I punched into my TI-84 the following, using the "<" function from the logic test: $$1<1<1$$ I'm sure that this statement must be false, for example it's false that $x<x$, so why does the calculator evaluate the expression as true, particularly if I also tried $1<1$ and it returned the right answer, that the statement is false?
According to the convention used by the calculator, a false statement evaluates to 0 and a true one evaluates to 1.
Perhaps (in an Occamistic sense) "$<$" is a binary operation that returns $0$ when the thing on the left is larger or equal than the thing on the right, and $1$ otherwise. Therefore, it is safe to assume that the calculator evaluated $$(1<1)<1=0<1=1$$
To test it, try $1<1<-1$.