Non-Aristotelian logic
The term non-Aristotelian logic, sometimes shortened to null-A, means any non-classical system of logic which rejects one of Aristotle's premises.

# History
Nicolai A. Vasiliev since 1910 and Jan ukasiewicz called their own work non-Aristotelian logic. Alfred Korzybski carried the term into his system of General Semantics in 1933 (citing ukasiewicz), and science fiction writer A. E. van Vogt later helped popularize it.  Korzybski focused on the use of three or more truth values in the new systems of logic, although he connected this to his own rejection of Aristotle's principle of identity. Following ukasiewicz's early work, Korzybski and later proponents of General Semantics associate these truth values with probabilities and the use of scientific induction. ukasiewicz later seemed more cautious about this connection.
While ukasiewicz seems to have spent more time on three-valued logic than any other system, he said that one could keep increasing the number of truth values indefinitely. Thus, he wrote: "if 0 is interpreted as falsehood, 1 as truth, and other numbers in the interval 0-1 as the degrees of probability corresponding to various possibilities, a many-valued logic is obtained which is expansion of three-valued logic and differs from the latter in certain details." 
Nicolai A. Vasiliev in 1910 rejected the law of contradiction as well as law of the excluded middle and proposed a logic he called imaginary which is tolerant to contradiction.
Hans Reichenbach described a system of logic that he explicitly linked with probability theory. He called his probability logic a generalization of two-valued logic. Reichenbach also suggested applying a three-valued logic to quantum mechanics. His probability logic does not receive much attention from modern logicians.
Aristotle allowed for the possibility of all these logics in De Interpretatione, Chapter 9. He wrote here that when it comes to statements about the future, "it is not necessary that of every affirmation and opposite negation one should be true and the other false." (Revised Oxford translation)
Lotfi Zadeh developed a system of "fuzzy logic" using a range of truth values from 0 to 1, but distinguished it sharply from probability theory.
Robert Anton Wilson in The New Inquisition developed a non-Aristotelian system of classification in which propositions can be assigned one of 7 values:  true, false, indeterminate, meaningless, self-referential, game rule, or strange loop.  Wilson did not devise a formal system for manipulating propositions once classified, but suggested that we can clarify our thinking by not restricting ourselves to simplistic true/false binaries.
Alternative terms for these logics in common academic usage include deviant logic and multi-valued logic (see Haack, 'Philosophy of Logic', 1980).  Not all non-classical logics fall into this class, e.g. Modal logic is a non-classical logic which, however, has only two truth values.

# Use in science fiction
The concept of non-Aristotelian logic was used by A. E. van Vogt as the central theme in his The World of Null-A novels, based on his interest in Alfred Korzybski's General Semantics, stories tinged by van Vogt's reflections upon revelations of police state conditions enforced by totalitarian regimes after World War II.
Van Vogt generally shortened non-Aristotelian logic to null-A in his description of logic systems incorporating three or more values, to represent relatively 'subjective' conclusions from inductive logic, rather than relying strictly on the binary, deductive reasoning. The null-A concept as depicted by van Vogt is complementary to Aristotle's system of two-valued, true/false logic, i.e., "A is either B, or it is not B."
Van Vogt's portrayals of General Semantics in sci-fi stories (wherein heroic characters struggled against incrementally stemwinding tactics used by minions of authoritarian entities), was somewhat different from its originator's, as Korzybski developed and described General Semantics not as a 'logic', but as a non-Aristotelian system of evaluation.  On the other hand, van Vogt also depicted General Semantics as a method of evaluation used to analyze the reasoning of others.  Protagonists in van Vogt's science fiction novels typically used null-A reasoning in almost dream-like settings to outwit villains who relied almost exclusively upon decision-tree, algorithmic reasoning, akin to the use of Aristotelian logic.