15.00 - Number of Logical Operators Supported for Bivalent and Trivalent Logics - Teradata Database

Teradata Database Design

prodname
Teradata Database
vrm_release
15.00
category
User Guide
featnum
B035-1094-015K

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.

  • ‘value’ = ‘value’
  • 2 = 2
  • 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.