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)
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?
Looks right to me, except that you should put in a step:
7. K & ~K
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.
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.
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.
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.
Take the indirect proof for example.
This is the indirect proof in proof format.
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.
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.
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.