Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition.
It is wrong that the first known classical logician who did not fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of [two-valued] logic"[1]).
In fact, Aristotle did not contest the universality of the law of excluded middle, but the universality of the bivalence principle: he admitted that this principle did not all apply to future events (De Interpretatione, ch.
IX),[2] but he didn't create a system of multi-valued logic to explain this isolated remark.
Until the coming of the 20th century, later logicians followed Aristotelian logic, which includes or assumes the law of the excluded middle.
The Polish logician and philosopher Jan Łukasiewicz began to create systems of many-valued logic in 1920, using a third value, "possible", to deal with Aristotle's paradox of the sea battle.
Later, Jan Łukasiewicz and Alfred Tarski together formulated a logic on n truth values where n ≥ 2.
) and Priest's "logic of paradox" add a third "undefined" or "indeterminate" truth value I.
The truth functions for negation (¬), conjunction (∧), disjunction (∨), implication (→K), and biconditional (↔K) are given by:[3] The difference between the two logics lies in how tautologies are defined.
In a similar manner he defined a logic with infinitely many truth values,
The implication above is the unique Heyting implication defined by the fact that the suprema and minima operations form a complete lattice with an infinite distributive law, which defines a unique complete Heyting algebra structure on the lattice.
[7] By adopting truth values defined in the same way as for Gödel logics
[9] Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations.
Multi-valued logics are intended to preserve the property of designationhood (or being designated).
For example, in a three-valued logic, sometimes the two greatest truth-values (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values.
Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion.
For example, the preserved property could be justification, the foundational concept of intuitionistic logic.
A key difference between justification and truth, in this case, is that the law of excluded middle doesn't hold: a proposition that is not flawed is not necessarily justified; instead, it's only not proven that it's flawed.
Functional completeness is a term used to describe a special property of finite logics and algebras.
[10] An adequate algebra is one in which every finite mapping of variables can be expressed by some composition of its operations.
[11][12] We can define a finitely many-valued logic as being Ln ({1, 2, ..., n} ƒ1, ..., ƒm) where n ≥ 2 is a given natural number.
Post (1921) proves that assuming a logic is able to produce a function of any mth order model, there is some corresponding combination of connectives in an adequate logic Ln that can produce a model of order m+1.
[14] The first group uses many-valued logic to solve binary problems more efficiently.
Other applications of many-valued logic include design of programmable logic arrays (PLAs) with input decoders, optimization of finite-state machines, testing, and verification.
The second group targets the design of electronic circuits that employ more than two discrete levels of signals, such as many-valued memories, arithmetic circuits, and field programmable gate arrays (FPGAs).
For example, the interconnect on and off chip can be reduced if signals in the circuit assume four or more levels rather than only two.
Applications using arithmetic circuits often benefit from using alternatives to binary number systems.
For example, residue and redundant number systems[15] can reduce or eliminate the ripple-through carries that are involved in normal binary addition or subtraction, resulting in high-speed arithmetic operations.
However, the practicality of these potential advantages heavily depends on the availability of circuit realizations, which must be compatible or competitive with present-day standard technologies.
An IEEE International Symposium on Multiple-Valued Logic (ISMVL) has been held annually since 1970.