Afour-valued logic is anylogic with four truth values. Several types of four-valued logic have been advanced.
Nuel Belnap considered the challenge ofquestion answering by computer in 1975. Noting human fallibility, he was concerned with the case where two contradictory facts were loaded into memory, and then a query was made. "We all know about the fecundity of contradictions in two-valued logic: contradictions are never isolated, infecting as they do the whole system."[1] Belnap proposed a four-valued logic as a means of containing contradiction.[2][3]
He called the table of valuesA4: Its possible values aretrue,false,both (true and false), andneither (true nor false). Belnap's logic is designed to cope with multiple information sources such that if only true is found thentrue is assigned, if only false is found thenfalse is assigned, if some sources say true and others say false thenboth is assigned, and if no information is given by any information source thenneither is assigned. These four values correspond to the elements of thepower set based on {T, F}.
T is the supremum andF the infimum in the logicallattice where None and Both are in the wings. Belnap has this interpretation: "The worst thing is to be told something is false simpliciter. You are better off (it is one of your hopes) in either being told nothing about it, or being told both that it is true and also that it is false; while of course best of all is to be told that it is true." Belnap notes that "paradoxes of implication" and are avoided in his 4-valued system.
Belnap addressed the challenge of extendinglogical connectives toA4. Since it is the power set on {T, F}, the elements ofA4 are ordered byinclusion making it alattice withBoth at the supremum andNone at the infimum, andT andF on the wings. Referring toDana Scott, he assumes the connectives areScott-continuous ormonotonic functions. First he expandsnegation by deducing that and. To expandAnd andOr the monotonicity goes only so far. Belnap uses equivalence () to fill out the tables for these connectives. He findsF whileT.
| N | F | T | B | |
|---|---|---|---|---|
| N | N | F | N | F |
| F | F | F | F | F |
| T | N | F | T | B |
| B | F | F | B | B |
| N | F | T | B | |
|---|---|---|---|---|
| N | N | N | T | T |
| F | N | F | T | B |
| T | T | T | T | T |
| B | T | B | T | B |
The result is a second latticeL4 called the "logical lattice", whereA4 is the "approximation lattice" determining Scott continuity.
Let onebit be assigned for each truth value: 01=T and 10=F with 00=N and 11=B.[4]
Then thesubset relation in thepower set on {T, F} corresponds to order in two-bit representation. Belnap calls the lattice associated with this order the "approximation lattice".
The logic associated with two-bit variables can be incorporated into computer hardware.[5]
As adiscrete system, the four-valued logic illustrates a set ofstates subject to transitions bylogical matrices to form atransition system. An input of two bits transitions to an output of two bits throughmatrix multiplication.
There are sixteen logical matrices that are2 × 2, and four logical vectors that act as inputs and outputs of the matrix transitions:
When C is input, the output is always C. Four of the sixteen have zero in one corner only, so the output of vector-matrix multiplication with Boolean arithmetic is always D, except for C input.
Nine further logical matrices need description to fill out the labelled transition system where the matrices label the transitions. Excluding C, inputs A, B, and D are considered in order and the output in X expressed as a triple, for example ABD for commonly known as theidentity matrix.
The asymmetric matrices differ in their action on row versus column vectors. The row convention is used here:
The remaining operations on X are expressed with matrices with three zeros, so outputs include C for a third of the inputs. The codes are CAA, BCA, ACA, and CBB in these cases.
A four-valued logic was established byIEEE with the standardIEEE 1364: It models signal values in digital circuits. The four values are1,0,Z andX. 1 and 0 stand forBoolean true and false, Z stands forhigh impedance or open circuit and X stands fordon't care (e.g., the value has no effect). This logic is itself a subset of the 9-valued logic standard calledIEEE 1164 and implemented in Very High Speed Integrated Circuit Hardware Description Language,VHDL'sstd_logic.
One should not confuse four-valued mathematical logic (using operators, truth tables,syllogisms, propositional calculus, theorems and so on) with communication protocols built using binary logic and displaying responses with four possible states implemented with Boolean-like type of values : for instance, theSAE J1939 standard, used forCAN data transmission in heavy road vehicles, which has four logical (Boolean) values:False,True,Error Condition, andNot installed (represented by values 0–3).Error Condition means there is a technical problem obstructing data acquisition. The logics for that is for exampleTrue andError Condition=Error Condition.Not installed is used for a feature that does not exist in this vehicle, and should be disregarded for logical calculation. On CAN, usually fixed data messages are sent containing many signal values each, so a signal representing a not-installed feature will be sent anyway.
Creation ofcarbon nanotubes forlogical gates has usedcarbon nanotube field-effect transistors (CNFETs). An anticipated demand fordata storage in theInternet of Things (IoT) provides a motivation. A proposal has been made for32 nm process application using a split bit-gate: "By using CNFET technology in 32 nm node by the proposed SQI gate, two split bit-lines QSRAM architectures have been suggested to address the issue of increasing demand for storage capacity in IoT/IoVT applications. Peripheral circuits such as a novel quaternary to binary decoder for QSRAM have been offered."[6]
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