protoman2050,
Oops! Sorry about that, I did it but I got side tracked.

This is one way. But this is one of the simpler ways to do it, which also reveals the problems in the argument. Usually if you have to use double negation or communication, there is room for refinement in the conveyance. Those two are also severe word play when you see the literal structure of the proof.
http://i36.tinypic.com/2887kn.jpg
 
Protoman2050
 
Reply Tue 22 Jul, 2008 10:48 am
@VideCorSpoon,
VideCorSpoon wrote:
Oops! Sorry about that, I did it but I got side tracked.

This is one way. But this is one of the simpler ways to do it, which also reveals the problems in the argument. Usually if you have to use double negation or communication, there is room for refinement in the conveyance. Those two are also severe word play when you see the literal structure of the proof.
http://i36.tinypic.com/2887kn.jpg


I see. Another way?
 
VideCorSpoon
 
Reply Tue 22 Jul, 2008 11:35 am
@Protoman2050,
Suffice to say there a number of ways to do it. It's all how you manipulate the rules and construct your argument. You can expand the argument by extrapolating from the rules you used to derive excess steps from them, but then that wouldn't be logical.

To further your own argument (with the proof that I did) you would have to revise #3 K (because a double negative is the same as if there were no negative at all) and flip #1 juxtaposition to have a logically flowing argument.

This would mean that the unknown variable, syntax, has to be accounted for in truth functional logic.
 
Protoman2050
 
Reply Tue 22 Jul, 2008 02:46 pm
@VideCorSpoon,
VideCorSpoon wrote:
To further your own argument (with the proof that I did) you would have to revise #3 K (because a double negative is the same as if there were no negative at all) and flip #1 juxtaposition to have a logically flowing argument.

This would mean that the unknown variable, syntax, has to be accounted for in truth functional logic.


Elaborate, please.
 
VideCorSpoon
 
Reply Tue 22 Jul, 2008 04:17 pm
@Protoman2050,
elaborate on your request for an elaboration. LOL! just kidding! what exactly are you unclear about? The argument honing or the syntax comment?
 
 

 
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