2011-07-29

What We Can Learn About Life From Curry's Paradox

Curry's Paradox is a fascinating piece of linguistic confusion that has so far been unresolved to the satisfaction of most. Because it is a somewhat complex and abstract paradox, summarizing it succinctly on a casual blog is daunting if not impossible. Thus, I am going to lean heavily on Wikipedia's entry for the following discussion. 

What I hope to show by the end of this is that understanding the nature of Curry's Paradox can teach us a great deal about our own personal ideas.

The Paradox
Citing Wikipedia, we can express the paradox as follows:

If this sentence is true, then Germany borders China.

Think about this for a moment. When I first heard Curry's Paradox, I had trouble seeing what exactly was paradoxical about it. But the fact that it is not immediately obvious highlights the contradiction perfectly. Breaking the issue up into its components, we have the following:
  • We know in advance that Germany does not border China.
  • As a result of this knowledge, we know that the sentence is not true.
  • But wait... the statement doesn't say that Germany borders China, it merely states that Germany borders China if the sentence is true.
  • So the sentence is actually entirely true.
  • But wait... Germany doesn't border China...
If you're like me, the above statements more or less reflect your thought process upon first encountering Curry's Paradox.

The Main Issue
What's really going on here is not the contradiction in a statement that is simultaneously true and false. Instead, we're looking at a statement that forces us to accept a false conclusion by assumption. What I mean is, when you say that Germany borders China if this statement is true, you are expressing not a hypothesis, but an assumption

Compare that to the sentence "If I hurry, I will make it to the opera before they dim the lights." This latter sentence is still expressed using an "If P then Q" structure, but we're not assuming anything, we're merely providing conjecture as to the direct consequence of hurrying. To prove that statement, we'd have to demonstrate something about speed, distance, and time.

Curry's Paradox, by contrast, involves no exogenous information. It is an apparently false statement that can be proven true only by using the statement itself as the underlying assumption

It is for this reason that I believe that the crux of Curry's Paradox is the choice between a contradiction and a false assumption

That's Logic. What About Life?
Understanding Curry's Paradox gives us insight into debates we may have with others. Often, in the midst of a complicated discussion, people appeal to each other by citing examples and hypothetical illustrations. These are important argumentative tools. However, it's important to keep in mind that your debate counterpart may be tailoring his or her argument to suit.

For example, if I wanted to explain to you why high taxes for rich people hurt the national economy, I might describe a situation in which an over-taxed business owner facing a tax increase opts to decrease production or fire a proportion of her full time employees. There is nothing wrong with this example, other than the fact that its truth is only evident if we assume that it is a likely story in the first place. Of course, assuming it is a likely story is effectively the same as assuming it is true.

Keep in mind that the paradox still holds for statements that are objectively true. I am still engaging in Curry's Paradox if I say, "If this statement is true, then Odysseus is the central character in The Odyssey." The fact that the latter part of the sentence is indisputably true does not mean the sentence itself is any less of a paradox.

Uh, Okay... Therefore...?
Therefore Curry's Paradox teaches us a lesson about our beliefs. We have to be careful about how we construct the reasoning behind our beliefs. We may in fact be starting from the initial belief that our beliefs are correct, and then concluding from that assumption that we were right all along.

More practically useful is the fact that someone you talk to may be engaging in such a ploy. You'll be a better-equipped person if you are able to recognize such situations and demonstrate to your friends why what they're saying is objectionable.