This weekend I attended a really fascinating lecture titled Speaking of Paradoxes which was presented by philosophy professor Graham Priest and hosted by the New York City Skeptics. Priest began by positing the following scenario: "Suppose you have professor Smith, and he's completely at odds with Professor Brown down the hall. Smith tells all his students not to listen to anything that Brown says, and to emphasize the point he writes on his blackboard 'Everything written on the blackboard in Room 33 is false!' But Professor Smith has made a mistake. Actually he's the one lecturing in Room 33." So is the statement "Everything written on the board in Room 33 is false" a true or false statement? If it's true then it's false in which case it's true in which case it's false, etc. This is a classic conundrum called the Liar Paradox which has bedeviled philosophers for thousands of years.
The foundations of logic in Western philosophy were laid by Aristotle, who argued that all propositions must logically be either true or false. If a statement isn't true then it must be false and if it isn't false then it must be true. For a statement to be both true and false is a contradiction, and contradictions are impossible in formal logic, therefore if a line of reasoning leads to a contradiction, there must be something wrong with the line of reasoning. But in all the years that philosophers have been arguing about the Liar Paradox, no one's ever come up with a problem in the line of reasoning, so maybe, Priest argues, what we have to do is question Aristotle's Law of Non-Contradiction itself. This is apparently a somewhat heretical notion. But despite its long pedigree, Priest argues that Aristotle's defense of the law is actually pretty flimsy, resting as it does on the presumption that if a statement such as "A and not A" is true, then it follows that anything is true. Priest argues that this is nonsensical.
Instead of dividing all statements into "true" and "false," he proposes a grid consisting of "true," "false," "true and false," and "neither true nor false." Apparently some ancient Buddhists played around with this idea for a few centuries before ultimately abandoning it -- regrettably, in Priest's view. No matter how hard you shove, Liar Paradox arguments such as "this statement is false" just aren't going to fit into either the "true" or "false" categories, but "true and false" seems like a pretty good fit for it (or maybe "neither true nor false").
He also discussed a number of other classic paradoxes, such as: Are there fewer even numbers than there are numbers total? Common sense would seem to say yes -- that there are half as many even numbers as numbers total -- but then, aren't there an infinite number of even numbers and an infinite number of numbers total, and doesn't infinity = infinity? There's also Zeno's Paradox -- Before you can get to point A, don't you first have to pass through a point halfway between where you're standing and point A? And before you can get halfway to the halfway mark, don't you first have to pass through a point halfway there? And before that don't you have to pass through a point halfway there? And isn't it possible to keep iterating this process infinitely, and so how is it possible to move at all, if there's an infinitely expanding set of tasks you must accomplish before you can get anywhere? These ones Priest indicated have been settled.
One that hasn't been settled apparently is this: It seems sensible to say that one nanosecond in a person's maturation process can never make such a dramatic difference that it would be sensible to call that person a "child" at one instant and then an "adult" one nanosecond later. So take a small child and fast forward time one nanosecond and ask, "Is this person still a child?" Yes, according to the principle established above -- one nanosecond just can't make that much difference. The problem is, you can keep repeating that process and by that same seemingly firm logic the "child" will never become an "adult," even if the compounded nanoseconds add up to seventy years.
Really cool, thought-provoking stuff. Anyway, these lectures are all recorded and are eventually made available online, if anyone's interested.