zeroth order logic
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Note. This entry overlaps to some degree with other entries on boolean functions^{} (http://planetmath.org/BooleanValuedFunction) and propositional logic^{} (http://planetmath.org/PropositionalCalculus), 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 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 inessential differences among these subjects can be subsumed under the appropriate isomorphisms^{}.
Contents:
1 Propositional forms
Table 1 lists equivalent^{} expressions for the four functions of concrete type $X\to \mathbb{B}$ and abstract type $\mathbb{B}\to \mathbb{B}$ in a number of different languages^{} for zeroth order logic.
Table 1. Propositional Forms on One Variable ${\mathcal{L}}_{1}$ ${\mathcal{L}}_{2}$ ${\mathcal{L}}_{3}$ ${\mathcal{L}}_{4}$ ${\mathcal{L}}_{5}$ ${\mathcal{L}}_{6}$ $x=$ 1 0 ${f}_{0}$ ${f}_{00}$ 0 0 $()$ false $0$ ${f}_{1}$ ${f}_{01}$ 0 1 $(x)$ not $x$ $\mathrm{\neg}x$ ${f}_{2}$ ${f}_{10}$ 1 0 $x$ $x$ $x$ ${f}_{3}$ ${f}_{11}$ 1 1 $(())$ true $1$
Table 2 lists equivalent expressions for the sixteen functions of concrete type $X\times Y\to \mathbb{B}$ and abstract type $\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ in the same set of languages.
Table 2. Propositional Forms on Two Variables ${\mathcal{L}}_{1}$ ${\mathcal{L}}_{2}$ ${\mathcal{L}}_{3}$ ${\mathcal{L}}_{4}$ ${\mathcal{L}}_{5}$ ${\mathcal{L}}_{6}$ $x=$ 1 1 0 0 $y=$ 1 0 1 0 ${f}_{0}$ ${f}_{0000}$ 0 0 0 0 $()$ false $0$ ${f}_{1}$ ${f}_{0001}$ 0 0 0 1 $(x)(y)$ neither $x$ nor $y$ $\mathrm{\neg}x\wedge \mathrm{\neg}y$ ${f}_{2}$ ${f}_{0010}$ 0 0 1 0 $(x)y$ $y$ and not $x$ $\mathrm{\neg}x\wedge y$ ${f}_{3}$ ${f}_{0011}$ 0 0 1 1 $(x)$ not $x$ $\mathrm{\neg}x$ ${f}_{4}$ ${f}_{0100}$ 0 1 0 0 $x(y)$ $x$ and not $y$ $x\wedge \mathrm{\neg}y$ ${f}_{5}$ ${f}_{0101}$ 0 1 0 1 $(y)$ not $y$ $\mathrm{\neg}y$ ${f}_{6}$ ${f}_{0110}$ 0 1 1 0 $(x,y)$ $x$ not equal to $y$ $x\ne y$ ${f}_{7}$ ${f}_{0111}$ 0 1 1 1 $(xy)$ not both $x$ and $y$ $\mathrm{\neg}x\vee \mathrm{\neg}y$ ${f}_{8}$ ${f}_{1000}$ 1 0 0 0 $xy$ $x$ and $y$ $x\wedge y$ ${f}_{9}$ ${f}_{1001}$ 1 0 0 1 $((x,y))$ $x$ equal to $y$ $x=y$ ${f}_{10}$ ${f}_{1010}$ 1 0 1 0 $y$ $y$ $y$ ${f}_{11}$ ${f}_{1011}$ 1 0 1 1 $(x(y))$ not $x$ without $y$ $x\Rightarrow y$ ${f}_{12}$ ${f}_{1100}$ 1 1 0 0 $x$ $x$ $x$ ${f}_{13}$ ${f}_{1101}$ 1 1 0 1 $((x)y)$ not $y$ without $x$ $x\Leftarrow y$ ${f}_{14}$ ${f}_{1110}$ 1 1 1 0 $((x)(y))$ $x$ or $y$ $x\vee y$ ${f}_{15}$ ${f}_{1111}$ 1 1 1 1 $(())$ true $1$
The columns of Tables 1 and 2 are conveniently described in the following order:

•
Language ${\mathcal{L}}_{3}$.
In Table 1, ${\mathcal{L}}_{3}$ describes each boolean function $f:\mathbb{B}\to \mathbb{B}$ by means of the sequence of two boolean values $(f(1),f(0))$.
In Table 2, ${\mathcal{L}}_{3}$ describes each boolean function $f:{\mathbb{B}}^{2}\to \mathbb{B}$ 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 table^{} under the column head for $f$. 
•
Language ${\mathcal{L}}_{2}$ lists the functions in the form ${f}_{i}$, where the index $i$ is a bit string formed from the sequence of boolean values in ${\mathcal{L}}_{3}$.

•
Language ${\mathcal{L}}_{1}$ notates the functions ${f}_{i}$ with an index $i$ that is the decimal equivalent of the binary numeral index in ${\mathcal{L}}_{2}$.
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 ${\mathcal{L}}_{4}$ expresses the boolean functions in terms of two families of logical operations^{}:
Logical conjunctions written as continued products^{}. For example:
$\begin{array}{ccc}\hfill xy\hfill & \hfill =\hfill & \hfill x\wedge y\hfill \\ \hfill xyz\hfill & \hfill =\hfill & \hfill x\wedge y\wedge z\hfill \end{array}$
Minimal negation operators written as parenthesized lists. For example:
$\begin{array}{ccc}\hfill ()\hfill & \hfill =\hfill & \hfill 0\hfill \\ \hfill (x)\hfill & \hfill =\hfill & \hfill \mathrm{\neg}x\hfill \\ \hfill (x,y)\hfill & \hfill =\hfill & \hfill x\ne y\hfill \end{array}$

•
Language ${\mathcal{L}}_{5}$ lists ordinary language expressions for the propositional forms. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.

•
Language ${\mathcal{L}}_{6}$ 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  20130322 17:55:47 
Last modified on  20130322 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 