When logicians talk about contradictions, they mean claims in the form that something
is the case and is not the case.
So, I am in New York and not in New York.
The sun is shining and is not shining.
Three is a prime number and it is not a prime number.
That kind of thing.
And when we meet something like that, mostly our reaction is, "Hey, that can't be right."
So, we're dealing with something called the principle of noncontradiction or the law of noncontradiction.
And roughly formulated, that's the view that a contradiction can't be true.
You can't accept a contradiction.
And, it has to be said that that principle is high orthodoxy in Western philosophy.
The status in Eastern philosophy is somewhat different, but let's just stick to the West.
Aristotle enunciates and defends the law in The Metaphysics and it's clear that he is
attacking some people who he thinks accept some contradictions.
So for him it's a live issue, but he gives a bunch of arguments in favor of the law,
and those really determine the ethos of Western philosophy on this topic.
With very few exceptions, Western philosophers have endorsed the law.
It's not because the arguments are very good. Some of them are so tangled it's actually
hard to sort what they are.
And indeed in modern scholarship on that bit of the Metaphysics, there's a rough consensus
that these arguments don't really fly.
As all positions in philosophy, you have to make a case for it,
and that's what philosophers who've been doubting the law of non-contradiction have been doing
in the last thirty or forty years - making a case.
An important part of the case is to use examples,
but another part of the case is to show how these examples aren't really susceptible to other treatments.
So, let me give you an example. There's a paradox called the Liar Paradox, which is actually very old -
It goes back to Eubulides, 5th century, before the common era.
Suppose I say, "This very thing I am telling you now is false.”
Then you say, "Well is that true or false?"
Well, if it's true, it says "This very thing I'm telling you now is false." - So if it's true it's false...
Hey, but it's false, well then, I've just said that, so it's true.
So, it seems to be true and false.
This is the Liar Paradox, right? And to many people who have never seen this sort of stuff before
it can certainly appear like a party trick.
It's never been treated that way by philosophers and logicians, and it has a remarkable history that -
At the beginning of the 20th century it was discovered that this is one of a family of paradoxes
that turned up in the foundations of mathematics.
So, what you say about this paradox is entrenched in hard questions in mathematics.
But people have tried to analyze this paradox for two-and-a-half-thousand years and explain
why the conclusion doesn't follow.
And their success has been minimal.
There are thousands of solutions in both contemporary logic and medieval logic and ancient logic,
and they've all got obvious problems.
So, the only thing there is consensus about is that there is no consensus
One thing that some modern logicians such as myself say is: Hey, look at this paradox.
We've been trying to say what's wrong with it, we've been barking up the wrong tree.
What we should be doing is just accepting the conclusion.
Other ways are trying to diffuse the contradiction don't have a very successful track record.
So, the thought is we are better off just to accept the contradiction.
Take the contradiction that I am both in New York sitting here talking to you
and not in New York because I'm in Australia.
That doesn't sound very plausible.
So, the next question that anyone is going to ask about this is, well,
"How do you know which one's good?"
And the answer to that is how do you know that anything is true? You look at the evidence, right?
So, Hume said that the wise person is the one who apportions their beliefs according to the evidence.
Now, there's just no evidence that I am not in New York.
That's why it is crazy not to believe it.
But the Liar Paradox is different because we've got this first order evidence: Here's the argument.
We've got second order evidence: Hey, attempts to get out of this...
argument haven't been very successful for 2,000 years.
So again, you look at the evidence.
My view about these matters is unorthodox in terms of contemporary logic and philosophy.
There are very few logicians who agree with this. There are more now than there were thirty ago.
In the early years when we were talking about this stuff it was regarded as
completely off the table, and people like myself were regarded as just not worth talking seriously
or worth talking about just because it's "looney tune" stuff.
Now, the attitude is changing and people are realizing that the law of non-contradiction
is not so easy to defend.
After we've hammered through so many of the philosophical and logical issues for the last thirty years,
people have come to realize that the issues were not as simple as they thought.
So, people are prepared to engage with the issues now in a way that they weren't thirty years ago.
So, my position in the establishment
is of a bit of a heretic, but not a complete fruitcake.