zeroth order logic


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Note. This entry overlaps to some degree with other entries on boolean functionsMathworldPlanetmath ( and propositional logicPlanetmathPlanetmath (, but serves as a compact reference and a translation manual for several different styles of notation.

Zeroth order logic is a term in popular use among practitioners for the common principles underlying the algebra of setsMathworldPlanetmath, boolean algebraMathworldPlanetmath, boolean functions, logical connectives, monadic predicate calculus, propositional calculus, and sentential logic. The term serves to mark a level of abstraction in which the inessential differences among these subjects can be subsumed under the appropriate isomorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

1 Propositional forms

Table 1 lists equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath expressions for the four functions of concrete type X𝔹 and abstract type 𝔹𝔹 in a number of different languagesPlanetmathPlanetmath for zeroth order logic.

Table 1. Propositional Forms on One Variable
1 2 3 4 5 6
x= 1 0
f0 f00 0 0 () false 0
f1 f01 0 1 (x) not x ¬x
f2 f10 1 0 x x x
f3 f11 1 1 (()) true 1

Table 2 lists equivalent expressions for the sixteen functions of concrete type X×Y𝔹 and abstract type 𝔹×𝔹𝔹 in the same set of languages.

Table 2. Propositional Forms on Two Variables
1 2 3 4 5 6
x= 1 1 0 0
y= 1 0 1 0
f0 f0000 0 0 0 0 () false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x¬y
f2 f0010 0 0 1 0 (x)y y and not x ¬xy
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x(y) x and not y x¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x,y) x not equal to y xy
f7 f0111 0 1 1 1 (xy) not both x and y ¬x¬y
f8 f1000 1 0 0 0 xy x and y xy
f9 f1001 1 0 0 1 ((x,y)) x equal to y x=y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x(y)) not x without y xy
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x)y) not y without x xy
f14 f1110 1 1 1 0 ((x)(y)) x or y xy
f15 f1111 1 1 1 1 (()) true 1

The columns of Tables 1 and 2 are conveniently described in the following order:

  • Language 3.

    In Table 1, 3 describes each boolean function f:𝔹𝔹 by means of the sequence of two boolean values (f(1),f(0)).
    In Table 2, 3 describes each boolean function f:𝔹2𝔹 by means of the sequence of four boolean values (f(1,1),f(1,0),f(0,1),f(0,0)).
    Sequences of these forms, perhaps in another order and perhaps with the logical values F and T instead of the boolean values 0 and 1, would normally be displayed vertically in a truth tableMathworldPlanetmath under the column head for f.

  • Language 2 lists the functions in the form fi, where the index i is a bit string formed from the sequence of boolean values in 3.

  • Language 1 notates the functions fi with an index i that is the decimal equivalent of the binary numeral index in 2.

    Notice that the sense of the binary and decimal codings is highly dependent on context. One needs to know the number of variables in the function and the sequence of points over which it is evaluated in order to decode the indices properly.

  • Language 4 expresses the boolean functions in terms of two families of logical operationsMathworldPlanetmath:

    Logical conjunctions written as continued productsMathworldPlanetmathPlanetmath. For example:


    Minimal negation operators written as parenthesized lists. For example:


  • Language 5 lists ordinary language expressions for the propositional forms. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.

  • Language 6 expresses the propositional forms in one of the several notations that are commonly used in formal logic.

Title zeroth order logic
Canonical name ZerothOrderLogic
Date of creation 2013-03-22 17:55:47
Last modified on 2013-03-22 17:55:47
Owner Jon Awbrey (15246)
Last modified by Jon Awbrey (15246)
Numerical id 19
Author Jon Awbrey (15246)
Entry type Definition
Classification msc 03G05
Classification msc 03B05
Related topic PropositionalCalculus
Related topic LogicalConnective
Related topic LogicalGraph
Related topic LogicalGraphFormalDevelopment
Related topic TruthFunction
Related topic TruthTable
Related topic DifferentialLogic
Related topic DifferentialPropositionalCalculus
Related topic DifferentialPropositionalCalculusAppendices
Related topic DifferentialPropositionalCalculusAppendix2