zeroth order logic

zeroth order logic

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Calculus

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Column

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Columns

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Compact

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degree

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Degree

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formal logic

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Formal 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

1

Propositional forms

Table 1 lists

equivalent

expressions for the four

functions

of concrete type

X

→

𝔹

and abstract type

𝔹

→

𝔹

in a number of different

languages

for zeroth order logic.

Table 1. Propositional Forms on One

Variable

ℒ

1

ℒ

2

ℒ

3

ℒ

4

ℒ

5

ℒ

6

x

=

1 0

f

0

f

00

0 0

(

)

false

0

f

1

f

01

0 1

(

x

)

not

x

¬

⁢

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

×

Y

→

𝔹

and abstract type

𝔹

×

𝔹

→

𝔹

in the same set of languages.

Table 2. Propositional Forms on Two Variables

ℒ

1

ℒ

2

ℒ

3

ℒ

4

ℒ

5

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

¬

⁢

x

∧

¬

⁢

y

f

2

f

0010

0 0 1 0

(

x

)

⁢

y

y

and not

x

¬

⁢

x

∧

y

f

3

f

0011

0 0 1 1

(

x

)

not

x

¬

⁢

x

f

4

f

0100

0 1 0 0

x

⁢

(

y

)

x

and not

y

x

∧

¬

⁢

y

f

5

f

0101

0 1 0 1

(

y

)

not

y

¬

⁢

y

f

6

f

0110

0 1 1 0

(

x

,

y

)

x

not equal to

y

x

≠

y

f

7

f

0111

0 1 1 1

(

x

⁢

y

)

not both

x

and

y

¬

⁢

x

∨

¬

⁢

y

f

8

f

1000

1 0 0 0

x

⁢

y

x

and

y

x

∧

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

⇒

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

⇐

y

f

14

f

1110

1 1 1 0

(

(

x

)

⁢

(

y

)

)

x

or

y

x

∨

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

ℒ

3

.

In Table 1,

ℒ

3

describes each boolean function

f

:

𝔹

→

𝔹

by means of the

sequence

of two

boolean

values

(

f

⁢

(

1

)

,

f

⁢

(

0

)

)

.

In Table 2,

ℒ

3

describes each boolean function

f

:

𝔹

2

→

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

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

ℒ

3

.

•

Language

ℒ

1

notates the functions

f

i

with an index

i

that is the decimal equivalent of the binary numeral index in

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

ℒ

4

expresses the boolean functions in terms of two families of logical

operations

:

Logical conjunctions written as continued

products

. For example:

x

⁢

y

=

x

∧

y

x

⁢

y

⁢

z

=

x

∧

y

∧

z

Minimal negation operators

written as parenthesized lists. For example:

(

)

=

0

(

x

)

=

¬

⁢

x

(

x

,

y

)

=

x

≠

y

•

Language

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

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

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

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