aarjaneiro/planetmath huggingface dataset:
Skolemization

Skolemization

Skolemization

is a way of removing

existential quantifiers

from a

formula

.

Variables

bound by existential quantifiers which are not inside the scope of

universal quantifiers

can simply be replaced by constants:

∃

x

[

x

<

3

]

can be changed to

c

<

3

, with

c

a suitable constant.

When the existential quantifier is inside a universal quantifier, the

bound variable

must be replaced by a

Skolem function

of the variables bound by universal quantifiers. Thus

∀

x

[

x

=

0

∨

∃

y

[

x

=

y

+

1

]

]

becomes

∀

x

[

x

=

0

∨

x

=

f

(

x

)

+

1

]

.

In general, the

functions

and

constants symbols

are new ones added to the

language

for the purpose of satisfying these formulas, and are often denoted by the formula they realize, for

instance

c

∃

x

⁢

ϕ

⁢

(

x

)

.

This is used in

second order logic

to move all existential quantifiers outside the scope of

first order

universal quantifiers. This can be done since second order

quantifiers

can quantify over functions. For instance

∀

1

x

⁢

∀

1

y

⁢

∃

1

z

⁢

ϕ

⁢

(

x

,

y

,

z

)

is

equivalent

to

∃

2

F

⁢

∀

1

x

⁢

∀

1

y

⁢

ϕ

⁢

(

x

,

y

,

F

⁢

(

x

,

y

)

)

.

Title

Skolemization

Canonical

name

Skolemization

Date of creation

2013-03-22 12:59:13

Last modified on

2013-03-22 12:59:13

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03B15

Classification

msc 03B10

Defines

Skolem function

Defines

Skolem constant

Generated on Thu Feb 8 19:45:57 2018 by

LaTeXML
negative translation

negative translation

It is well-known that classical

propositional logic

PL

c

can be considered as a subsystem of intuitionistic propositional logic PL

i

by translating any wff

A

in PL

c

into

¬

⁢

¬

⁢

A

in PL

i

. According to Glivenko’s

theorem

,

A

is a theorem of PL

c

iff

¬

⁢

¬

⁢

A

is a theorem of PL

i

. This translation, however, fails to

preserve

theoremhood in the corresponding predicate logics. For example, if

A

is of the form

∃

x

⁢

B

, then

⊢

c

A

no longer implies

⊢

i

¬

¬

A

. A number of translations have been devised to overcome this defect. They are collective known as

negative translations

or

double negative translations

of

classical logic

into

intuitionistic logic

. Below is a list of the most commonly mentioned negative translations:

•

Kolmogorov negative translation

:

(a)

p

∘

:=

¬

⁢

¬

⁢

p

with

p

atomic and

⟂

∘

⁣

:=

⁣

⟂

(b)

(

A

∧

B

)

∘

:=

¬

⁢

¬

⁡

(

A

∘

∧

B

∘

)

(c)

(

A

∨

B

)

∘

:=

¬

⁢

¬

⁡

(

A

∘

∨

B

∘

)

(d)

(

A

→

B

)

∘

:=

¬

¬

(

A

∘

→

B

∘

)

(e)

(

∀

x

⁢

A

)

∘

:=

¬

⁢

¬

⁢

∀

x

⁢

A

∘

(f)

(

∃

x

⁢

A

)

∘

:=

¬

⁢

¬

⁢

∃

x

⁢

A

∘

•

Godël negative translation

:

(a)

p

-

:=

¬

⁢

¬

⁢

p

with

p

atomic and

⟂

-

⁣

:=

⁣

⟂

(b)

(

A

∧

B

)

-

:=

A

-

∧

B

-

(c)

(

A

∨

B

)

-

:=

¬

⁢

¬

⁡

(

A

-

∨

B

-

)

(d)

(

A

→

B

)

-

:=

A

-

→

B

-

(e)

(

∀

x

⁢

A

)

-

:=

∀

x

⁢

A

-

(f)

(

∃

x

⁢

A

)

-

:=

¬

⁢

¬

⁢

∃

x

⁢

A

-

•

Kuroda negative translation

:

(a)

p

u

:=

p

with

p

atomic and

⟂

u

⁣

:=

⁣

⟂

(b)

(

A

∧

B

)

u

:=

A

u

∧

B

u

(c)

(

A

∨

B

)

u

:=

A

u

∨

B

u

(d)

(

A

→

B

)

u

:=

A

u

→

B

u

(e)

(

∀

x

⁢

A

)

u

:=

∀

x

⁢

¬

⁢

¬

⁢

A

u

(f)

(

∃

x

⁢

A

)

u

:=

∃

x

⁢

A

u

And then

A

u

:=

¬

⁢

¬

⁢

A

u

.

•

Krivine negative translation

:

(a)

p

r

:=

¬

⁢

p

with

p

atomic and

⟂

r

:=

¬

⟂

(b)

(

A

∧

B

)

r

:=

A

r

∨

B

r

(c)

(

A

∨

B

)

r

:=

A

r

∧

B

r

(d)

(

A

→

B

)

r

:=

¬

A

r

∧

B

r

(e)

(

∀

x

⁢

A

)

r

:=

∃

x

⁢

A

r

(f)

(

∃

x

⁢

A

)

r

:=

¬

⁢

∃

x

⁢

¬

⁢

A

r

And then

A

r

:=

¬

⁢

A

r

.

Remark

. It can be shown that for any wff

A

:

⊢

i

A

*

↔

A

#

and

⊢

c

A

iff

⊢

i

A

*

where

*

,

#

∈

{

∘

,

-

,

u

,

r

}

.

Title

negative translation

Canonical

name

NegativeTranslation

Date of creation

2013-03-22 19:35:48

Last modified on

2013-03-22 19:35:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B20

Classification

msc 03F55

Synonym

double negative translation

Defines

Kolmogorov negative translation

Defines

Godel negative translation

Defines

Kuroda negative translation

Defines

Krivine negative translation

\@unrecurse

Generated on Thu Feb 8 19:46:47 2018 by

LaTeXML
finiteness of C_∪ and C_∩

finiteness of

C

∪

and

C

∩

Theorem 1

For any set

L

and any subsets

X

and

Y

of

L

, the

consequence
operator

C

∩

⁢

(

X

,

Y

)

is finitary.

Theorem 2

For any set

L

, any subsets

X

of

L

, and any

finite

subset

Y

of

L

, the consequence operator

C

∩

⁢

(

X

,

Y

)

is finitary.

Title

finiteness of

C

∪

and

C

∩

Canonical name

FinitenessOfCcupAndCcap

Date of creation

2013-03-22 16:30:04

Last modified on

2013-03-22 16:30:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03B22

Classification

msc 03G10

Classification

msc 03G25

Generated on Thu Feb 8 19:47:35 2018 by

LaTeXML
modus ponens

modus ponens

Modus ponens

is a

rule of inference

that is commonly found in many logics where the binary

logical connective

→

(sometimes written

⇒

or

⊃

) called

logical implication

are defined. Informally, it states that

from

A

and

A

→

B

, we may infer

B

.

Modus ponens is also called the

rule of detachment

: the

theorem

b

can be “detached” from the theorem

A

→

B

provided that

A

is also a theorem.

An example of this rule is the following: From the premisses “It is raining”,
and “If it rains, then my laundry will be soaked”, we may draw the

conclusion

“My laundry will be soaked”.

Two common ways of mathematically denoting modus ponens are the following:

A

A

→

B

B

or

{

A

,

A

→

B

}

⊢

B

.

One formal way of looking at modus ponens is to define it as a

partial function

⊢

:

F

×

F

→

F

,

where

F

is a set of

formulas

in a

language

L

where a

binary operation

→

is defined, such that

1.

⊢

(

A

,

B

)

is defined whenever

A

,

B

∈

F

and

B

≡

(

A

→

C

)

for some

C

∈

L

, and

2.

when this is the case,

C

∈

F

and

⊢

(

A

,

B

)

:=

C

;

3.

⊢

is not defined otherwise.

Remark

. With modus ponens, one can easily prove the

converse

of the

deduction theorem

(see this link (

http://planetmath.org/DeductionTheorem

)). Another easily proven fact is the following:

If

Δ

⊢

A

and

Δ

⊢

A

→

B

, then

Δ

⊢

B

, where

Δ

is a set of formulas.

To see this, let

A

1

,

…

,

A

n

be a

deduction

of

A

from

Δ

, and

B

1

,

…

,

B

m

be a deduction of

A

→

B

from

Δ

. Then

A

1

,

…

,

A

n

,

B

1

,

…

,

B

m

,

B

is a deduction of

B

from

Δ

, where

B

is inferred from

A

n

(which is

A

) and

B

m

(which is

A

→

B

) by modus ponens.

Title

modus ponens

Canonical

name

ModusPonens

Date of creation

2013-03-22 16:50:48

Last modified on

2013-03-22 16:50:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B22

Classification

msc 03B05

Classification

msc 03B35

Synonym

rule of detachment

Synonym

detachment

Synonym

modus ponendo ponens

Generated on Thu Feb 8 19:47:56 2018 by

LaTeXML
modus tollens

modus tollens

The law of

modus tollens

is the

inference rule

which allows one to
conclude

¬

⁢

P

from

P

⇒

Q

and

¬

⁢

Q

. The name “modus
tollens” refers to the fact that this rule allows one to take away the

conclusion

of a

conditional

statement and conclude the

negation

of the
condition. As an example of this rule, we may cite the following:

If the postman is at the door, the doorbell will ring twice

The bell is not ringing.

The postman is not at the door.

The

validity

of this rule may be established by means of the following

truth table

:

P

Q

P

⇒

Q

¬

⁢

P

¬

⁢

Q

F

F

T

T

T

F

T

T

T

F

T

F

F

F

T

T

T

T

F

F

This rule can be used to justify the popular technique of

proof by
contradiction

. In this technique, one assumes a

hypothesis

P

and
then derives a conclusion

Q

. This is tantamount to showing that

P

⇒

Q

. Next one demonstrates

¬

⁢

Q

. Applying modus
tollens, one then concludes

¬

⁢

P

.

Title

modus tollens

Canonical

name

ModusTollens

Date of creation

2013-03-22 16:56:03

Last modified on

2013-03-22 16:56:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B22

Classification

msc 03B35

Classification

msc 03B05

Generated on Thu Feb 8 19:47:58 2018 by

LaTeXML
recursive set

recursive set

A subset

S

of the

natural numbers

ℕ

is said to be

recursive

if its

characteristic function

χ

S

⁢

(

x

)

:=

{

1

if

⁢

x

∈

S

0

if

⁢

x

∈

ℕ

-

S

is recursive (

computable

). In other words, there is an

algorithm

(via

Turing machine

for example) that determines whether an

element

is in

S

or not in

S

.

More generally, a subset

S

⊆

ℕ

n

is

recursive

if its characteristic function

f

S

is recursive.

A

recursive set

is also known as a

decidable set

or a

computable set

.

Examples of recursive sets are finite subset of

ℕ

, the set

ℕ

itself, the set of

even integers

, the set of

Fibonacci numbers

, the set of pairs

(

a

,

b

)

where

a

divides

b

, and the set of

prime numbers

. In the last example, one may use the

Sieve of Eratosthenes

as an algorithm to determine the

primality

of an

integer

.

A set

S

⊆

ℕ

is

recursively enumerable

if the

partial function

f

⁢

(

x

)

:=

{

1

if

⁢

x

∈

S

undefined

if

⁢

x

∈

ℕ

-

S

is computable. In other words, there is an algorithm that halts (and returns

1

) only when an element in

S

is used as an input.

Remarks

•

A special case of a recursive set is that of a

primitive recursive set

. A set is

primitive recursive

if its characteristic function is primitive recursive (

http://planetmath.org/PrimitiveRecursive

). All of the examples cited above are primitive recursive.

•

On the other hand, one can broaden the scope of recursiveness to sets which are not necessarily subsets of

ℕ

n

. Below are two examples:

–

Since

ℤ

can be effectively embedded in

ℕ

, so the notion of recursive sets be extended to subsets of

ℤ

.

–

Since every

finite set

Σ

can be encoded by the natural numbers, we can define a

recursive language

over

Σ

to be a subset

L

⊆

Σ

*

such that

L

, when encoded by the natural numbers, is a recursive set. Equivalently,

L

is recursive iff there is a Turing machine that decides

L

(accepts

L

and rejects

Σ

*

-

L

).

•

Similarly, recursive enumerability can be defined on

languages

: a language

L

over

Σ

is

recursively enumerable

if its

encoding

by the natural numbers is a

recursively enumerable set

. This is

equivalent

to saying that

L

is accepted by a Turing machine.

•

Using the above definition, one can define a

recursive predicate

or a

recursively enumerable predicate

φ

⁢

(

x

)

according to whether

{

x

∣

φ

⁢

(

x

)

}

is a recursive or recursively enumerable set respectively.

Title

recursive set

Canonical name

RecursiveSet

Date of creation

2013-03-22 17:34:52

Last modified on

2013-03-22 17:34:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B25

Classification

msc 03D20

Synonym

decidable set

Synonym

computable set

Synonym

decidable predicate

Synonym

computable predicate

Defines

recursively enumerable set

Defines

recursive predicate

Defines

recursively enumerable predicate

Defines

recursive language

Generated on Thu Feb 8 19:48:08 2018 by

LaTeXML
inference rule

inference rule

In logic, an

inference rule

is a rule whereby one may correctly
draw a

conclusion

from one or more

premises

. For example, the law of
the

contrapositive

allows one to conclude a statement of the form

¬

⁢

Q

⇒

¬

⁢

P

from a premise of the form

P

⇒

Q

.

Here, ‘

P

’ and ‘

Q

’ are

propositional variables

, which can stand for
arbitrary

propositions

. A popular way to indicate applications of

rules
of inference

is to list the premises above a line and write the
conclusions below the line. For

instance

, we might indicate the law
of the contrapositive thus:

P

⇒

Q

¬

⁢

Q

⇒

¬

⁢

P

A typical application of the law of
contrapositive would be to conclude ”If my clothes are dry, then it is not
raining”, from ”If it rains, then my clothes will be wet.” which could be
expressed as follows using the notation described above:

If it rains, then my clothes will be wet.

If my clothes are dry, then it is not raining.

(In this
instance,

P

is “It is raining” and

Q

is “My clothes are dry”.

An important feature of rules of inference is that they are purely formal,
which means that all that matters is the form of the

expression

;
meaning is not a consideration in applying a rule of inference.
Thus, the following are equally valid applications of the rule of
the contrapositive:

If the jabberwocky is mimsy, then the toves blithe.

If the toves are not blithing, then the jabberwocky is not mimsy.

If my cat has a tail, then my cat is a dog.

If my cat is not a dog, then my cat does not have a tail.

In the first example, the statements are nonsense and in the second
example, the statements are false, but this doesn’t matter — both
examples constitute valid apllications of the rule of the contrapositive.
Of course, in order to draw valid conclusions, we need to start with
valid premises, but the point of these examples is clarify the
distinction between valid statements and valid applications of
rules of inference.

Title

inference rule

Canonical

name

InferenceRule

Date of creation

2013-03-22 16:50:51

Last modified on

2013-03-22 16:50:51

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B35

Classification

msc 03B22

Classification

msc 03B05

Synonym

rule of inference

Related topic

ModusPonens

Related topic

LogicalAxiom

Related topic

DeductiveSystem

Generated on Thu Feb 8 19:48:12 2018 by

LaTeXML
Church integer

Church integer

A

Church integer

is a representation of integers as

functions

, invented by Alonzo Church. An integer

N

is represented as a

higher-order function

, which applies a given function to a given

expression

N

times.

For example, in the programming language

Haskell

, a function that returns a particular Church integer might be

church

⁡

0

=

f

⁢

x

→

x

church

⁡

n

=

c

where

:

c

⁢

f

⁢

x

=

c

′

⁢

f

⁢

(

f

⁢

x

)

⁢

where

:

c

′

=

church

⁡

(

n

-

1

)

The

transformation

from a Church integer to an integer might be

unchurch n = n (+1) 0

Thus we can generate the integers–the

(+1)

function would be applied to an initial value of

0

n

times, yielding the ordinary integer

n

.

Title

Church integer

Canonical

name

ChurchInteger

Date of creation

2013-03-22 12:32:31

Last modified on

2013-03-22 12:32:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03B40

Classification

msc 68N18

Related topic

LambdaCalculus

Generated on Thu Feb 8 19:48:15 2018 by

LaTeXML
combinatory logic

combinatory logic

Combinatory logic

was invented by Moses Schönfinkel in the early 1920s, and was mostly developed by Haskell Curry. The idea was to reduce the notation of logic to the simplest terms possible. As such, combinatory logic consists only of

combinators

,

combination

operations

, and no

free variables

.

A

combinator

is simply a

function

with no

free variables

. A

free variable

is any variable referred to in a function that is not a parameter of that function. The operation of

combination

is then simply the application of a combinator to its parameters. Combination is specified by simple

juxtaposition

of two terms, and is left-associative. Parentheses may also be present to override

associativity

. For example

f

⁢

g

⁢

x

⁢

y

=

(

f

⁢

g

)

⁢

x

⁢

y

=

(

(

f

⁢

g

)

⁢

x

)

⁢

y

All combinators in combinatory logic can be derived from two basic combinators,

S

and

K

. They are defined as

S

⁢

f

⁢

g

⁢

x

=

f

⁢

x

⁢

(

g

⁢

x

)

K

⁢

x

⁢

y

=

x

Reference is sometimes made to a third basic combinator,

I

, which can be defined in terms of

S

and

K

.

I

⁢

x

=

S

⁢

K

⁢

K

⁢

x

=

x

Combinatory logic where

I

is considered to be derived from

S

and

K

is sometimes known as

pure combinatory logic

.

Combinatory logic and

lambda calculus

are

equivalent

. However, lambda calculus is more concise than combinatory logic; an expression of size

𝒪

⁢

(

n

)

in lambda calculus is equivalent to an expression of size

𝒪

⁢

(

n

2

)

in combinatory logic.

For example,

S

⁢

f

⁢

g

⁢

x

=

f

⁢

x

⁢

(

g

⁢

x

)

in combinatory logic is equivalent to

S

=

(

λ

⁢

f

⁢

(

λ

⁢

g

⁢

(

λ

⁢

x

⁢

(

(

f

⁢

x

)

⁢

(

g

⁢

x

)

)

)

)

)

, and

K

⁢

x

⁢

y

=

x

is equivalent to

K

=

(

λ

⁢

x

⁢

(

λ

⁢

y

⁢

x

)

)

.

Title

combinatory logic

Canonical name

CombinatoryLogic

Date of creation

2013-03-22 12:32:29

Last modified on

2013-03-22 12:32:29

Owner

Logan (6)

Last modified by

Logan (6)

Numerical id

5

Author

Logan (6)

Entry type

Definition

Classification

msc 03B40

Related topic

LambdaCalculus

Related topic

Substitutability

Defines

combinator

Defines

combination

Defines

free variable

Defines

pure combinatory logic

Generated on Thu Feb 8 19:48:17 2018 by

LaTeXML
lambda calculus

lambda calculus

Lambda calculus

(often referred to as

λ

-calculus) was invented in the 1930s by Alonzo Church, as a form of mathematical logic dealing primarly with

functions

and the application of functions to their

arguments

.
In

pure lambda calculus

, there are no

constants

. Instead, there are only

lambda abstractions

(which are simply

specifications

of functions),

variables

, and applications of functions to functions. For

instance

,

Church integers

are used as a substitute for actual constants representing integers.

A lambda abstraction is typically specified using a

lambda expression

, which might look like the following.

λ

⁢

x

.

f

⁢

x

The above specifies a function of one argument, that can be

reduced

by applying the function

f

to its argument (function application is left-associative by default, and parentheses can be used to specify associativity).

The

λ

-calculus is

equivalent

to

combinatory logic

(though much more concise). Most functional programming

languages

are also equivalent to

λ

-calculus, to a degree (any imperative features in such languages are, of course, not equivalent).

Examples

We can specify the Church integer

3

in

λ

-calculus as

3

=

λ

⁢

f

⁢

x

.

f

⁢

(

f

⁢

(

f

⁢

x

)

)

Suppose we have a function

inc

, which when given a string representing an integer, returns a new string representing the number following that integer.
Then

3

⁢

inc

⁡

"0"

=

"3"

Addition

of Church integers in

λ

-calculus is

add

=

λ

x

y

.

(

λ

f

z

.

x

f

(

y

f

z

)

)

add

⁡

2 3

=

λ

⁢

f

⁢

z

⁢

. 2

⁢

f

⁢

(

3

⁢

f

⁢

z

)

=

λ

⁢

f

⁢

z

⁢

. 2

⁢

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

z

)

)

)

=

λ

⁢

f

⁢

z

.

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

z

)

)

)

)

=

5

Multiplication

is

mul

=

λ

x

y

.

(

λ

f

z

.

x

(

λ

w

.

y

f

w

)

z

)

mul

⁡

2 3

=

λ

⁢

f

⁢

z

⁢

. 2

⁢

(

λ

⁢

w

⁢

. 3

⁢

f

⁢

w

)

⁢

z

=

λ

f

z

. 2

(

λ

w

.

f

(

f

(

f

w

)

)

)

z

=

λ

⁢

f

⁢

z

.

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

(

f

⁢

z

)

)

)

)

)

=

6

.

Russell’s Paradox in

λ

-calculus

The

λ

-calculus readily admits Russell’s

Paradox

.
Let us define a function

r

that takes a function

x

as an argument, and is reduced to the application of the logical function

not

to the application of

x

to itself.

r

=

λ

⁢

x

.

not

⁡

(

x

⁢

x

)

Now what happens when we apply

r

to itself?

r

⁢

r

=

not

⁡

(

r

⁢

r

)

=

not

⁡

(

not

⁡

(

r

⁢

r

)

)

⋮

Since we have

not

⁡

(

r

⁢

r

)

=

(

r

⁢

r

)

, we have a paradox.

Title

lambda calculus

Canonical

name

LambdaCalculus

Date of creation

2013-03-22 12:32:39

Last modified on

2013-03-22 12:32:39

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

9

Author

ratboy (4018)

Entry type

Definition

Classification

msc 03B40

Related topic

CombinatoryLogic

Related topic

ChurchInteger

Related topic

RussellsParadox

Defines

pure lambda calculus

Defines

lambda abstraction

Defines

lambda expression

Generated on Thu Feb 8 19:48:20 2018 by

LaTeXML
prescisive abstraction

prescisive abstraction

Prescisive abstraction

or

prescision

, variously spelled as

precisive abstraction

and

prescission

, is a formal

operation

that marks, selects, or singles out one feature of a concrete experience to the disregard of others.

The above

definition

derives from one given by

Charles Sanders Peirce

(CP 4.235) in the

context

of distinguishing two kinds of abstraction, the other being

hypostatic abstraction

.

1

References

•

Peirce, Charles Sanders (1902), “The Simplest Mathematics”, CP 4.227–323 in

Collected Papers of Charles Sanders Peirce

, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as (CP volume.paragraph).

Title

prescisive abstraction

Canonical

name

PrescisiveAbstraction

Date of creation

2013-03-22 17:54:03

Last modified on

2013-03-22 17:54:03

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

5

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03B42

Classification

msc 03B15

Classification

msc 03B22

Classification

msc 03B30

Classification

msc 03A05

Classification

msc 00A30

Synonym

simple abstraction

Related topic

HypostaticAbstraction

Generated on Thu Feb 8 19:48:35 2018 by

LaTeXML
filtration

filtration

Let

M

=

(

W

,

R

,

V

)

be a Kripke model for a

modal logic

L

. Let

Δ

be a set of wff’s. Define a

binary relation

∼

Δ

on

W

:

w

∼

Δ

u

iff

⊧

w

A

iff

⊧

u

A

for any

A

∈

Δ

.

Then

∼

Δ

is an

equivalence relation

on

W

. Let

W

′

be the set of

equivalence classes

of

∼

Δ

on

W

. It is easy to see that if

Δ

is finite, so is

W

′

. Next, let

V

′

⁢

(

p

)

:=

{

[

w

]

∈

W

′

∣

w

∈

V

⁢

(

p

)

}

.

Then

V

′

is a well-defined

function

. We call a binary relation

R

′

on

W

′

a

filtration

of

R

if

•

w

⁢

R

⁢

u

implies

[

w

]

⁢

R

′

⁢

[

u

]

•

[

w

]

⁢

R

′

⁢

[

u

]

implies that for any wff

A

with

□

⁢

A

∈

Δ

, if

⊧

w

□

⁢

A

, then

⊧

u

A

.

The triple

M

′

:=

(

W

′

,

R

′

,

V

′

)

is called a

filtration

of the model

M

.

Proposition 1

.

(Filtration Lemma) Let

Δ

be a set of wff’s

closed under

the formation of

subformulas

: any subformula of any

formula

in

Δ

is again in

Δ

. Then

M

′

⊧

[

w

]

A

𝑖𝑓𝑓

M

⊧

w

A

.

Title

filtration

Canonical name

Filtration1

Date of creation

2013-03-22 19:35:39

Last modified on

2013-03-22 19:35:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

\@unrecurse

Generated on Thu Feb 8 19:49:18 2018 by

LaTeXML