Many-valued logic

Multi-valued logic is a generic term for all logical systems that use more than two truth values ​​.

Starting point for the development of multivalent logics was the epistemological question of whether the principle of bivalence plays except logical truth. For statements about the future already represents Aristotle this question by arguing that the truth of a statement such as "Tomorrow there will be a sea battle " until the evening of tomorrow will not be known and that she contingent up to this point yet as indeterminate and thus as must be considered possible.

The first formalized in the modern sense -valued logic is that in 1920 presented by Jan Łukasiewicz three-valued logic L3. Her three truth values ​​interpreted Łukasiewicz, citing the example of the naval battle Aristotle as "true", "false" and - for future statements whose truth is not yet known - "( contingent ) possible."

In recent times, many-valued logics have gained great practical importance in the field of computer science. They allow to deal with the fact that databases can contain not only uniquely determined, but also indefinite, missing, or even contradictory information.

Fundamentals of polyvalent logic

While multi-valued logic with the principle of bivalence one of the two basic principles of classical logic gives up, it retains the other basic principle of extensionality, in: The truth value of each composite statement is still uniquely determined by the truth values ​​of its sub- statements.

In contrast to classical logic in the interpretation of truth values ​​in many-valued logics is less natural given. There have been proposed many different interpretations. For this reason, and because many interpretations that do not watch more than two values ​​as shades or types of truth and falsehood, but as epistemically as gradation of knowledge or certainty (for example, with the three values ​​" known to be true ", " unknown " and " known to be false " ), the values ​​of polyvalent logic are often not referred to as truth values, but as a pseudo logical values, or as a quasi- truth values ​​. For the sake of compactness of this article nevertheless used throughout, the term " truth value ".

Besides the problem of interpretation of truth values ​​arise when dealing with multivalued logic numerous tasks of a technical nature and there are more interpretation problems: Basic concepts such as that of tautology, that of contradiction or that the conclusion must be redefined and interpreted.

Systems polyvalent propositional logic

Kleene logic

The Kleene logic contains three truth values, namely 1 for " true ", 0 for "false" and that is also referred to as i and is " neither true nor false." Kleene defined the negation, conjunction, disjunction and implication by the following boolean functions:

This forms - for example, even at Łukasiewiczs trivalent logic, see there - the disjunction of the maximum and the minimum of the conjunction of the associated truth values, and the negation of a statement with truth value is calculated as v 1 -v.

Considering 1 as the only designated truth value, then there is in no tautologies; Looking at both 1/2 and 1 as designated, then the amount of Tautologies is identical to the amount of the classic divalent statements logic.

Gödel logics and

Gödel defined in 1932 a family of multivalent logics with finitely many truth values ​​, so that, for example, includes the truth-values ​​and truth values ​​. He defines analog logic with infinitely many truth values ​​, in which are used to 1 as truth values ​​the real numbers in the interval of 0. Designated truth value in each of these logics 1

The conjunction and disjunction he defined as the maxima and minima of the formula truth values:

The negation and implication are defined by the following truth-value functions:

The Gödel's systems are completely axiomatizable, ie it can be set up calculi in which all tautologies of the respective system are derived.

Łukasiewicz logics

The implication and the negation defined by the following January Łukasiewicz truth-value functions:

First Łukasiewicz developed according to this scheme in 1920 his three-valued logic, the system with the truth values ​​and truth-value 1 -designate in 1922 following his unendlichwertige logic in which he extends the set of truth values ​​to the interval of real numbers from 0 to 1. Designated truth value in both cases is 1

Generalizing to disintegrate the Łukasiewiczen logics in the finitely systems ( truth value amount as in Gödel ), in the previously mentioned and, in truth values ​​as the rational numbers (ie, fractions) are used in the interval from 0 to 1. The set of tautologies, that is, the statements with truth value -designate, is identical with and.

Product logic

The product logic contains a conjunction and an implication which are defined as follows:

• For:

In addition, the product logic contains a truth constant, the " wrong " means the truth value.

By means of additional constants a negation and a further conjunct can be defined as follows:

Post- logics

Post 1921 defines a family of logics (as in and ) with the truth values ​​. Negation and disjunction defined mail as follows:

Tetrahydric logic of Belnap

Nuel Belnap developed in 1977 with his four-valued logic the truth values ​​t (true, true), f (false ), u ( unknown) and b ( both, so a contradictory information).

Bočvar logic

The Bočvar logic ( especially in English-speaking countries also Bochvar logic) contains two classes of connectives, namely the inner connectives one hand and the outer connectives other. The inner connectives negation, implication, disjunction, conjunction and Bisubjunktion correspond to those of classical logic. The outer connectives negation, implication, disjunction, conjunction and Bisubjunktion are metalinguistic in nature and are the following:

• ( Is wrong)
• ( is true, as well )
• ( True or is true)
• ( Is true and is true)
• ( Is true iff is true)

The truth-value functions correspond to those of Kleene logic.

For the definition of the outer connectives another digit is added connective taken, namely the external confirmation with the truth function

This allows the outer connectives, as follows, define:

The logic of the outer connectives, which makes a distinction between 0 and i corresponds exactly to the classical logic.

Fuzzy Logic

In fuzzy set theory, often referred to as fuzzy logic, ambiguous statements are also covered. An example is the statement that " the weather is very warm ." This statement is dependent on the actual temperature be true in varying degrees: 35 degrees with certainty, to some extent at 25 degrees, 0 degrees in any case. The arbitrarily determined degree of applying to be represented by a real number between 0 and 1.

The above -valued logics treat individual statements to be atomic, so do not know their internal structure. In contrast, the fuzzy set theory treats only statements with a very special internal structure: One basic set of possible observations (eg, the temperatures occurring ) are degrees of applying a statement ( " very warm " ) is assigned. The term quantity in fuzzy set refers to the set of observations, in which the degree of applying the statement is > 0; the concept of fuzzy points to the variable degree of applying.

The fuzzy set theory does not address the question of whether it is unknown or doubtful whether a statement is true. A statement as such is associated with absolutely no truth value; the degrees of applying are no truth-values ​​, but rather interpretations of an original measured value.

The fuzzy set theory provides also methods, the degree of applying for statements in which several elementary statements are linked ( " the weather is warm and dry" ) to be determined. The method of combining Applying degrees can be partially applied to the combination of truth values ​​of many-valued logics.

Application of multivalent logics

In the development of hardware logic circuits are used to simulate multi-valued logic, to present different states, as well as the tri-state gate and model buses. In the hardware description language VHDL as defined in the IEEE Standard with the number 1164 neunwertige logic is often used, the default Logic 1164. Has the values

Standard Logic 1164 neunwertige a logic for hardware simulation

In a real circuit occur only 1, 0 and on (with inputs and outputs ) Z. In the simulation, the state U occurs in signals, which so far has not yet been assigned another value. The value - ( Don't-Care is outside of VHDL often depicted with X) is only used for synthesis; it signals the translation program that a certain condition is not provided and therefore it does not matter how the synthesized circuit deal with this condition.

The distinction between strong and weak drivers is used in a case of conflict (when two outputs are connected together on a single line and provide different values ​​) to decide which signal the corresponding line is attributed. This conflict often occurs in bus systems, where several bus nodes start at the same time to send data. Will now face a 1 (high) to an L (low ), then the strong signal prevails, and the signal line, the value 1 is attributed. Meeting, however, equally strong sequential signals, the signal line is in an undefined state. These states are X (for conflict between 1 and 0 ) and W (at conflict between H and L).

Demarcation

Multi-valued logics are often discussed under metaphysical or epistemological issues. This includes, for example, the common question which logical system " right," ie, which logical system, the reality correctly (or rather best) describes. Different philosophical currents give different answers to this question; some trends, such as positivism, reject even the question itself as meaningless from.