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Zeroth order logic is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, boolean functions, logical connectives, monadic predicate calculus, propositional calculus, and sentential logic. The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type 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 by means of the sequence of four boolean values, Such a sequence, perhaps in another order, and perhaps with the logical values and instead of the boolean values and respectively, would normally be displayed as a column in a truth table.
Language lists the sixteen functions in the form where the index is a bit string formed from the sequence of boolean values in
Language notates the boolean functions with an index that is the decimal equivalent of the binary numeral index in
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 operators, the first few of which are given in the following variant notations:
It may be noted that is the same function as and The inclusive disjunctions indicated for and for may be replaced with exclusive disjunctions without affecting the meaning, since 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.