"The Liar's Paradox [Epimenides' "pseudomenon"] and other '''Self-Reflexive''' Paradoxes of Logic and Set Theory.

The ancient «arché», and epitome, of all of these “self-referential” paradoxes, is the Epimenides “peudomenon”, or “Liar Paradox”.

This ‘‘‘[self-]reflexive’’’ [cf. Bertrand Russell], “self-referential” sentence, epitomized as “This sentence is false.”, can be translated, into original-Boolean algebra, as the logical equation 

t  =  (0/0)(1 - t). 

One of the two branches of this equation is t  =  (1)(1 - t), which simplifies to t   =  1 - t, so that 2t = 1.   

So t = 1/2  for that branch.

In original-Boolean, this means that the quoted sentence is “true half of the time, false the other half of the time”, as in a cyclical, “infinite” frequency, “infinitesimal” wave-length oscillation between 0 and 1. 

Of course, many more meaningful sentences exhibit truth-value fluctuation, in an aperiodic way, more like a “strange attractor”, less like a “limit-cycle attractor”, in terms of the 'solution-geometries' of nonlinear differential equations in their state-spaces.   

For example, the sentence, “It is raining.”, is true, lately, on Earth’s surface, some fraction of the time.    

It’s negation, “It is not raining.”, is true ‘1 minus that fraction’ of the time.

Sentences like, “It is sunny.”, are true during some fraction of that ‘1 minus that fraction’ of the time.