By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type ''X'' × ''Y'' → and abstract type × → in a number of different languages for zeroth order logic.
These six languages for the sixteen boolean functions are conveniently described in the following order:
- Language describes each boolean function ''f'' : '''B'''2 → '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a Truth Table .
- Language lists the sixteen functions in the form '''fi''', where the index '''i''' is a Bit String formed from the sequence of boolean values in '''L3'''.
- Language notates the boolean functions '''fi''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L2'''.
- Language expresses the sixteen functions in terms of logical Conjunction , indicated by concatenating function names or proposition expressions in the manner of products, plus the family of '' Minimal Negation Operator s'', the first few of which are given in the following variant notations:
:
It may also be noted that is the same function as and , and that the inclusive disjunctions indicated for and for may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function is not the same thing as the function .
- Language lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
- Language expresses the sixteen functions in one of several notations that are commonly used in formal logic.
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