The Liar statement, “This statement is false”, is one of the oldest unresolved
paradoxes. To see why it is paradoxical, consider whether it is true or false. On
the one hand, if it is true, then since it says it is false, it must be false. But surely
it cannot be both true and false! So, on the other hand, maybe it is false. But
then, since it simply asserts its own falsity, it must be true. It seems that we are
forced into a contradiction. On first glance, this may seem like a cute puzzle or a
mere oddity, but the Liar paradox indicates that there is a deep flaw in our
ordinary conception of truth.
I’m interested in “gappy” solutions to the paradox– solutions that hold that some
statements (like the Liar) are neither true nor false. The statements falling in this
gap between truth and falsity might be called pathological. Introducing this new
category of statements lets us deal with the Liar statement above, but the Liar
gets its revenge in a strengthened form: “This statement is false or pathological”.
This strengthened Liar statement cannot consistently be considered true, false, or
pathological, so it seems that we need to introduce yet another category of
statements. Introducing the new category leads to another paradoxical
statement, which leads to another category, which leads to another paradoxical
statement, and so on. The result is a theory of truth with infinitely many
categories of statements. I’m working on developing the philosophical
motivations for such a theory.