# Mathematical Digest

### Short Summaries of Articles about Mathematics

in the Popular Press

"This headline is (half) false." *The Economist,* October 2003, page 81.

The self-referential statement "this sentence is false"---known as the Liar paradox---is not resolvable using standard logic: if it is false, then it must be true, and if it is true, then it must be false. In the 1960s, Lotfi Zadeh came up with the idea that things can be partially true or false---known as "fuzzy logic"---and was able to resolve simple self-referential paradoxes like the liar paradox: it is half-true. More complicated paradoxes could only be described by non-linear systems of equations which were much more difficult to solve.

Now, Kostis Vezerides of the American College of Thessaloniki and Athanasios Kehagias of the Aristotle University of Thessaloniki report that most self-referential paradoxes can be resolved consistently using fuzzy logic. They used the "Brouwer Fixed-Point Theorem" from topology to prove that at least one solution had to exist for each paradox. They then looked to "control theory"---the science of how to operate complicated systems---for the means to best approximate, then solve, the equations that describe the paradox. One of their results: a set of self-referential statements that assert absolute falsity or veracity about each other---Grace says "Susan always lies," while Susan says "Grace always tells the truth"---must be precisely half-true. Possible directions for future research include developing a new form of logic, somewhere between fuzzy and binary logic.

*--- Claudia Clark*