Example of mathematical antilogies involving the equality symbol ( always false statements, for all permissible values of the variables). Logicians ( in propositional calculus) classify statements/formulas into 3 categories : tautologies ( always true) , contingent statements ( sometimes true, sometimes false) , antilogies ( always false). I can find examples of mathematical "tautologies" , like (a+b)²=a²+b²+2ab. I can find an example of mathematical contingent statement: a+a=a ( which is true if x=0, false otherwise), or a²=a ( true iff x=0, x=1) But I cannot find an example of mathematical "antilogy" ( a statement that would be false for all permissible values of the variables) that would be an equality.

Since this was received well in the comments, and since it's generally considered better to have answers posted as answers: $$a=a+1$$

Digression: Of course, a lot depends on the phrase, "permissible values of the variables". If only natural numbers are permissible values, then $a+1=0$ answers the question. If only integers are permissible, $a+a=1$. If only rationals are permissible, $a^2=2$. If only reals, $a^2+1=0$. One might even object that $a=a+1$ is not an antilogy, if infinite cardinals are permissible. So perhaps one has to go to $a-a=1$ for an example of a one-variable equation that is an antilogy in any theory in which subtraction is a binary operation and $0\
e1$.