NationMaster - Encyclopedia: 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 logic, 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. In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is... In mathematics, a monadic logic is one that employs only unary relations. ... In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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

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

f₄

f₀₁₀₀

0 1 0 0

x (y)

x and not y

x & ~y

f₅

f₀₁₀₁

0 1 0 1

(y)

not 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

f₁₁

f₁₀₁₁

1 0 1 1

(x (y))

not x without y

x => y

f₁₂

f₁₁₀₀

1 1 0 0

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

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

It may also be noted that $(x, y)!$ is the same function as $x + y!$ and $x ne y$ , and that the inclusive disjunctions indicated for $(x, y)!$ and for $(x, y, z)!$ may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function $(x, y, z)!$ is not the same thing as the function $x + y + z!$ . Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

See also

Categories: Logic | Mathematical logic Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is... Boolean logic is a system of syllogistic logic invented by 19th-century British mathematician George Boole, which attempts to incorporate the empty set, that is, a class of non-existent entities, such as round squares, without resorting to uncertain truth values. ... A boolean-valued function is a function of the type , where is an arbitrary set, where is a generic 2-element set, typically , and where the latter is frequently interpreted for logical applications as . ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... 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 mathematics, a monadic logic is one that employs only unary relations. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ...

See also GA_googleFillSlot("encyclopedia_square");

See also