Number of Logical Operators Supported for Bivalent and Trivalent Logics
In a bivalent logic, there are exactly four possible monadic, or single-operand, logical operators, as indicated in the following table. Note that the numberings have no meaning other than to distinguish the operators from one another.
|
This operator number … |
Performs these mappings of TRUE and FALSE values … |
|
1 |
T → T F → T |
|
2 |
T → F F → F |
|
3 |
T → F F → T |
|
4 |
T → T F → F |
By generalization, it is also true that there are exactly 16 possible dyadic, or two-operand, logical operators.
The formulas that produce the numbers of monadic and dyadic operators are provided as follows, where n represents the valence of the logic in question. For a bivalent logic, its value is 2; for a trivalent logic, its value is 3.
By performing the calculations for a trivalent logic, you find that 3VLs have 27 possible monadic operators and 19,683 possible dyadic operators. These figures, of course, are redundant: they represent the number of possible operators, not the number of useful operators.
SQL, like most programming languages, overdetermines its operators. For example, the function performed by the = operator in the following two equations is not identical internally, but the language uses the same symbol for the operation in both cases.
The issue raised here is whether SQL supports a sufficient number of trivalent logical operators to fully support a 3VL, and the answer to that question is negative.
Nevertheless, they indicate that SQL does not support anywhere near the number of logical operators required to support a coherent 3VL. Imagine the difficulty of trying to support the 4VL that Codd (1990) advocates, for which there is a possible 256 monadic operators and over four billion dyadic operators (4,294,967,296 to be exact). See Bolc and Borowik (1992, 2000) for examples of how complicated many‑valued logics are.