Көпмәнді логика: екіден артық шындық мәнінің логикасы

Propositional calculus in which there are more than two truth values

Many valued logic (also multi or multiple valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i. e., "true" and "false") for any proposition. Classical two valued logic may be extended to n valued logic for n greater than 2. Those most popular in the literature are three valued (e. g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), four valued, nine valued, the finite valued (finitely many valued) with more than three values, and the infinite valued (infinitely many valued), such as fuzzy logic and probability logic.

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"). 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), 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 20th century brought back the idea of multi valued logic. 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. Meanwhile, the American mathematician, Emil L. Post (1921), also introduced the formulation of additional truth degrees with n ≥ 2, where n are the truth values. Later, Jan Łukasiewicz and Alfred Tarski together formulated a logic on n truth values where n ≥ 2. In 1932, Hans Reichenbach formulated a logic of many truth values where n→∞. Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.

In 1921 Post defined a family of logics with (as in and ) the truth values Negation and conjunction and disjunction are defined as follows:

In 1951, Alan Rose defined another family of logics for systems whose truth values form lattices.

Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is "truth." In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn't have to be that of "truth"; instead, it can be some other concept. Multi valued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. 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. Thus, a proposition is not true or false; instead, it is justified or flawed. 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. The key difference is the determinacy of the preserved property: One may prove that P is justified, that P is flawed, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.

Functional completeness is a term used to describe a special property of finite logics and algebras. A logic's set of connectives is said to be functionally complete or adequate if and only if its set of connectives can be used to construct a formula corresponding to every possible truth function. An adequate algebra is one in which every finite mapping of variables can be expressed by some composition of its operations. Classical logic: CL = ({0,1}, ¬, →, ∨, ∧, ↔) is functionally complete, whereas no Łukasiewicz logic or infinitely many valued logics has this property. 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.

Known applications of many valued logic can be roughly classified into two groups. The first group uses many valued logic to solve binary problems more efficiently. For example, a well known approach to represent a multiple output Boolean function is to treat its output part as a single many valued variable and convert it to a single output characteristic function (specifically, the indicator function). 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). Many valued circuits have a number of theoretical advantages over standard binary circuits. 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. In memory design, storing two instead of one bit of information per memory cell doubles the density of the memory in the same die size. Applications using arithmetic circuits often benefit from using alternatives to binary number systems. For example, residue and redundant number systems can reduce or eliminate the ripple through carries that are involved in normal binary addition or subtraction, resulting in high speed arithmetic operations. These number systems have a natural implementation using many valued circuits. 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. In addition to aiding in the design of electronic circuits, many valued logic is used extensively to test circuits for faults and defects. Basically all known automatic test pattern generation (ATG) algorithms used for digital circuit testing require a simulator that can resolve 5 valued logic (0, 1, x, D, D'). The additional values—x, D, and D'—represent (1) unknown/uninitialized, (2) a 0 instead of a 1, and (3) a 1 instead of a 0.

An IEEE International Symposium on Multiple Valued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and verification. There is also a Journal of Multiple Valued Logic and Soft Computing.