Fuzzy logic is a sort of computer logic that is different from boolean algebra. It is different in the way that it allows values to be more accurate than on or off. While boolean logic only allows true or false, fuzzy logic allows all things in between. An example of this could be a computer game: A person is standing in a doorway while a thing explodes. The character is hit or not hit if boolean logic is used, but the doorway protects him from the explosion. Therefore, he might only be hit 20%, and takes less damage.
To put it in more precise mathematical terms, classical logic has two values. These values are usually called false (0) or true (1). With fuzzy logic, a (calculated) value of 0.8 or 0.971 is possible. It is important to know the difference between fuzzy logic and chance. A coin that is thrown has a chance of 0.5 for landing heads up. If it is thrown 1000 times, it is expected that it will land with heads side up 500 times. With fuzzy logic, a thing with a "truth value" of 0.5 will have a value of 0.5 no matter how many times it is done. It is not a 50% chance of true or untrue, it is 50% true and 50% untrue at the same time. Fuzzy logic is used a lot in expert systems and neural networks.
Humans tend to use a combination of predicate logic and fuzzy logic. If you are an outfielder catching a baseball hit into the air; then your precise logic will calculate trajectory and start you running to the point of intercept (catching). However, once close to the ball the eyes and brain of the outfielder lacks the ability to accurately estimate distance and speed because the ball is coming straight at the outfielder. The human brain switches to fuzzy logic that says "get me closer", "get me closer", and so on. That is why you see outfielders in baseball run to a spot and then move around as the ball gets closer.
Predicate logic says calculate the point to be at to catch the ball. Fuzzy logic says because of wind or other things you might not be in the correct place so just keep getting closer until you catch the ball.
In predicate logic it is the mathematics of calculating the path of the ball that determines your action. In fuzzy logic it is the error of your calculations that determines your action.
In effect, it's like your brain trying to control your finger to pick your nose while traveling down a bumpy road.