I encountered an extremely crappy entry on logical validity, and made a quick edit to make it at least barely passable. It is by no means good yet, but it's at least better than the text it replaced. — Unsigned, by: G.D. / talk / contribs
Valid argument with "irrelevant" premisses
While it may appear that an argument with irrelevant premisses cannot be valid, in the technical sense of formal logic, this is not so. In formal logic, it is very difficult to specify what irrelevant premises are, and relevance, whatever it means, is not part of the definition of validity. There is a story (I cannot, on the spur of the moment, get a good citation for it) that someone asked Bertrand Russell to show the validity of the argument from a false premise to a false conclusion, using the example: "1 = 0. Therefore I am the Pope." Russell responded: "1 = 0. Therefore 2 = 1. (addition of 1 to each side) The Pope and you are 2. The Pope and you are 1. (from the preceding that 2=1) Therefore you are the Pope." While this example is not immediately relevant, I think that we can follow the same procedure with an example like yours: "The sky is not blue. Therefore 1=1." is a valid argument. I show it thus: "Assume that the conclusion is false and the premise is true; that is, that 1 !=1 and that the sky is blue. Therefore 1 !=1; and that is false. Therefore the assumption is false; which means that the argument is valid." TomS TDotO (talk) 16:26, 31 May 2013 (UTC)
Response: In traditional first order logic validity is a property of logical form. A valid argument has a form such that every substitution instance of that form will also be valid. This comes strait from standard textbooks such as (http://www.amazon.com/Introduction-Logic-14th-Edition-Irving/dp/0205820379) and also the SEP article I used as a citation.
The alternative definition presented by this article, which I admit to be very useful for giving people the general idea of validity, is subject to two problematic absurdities. First, any argument with a necessary conclusion is valid no matter it's premise. The definition of relevance is not relevant here (no pun intended): just assume that there is some p such that it is irrelevant to a necessarily true q. As long as there is such a pair we shouldn't accept a definition of validity that makes the argument
- therefore q
valid. The definition the article suggests unambiguously does that. Perhaps more compelling, given where we are, would you like a theist to be able to maintain that the following is a valid argument for God's existence:
- The sky is blue. [or anything whatever, really]
- Therefore God exists.
Even if God's existence were necessary, as theists claim, that would still be an illogical argument, not a valid one.
The second problem is necessarily false premises. Any argument with a necessarily false premise will also be valid, no matter it's conclusion, because it will be impossible for the premises to be true, and so impossible for them to be true while the conclusion is false.
(A possible third absurdity is what such a definition would force you to say about formal fallacies. The definition you're defending renders affirming the consequent only sometimes an invalid argument. But
- p then q
- therefore p
will be valid if p is necessary. So also for the other formal fallacies. This contradicts much that's others have already said on this wiki.)
The Russell example is interesting, but short a bunch of mathematical axioms being inserted into the system, plus some metaphysical apparatus having to do with being identical, I'm pretty sure that what Russel gave isn't a valid first order argument. Put differently, if it is valid it is so in virtue of assumed premises not shown. You seem mainly to just be arguing that if we apply the definition given in the article it shows the argument I gave to be valid. That's unarguable, but you can't support the correctness of definition itself that way. (Joseph, a logic teacher in TX)