Monday, 26 October 2015

The Nature of Paradox


At first my reaction involved thinking that this person was clearly confused, but then I wondered if, perhaps, I think about paradoxes in a slightly different way to most people.  If that is so, then I should clarify what I mean when I use the term "paradox".

I've actually written about paradoxes a few times (Patently Paradoxical Pabst's Perplexing Performance, WLC Takes Us for a Ride, There is no Twin Paradox, Immovability and a series on the Bertrand Paradox) but it was in my response to Melchior regarding the Bertrand Paradox that, possibly, I have most clearly articulated my position on what a paradox means.

Let me try again.

As far as I am concerned, thinking only about the strictest meaning of the term "paradox", if a statement is paradoxical then it is:

  • wrong,
  •  self-referential, or
  •  self-referential and wrong

If you are thinking through the logic associated with a proposition and you come across a paradox, then there is something wrong with either the proposition or your thinking about it.  (Note that we can use paradoxes to identify where our thinking is incorrect, but we can't use them to bootstrap the non-existent into existence.)  For this reason, I tend to think in terms of resolving a paradox - which means identifying the problem in thinking that leads to the appearance of a paradox.  Once you've eliminated the problem, then you no longer have a paradox.

There are some paradoxes for which the problem cannot really be eliminated, because a statement is in some sense self-referential, but these tend to be either meaningless or vague.  An example is the classic "this statement is false".  Sure, it's paradoxical, but it's also meaningless, since it refers only to itself.  Another is the even more classic "all Cretans are liars" (as spoken by a Cretan).  It's only paradoxical if you define "liar" to mean a person who always lies, as opposed to the rather more accurate, if also somewhat vague definition - namely someone who lies (with some undefined frequency).

Where a paradox is meaningful (at least in some sense), it tends to arise because of limitations on logic.  Russell's paradox, for example, is self-referential, but it's not meaningless because of its application to set theory. That said, it did show that na├»ve set theory was flawed, so it is amenable to a trivial resolution.  Another paradox that can be trivially resolved is the paradox of the stone.  The paradox hangs on the notion of omnipotence.  Once you accept the fact that omnipotent beings can't exist, the "paradox" dissipates.

It's worth nothing that logic works within a framework.  For example, we could look at a simple syllogism:

(Major Premise) if A then B → (Minor Premise) A → (Conclusion) therefore B

Using this form, we could conclude that, given that I have walked the dogs, the dogs will be tired.  What we can't conclude, using this syllogism, is that the form of the syllogism is true and valid.  Trying to avoid the assumption that the form of the syllogism is true and valid leads to a sort of paradox:
if a syllogism of the form
  • if A then B → A → therefore B
is true and valid then the syllogism
  • if I have walked the dogs then the dogs will be tired → I have walked the dogs → therefore the dogs will be tired
will be true and valid
a syllogism of the form
  • if A then B A therefore B
is true and valid 
therefore the syllogism
  • if I have walked the dogs then the dogs will be tired → I have walked the dogs → therefore the dogs will be tired
will be true and valid

While this seems to be saying that the conclusion is conditional on the truth of the minor premise, which is always the case for syllogisms of this form, the whole structure itself is in the form of the syllogism that is the subject of the minor premise (as shown by the colour coding, showing Major Premise, Minor Premise and Conclusion).

Now when I say this is a "sort of" paradox, I don't mean that it is necessarily an "actual" paradox.  Remember I said that we can use paradoxes to identify where our thinking is incorrect.  What this means is that we have falsifiability.  If this structure ever fails, then we say that we have falsified this form of syllogism.  It's about as scientifically rigorous as you can get, as well as being logically rigorous.

Similarly, we can test science scientifically and we do so all the time.  Our working hypothesis is that the scientific method always works - and this is a falsifiable hypothesis.  If we come across any situation in which rigorous application of the scientific method doesn't work, then (pseudo-paradoxically) we will have used the scientific method to show that the scientific method doesn't always work.  Good luck with that!

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