A Toy Windmill said:
As I said, there are ways to save the idea that you have proven the irrationality of the square root of 2 from inconsistent statements, but in the way suggested, you conceded that it meant stepping outside the formalism, which is to say that what is "really going on" isn't captured formally anymore. I suggest that's a weak point, given that you're arguing with someone with serious disagreements about the nature of logical validity.
I don't think the idea actually needs "saving"
, so we disagree about that (and, it seems, about the results of the previous exchange), and I was trying to put it in a way that might be hopefully persuasive to you. So, that did not work, but I was not trying to persuade Speakpigeon of anything - I've already reckoned that there is no point in trying -, and while the thread is mostly meant to show in yet another way the untenability of Speakpigeon's positions, I was talking to you in that reply, not trying to convince readers of the failure of Speakpigeon's position.
When it comes to Speakpigeon's position, I do not think that the problem would be to step out of the formalism, but
if there is a problem, the problem would be to step inside it in the first place - i.e., to use the formalism at all -, since Speakpigeon rejects the formalism altogether.
In any case, also when it comes to Speakpigeon's position, I started this thread not to interpret things in a first-order context, but rather, to show a case in which a contradiction is derived in the context of a standard mathematical argument, from an inconsistent set of premises. If they're not called premises, then from an inconsistent set of statements a contradiction is derived. In that context, the argument succeeded already - well, actually, it was not even necessary, since a similar argument succeeded in the other thread, but this one works too, for the following reason: In reply to my points, Speakpigeon made a distinction between an inconsistent set of premises and contradictory premises:
https://talkfreethought.org/showthr...Squid-Argument&p=694447&viewfull=1#post694447
One derives a contradiction.
The right word is indeed "inconsistent". The premises are inconsistent, i.e. one premise implies the negation of the other premise.
Contradictory premises would be p and not p and that's not what we have here.
The premises here are not contradictory.
You need to make sure you know the basics before posting silly arguments.
EB
So, Speakpigeon does not raise the same objections that you do (Speakpigeon clearly misunderstands what you're saying), and
concedes that this - and similar cases - is an argument with inconsistent premises, but somehow claims that it is not a problem when "one premise implies the negation of the other premise.", but rather, the problem is with premises like "p and not p". Of course, that already debunks Speakpigeon's own claim that the
"Improved Squid Argument" (and several similar arguments Speakpigeon asks about) is invalid, since that is precisely a case in which no premise contradicts itself, but one or more premises imply the negation of another.
So, this already worked! (by "worked" of course, I mean it worked in exposing Speakpigeon's position as absurd, not that it would do anything to persuade Speakpigeon, which I would not try).
A Toy Windmill said:
As a final remark, and to throw in some real philosophy of mathematics, how does a logicist interpret your proof? They take logical validity to be powerful enough to cover all the mathematics we care about, and can show in their logics that arithmetical facts are just a subset of logical truths. All the ambient mathematical facts are now logical truths, and your proof shows that the statement that the square root of 2 is rational is analytically inconsistent with itself!
I'm not sure I get the terminology, but
if I'm reading this right, then it seems under this view of the philosophy of mathematics, this is even better, for the following reasons:
1. This would be an example of a standard mathematical argument with an inconsistent set of premises. Sure, the set's cardinality would be 1, and the only premise would be, as you say, the premise that the square root of two is rational. But then, as you say, the proof under this interpretation shows that the statement the square root of 2 is rational is inconsistent with itself.
2. In addition to debunking Speakpigeon's position for the reason I explained above in this post, it would debunk Speakpigeon's position in yet another way: Speakpigeon accepts the proof as valid, but claims that arguments with contradictory premises are not valid. But alas, this is an argument with one contradictory premise, it seems.
That said, there might be a difficulty, not with regards to debunking Speakpigeon's position - that's already done -, but with regard to points 1. and 2. above (which is why I mentioned that I'm not sure I get the terminology): maybe your "analytically" qualifier makes a distinction between "inconsistent" and "analytically inconsistent", so I would like to ask whether you think that that would prevent 1. and/or 2. above (and if so, I'd like to ask for more details). My impression is that the "analytically" qualifier probably causes no trouble for either 1. or 2. above, since the logicist seems to consider analytical truths to be logical truths as you say (personally, I make a distinction between analytic truths and logical truths. For example, 'No bachelors are married' would be an example of the former, but not the latter, whereas any instance of a logically valid formula (e.g., ( P v ¬P)) would be an example of the latter).