# Zeroth-order logic

Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate calculus, propositional calculus, or sentential calculus. One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent isomorphisms.

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type X × Y → B and abstract type B × B → B in a number of different languages for zeroth order logic.

Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
x :1 1 0 0
y :1 0 1 0
f0f00000 0 0 0( )false0
f1f00010 0 0 1(x)(y)neither x nor y¬x ∧ ¬y
f2f00100 0 1 0(x) yy and not x¬x ∧ y
f3f00110 0 1 1(x)not x¬x
f4f01000 1 0 0x (y)x and not yx ∧ ¬y
f5f01010 1 0 1(y)not y¬y
f6f01100 1 1 0(x, y)x not equal to yx ≠ y
f7f01110 1 1 1(x y)not both x and y¬x ∨ ¬y
f8f10001 0 0 0x yx and yx ∧ y
f9f10011 0 0 1((x, y))x equal to yx = y
f10f10101 0 1 0yyy
f11f10111 0 1 1(x (y))not x without yx → y
f12f11001 1 0 0xxx
f13f11011 1 0 1((x) y)not y without xx ← y
f14f11101 1 1 0((x)(y))x or yx ∨ y
f15f11111 1 1 1(( ))true1

These six languages for the sixteen boolean functions are conveniently described in the following order:
• Language L3 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)). 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 L2 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 L1 notates the boolean functions fi with an index i that is the decimal equivalent of the binary numeral index in L2.
• Language L4 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 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 L5 lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
• Language L6 expresses the sixteen functions in one of several notations that are commonly used in formal logic.

A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers.
In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. All atomic formulae have the form , where is a predicate letter and is a variable.
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
A truth table is a mathematical table used in logic — specifically in connection with Boolean algebra, boolean functions, and propositional calculus — to compute the functional values of logical expressions on each of their functional arguments, that is, on each
A bit array (or bitmap, in some cases) is an array data structure which compactly stores individual bits (boolean values). It implements a simple set data structure storing a subset of and is effective at exploiting bit-level parallelism in hardware to perform operations
Conjunction can refer to:
• Astronomical conjunction, an astronomical phenomenon
• Astrological aspect, an aspect in horoscopic astrology
• Grammatical conjunction, a part of speech
• Logical conjunction, a mathematical operator

In logic and mathematics, the minimal negation operator is a multigrade operator where each is a k-ary boolean function defined in such a way that if and only if exactly one of the arguments is 0.
Ampheck, from Greek ἀμφήκης 'double-edged', is a term coined by Charles Sanders Peirce for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or NAND

• Algebra of sets
• Algebraic normal form
• Ampheck
• And-inverter graph
• Boole, George
• Boolean algebra (structure)
• Boolean algebras canonically defined
• Boolean conjunctive query
• Boolean domain
• Boolean function
• Boolean algebra (logic)

A boolean domain B is a generic 2-element set, say, B = , whose elements are interpreted as logical values, typically 0 = false and 1 = true.

A boolean variable x is a variable that takes its value from a boolean domain, as x
A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers.

A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain.
An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic beginning in the 1880's, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are
An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914.
First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: first-order predicate calculus (FOPC), the lower predicate calculus,
indicator function or a characteristic function is a function defined on a set that indicates membership of an element in a subset of .

The indicator function of a subset of a set is a function

defined as

A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic.

In his papers on qualitative logic, entitative graphs, and existential graphs
In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is true.

In classical logic, the only possible truth values are true and false.
In logic and mathematics, the minimal negation operator is a multigrade operator where each is a k-ary boolean function defined in such a way that if and only if exactly one of the arguments is 0.
In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. All atomic formulae have the form , where is a predicate letter and is a variable.
In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary.