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Thursday, February 05, 2009


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Is a round square actually contradictory, Bill? I hope you will forgive me if I stray from the topic a bit, but this thought has exercised me in the past.

First, to be clear about my definition, a contradiction is a property of propositions, it is the opposite of a tautology. That is, if P is a contradiction, then ~P is a tautology.

Second, since contradiction is a property of propositions, not of geometric figures, the figures themselves cannot be contradictory. What I suppose "a round square is contradictory" means is that the proposition schema

1. X is a round square

is a contradiction for all X. But 1 is not a contradiction. It has two distinct predicates, "round" and "square". It has the form

p(X) and q(X)

which is not a contradiction. Now 1 combined with

2. forall X . X is round implies that X is not square

is a contradiction. But 2 is not included in the mere concept of a round square; 2 comes from an extra-logical intuition. 2 is a priori, but it is not logical. So although 1 is false via a priori intuitions, it is not a contradiction.

I would propose the resurrection of a grand old term for such situations: "absurd". This used to be used in old proofs as something similar to "contradictory", but was used more broadly than mere logical contradiction. I suggest that a proposition is absurd if it is a priori false. Therefore, round squares are absurd, 1+1=9 is absurd, time travel that changes time is absurd, the idea that borrowing hundreds of billions of dollars out an economy suffering from lack of available credit will help the economy is absurd.


I'd say that

P) x is a round square

isn't equivalent to

Q) p(x) and q(x).

So proving that (Q) is non-contradictory doesn't prove (P) is.

You say "But 2 is not included in the mere concept of a round square; 2 comes from an extra-logical intuition." I rather think that your (2) is included in (1) - so long as you understand what you are really asserting - an object is round if it has exactly 1 side, and an object is square if it has exactly four. Bearing in mind that to understand a concept is to recognise what that concept extends to and what it doesn't, it seems obvious that the set that is the extension of ROUND and the set that is the extension of SQUARE will never have a common member.


About the FBI: I think this example only works on the assumption that 2-dimensional semanticism (espoused by Chalmers' et al.) is false. They would say that your example confused what were really two distinct concepts.

"If I think the FBI as an object that has the phenomenal properties of iron but also floats, then those properties are combinable in my thought without contradiction."

Agreed, I guess this concept of iron would be our pre-scientific one. Call it iron1.

"But if I know more about iron, including its specific gravity, and I import this information into my concept of iron, then the concept of the FBI will harbor a contradiction."

I'm not exactly sure what this "concept importation" would amount to. Surely all that is happening here is that the cluster of concepts which composed iron1 is having a few more added. So I'm tempted to think what you end up with is strictly a different concept, call it iron2. (I'm not sure what the criteria of identity for concepts could be.) The 2Dists say that to get concepts like iron2 all you are doing is rigidifying from the actual referents of a concept like iron1. (Or such was my understanding.)

Anyway, once we distinguish between iron1 and iron2 your claim that "necessarily, iron doesn't float in water" can be seen to be self-evidently false if iron = iron1, and seen to be self-evidently true if iron=iron2. But then you no longer have any reason to doubt that conceivability is a good guide to possibility.


Matt, given that a circle has one side, that a square has four sides, that 1=/=4, and that if a figure has n sides then it cannot also have m sides where m=/=n, then you can find a contradiction. But all of those facts come from non-logical intuitions of mathematics and space --they are not deducible.

Just to clarify, I'm using "intuition" in the Kantian sense. In Kantian terms, it is common to conflate logic with the analytic. If you view it that way, then what I'm saying is that the absurdity of a round square is not analytic but that it is a priori.


I doubt that we have any substantive disagreement once we adjust for differences in terminology.


I don't have the time to be rigorous, but concepts can be more or less determinate. If my concept of X is not very determinate, then the impossibility of X might not show up within my concept. Suppose someone has the concept of a red point and infers that red points are possible because 'x is both red and a point' is not formally self-contradictory. But then this person comes to realize that points are unextended and that necessarily colors are extended. Importing this new information into his original concept and thus making it more determinate he comes to realize that nothing in reality can instantiate the concept red point.

Greetings Bill,

I've run into this argument on several occasions and while the author(s) insist theists will accept the premises it's more the validity I'd appreciate your take on.

1) If God is possible, then God is a necessary being.
2) If God is a necessary being, then unjustified evil is impossible.
3) Unjustified evil is possible.
Therefore, God is not possible.

Greetings and thanks. Response in a separate post on front page near the top.

Too much logic.....theology is not logic, faith....a mis understood concept of rational beings. Augustine was the man....flawed yes but on the right track.

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