Aristotle

Aristotle (384 - 322 BC) laid the foundation of western mathematics. He laid down the laws of mathematics and logic in "The Laws of Thought". One important law that had profound implications for later logic and philosophy was "The Law of the Excluded Middle" that states every proposition must be true or false. In other words, there is no middle ground; for example, 1 + 1 is either 2 or it is not 2.

Unfortunately, Aristotle's Low of the Excluded Middle can get us into trouble. The British mathematician Russell presented the following paradox.

The crew of a ship consists only of men who all all clean-shaven. A barber on board claims that he shaves only  those men who don't shave themselves. Who shaves the barber? If he doesn't shave himself then he does. And if he shaves himself then he does not.

Russell set down the laws of modem logic in his Principia Mathematica in 1910.

A similar paradox due to Epimenides goes:

All Cretans are liars
I am a Cretan.

Again we have a similar problem where the application of Aristotelian logic breaks down.

There have been attempts to deal with the paradoxes thrown up by Aristotle's Law of the Excluded Middle by the Polish logician Łukasiewicz and Knuth, both of whom suggested three-level logic.

Zadeh

In 1965 Lotfi Zadeh at UC Berkeley proposed a logic system that supported infinite value logic. Zadeh proposed that an element can have a membership function that describes its membership of a set. For example, the expression mA(x) is the membership function of x in A.

Zadeh's logic was called "Fuzzy set theory" which has proved a little unfortunate because some have taken "fuzzy" to mean imprecise or inaccurate and they regard the use of fuzzy logic rather like the fraudulent snake oil remedies of a hundred years ago.