Zeroth-order logic

Information about 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
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	¬x ∧ y
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	x ≠ y
f₇	f₀₁₁₁	0 1 1 1	(x y)	not both x and y	¬x ∨ ¬y
f₈	f₁₀₀₀	1 0 0 0	x y	x and y	x ∧ y
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	x → y
f₁₂	f₁₁₀₀	1 1 0 0	x	x	x
f₁₃	f₁₁₀₁	1 1 0 1	((x) y)	not y without x	x ← y
f₁₄	f₁₁₁₀	1 1 1 0	((x)(y))	x or y	x ∨ y
f₁₅	f₁₁₁₁	1 1 1 1	(( ))	true	1

These six languages for the sixteen boolean functions are conveniently described in the following order:

Language L₃ describes each boolean function f : B² → 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 L₂ lists the sixteen functions in the form f_i, where the index i is a bit string formed from the sequence of boolean values in L₃.
Language L₁ notates the boolean functions f_i 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:

$\text{[math]}$

It may also be noted that $\text{[math]}$ is the same function as $\text{[math]}$ and $\text{[math]}$ , and that the inclusive disjunctions indicated for $\text{[math]}$ and for $\text{[math]}$ may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function $\text{[math]}$ is not the same thing as the function $\text{[math]}$ .

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.

• • [ e] Logic
Navigation:	WikiProject Cleanup Noticeboard
Main Articles:	Reason History of Logic Philosophical Logic Philosophy of logic Mathematical Logic Metalogic Logic in computer science
Key Concepts	Reasoning: Deduction Induction Abduction Informal Logic: Proposition Inference Argument Validity Cogency Term logic Critical Thinking Fallacies Syllogism Mathematical logic: Set Syntax Semantics Wff Axiom Theorem Consistency Soundness Completeness Decidability Formal system Set theory Proof theory Model theory Recursion theory Zeroth-Order Logic: Boolean functions Monadic predicate calculus Propositional calculus Logical connectives Truth Tables Predicate Logic: First-Order Logic Quantifiers Second-order logic Modal Logic: Deontic logic Epistemic logic Temporal logic Doxastic logic Other non-classical logics: Computability logic Fuzzy logic Linear logic Relevance logic Non-monotonic logic
Controversies	Paraconsistent Logic Dialetheism Intuitionistic logic Paradoxes Antinomies Is logic empirical?
Key Figures	Aristotle Boole Cantor Carnap Church Frege Gentzen Gdel Hilbert Kripke Peano Peirce Putnam Quine Russell Skolem Tarski Turing Whitehead
Lists	Basic Topics in Logic Topics in Logic Logicians Rules of Inference Mathematical Logic Topics Basic Discrete Mathematics Topics Set theory topics Paradoxes Fallacies Logical Symbols

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.
..... Click the link for more information.

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.
..... Click the link for more information.

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
..... Click the link for more information.

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.
..... Click the link for more information.

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
..... Click the link for more information.

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
..... Click the link for more information.

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

..... Click the link for more information.

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.
..... Click the link for more information.

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
..... Click the link for more information.

..... Click the link for more information.

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)

..... Click the link for more information.

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 ∈
..... Click the link for more information.

..... Click the link for more information.

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.
..... Click the link for more information.

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
..... Click the link for more information.

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.
..... Click the link for more information.

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,
..... Click the link for more information.

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

..... Click the link for more information.

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
..... Click the link for more information.

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.
..... Click the link for more information.

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.
..... Click the link for more information.

Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
..... Click the link for more information.

In western philosophy, reason has had a twofold history. On the one hand, it has been taken to be objective and so to be fixed and discoverable by dialectic, analysis or study.
..... Click the link for more information.

The history of logic documents the development of logic as it occurs in various cultures and traditions in history. While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally
..... Click the link for more information.

This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.

EnglishInfo

Zeroth-order logic

Information about Zeroth-order logic

See also

Search Dictionary