syntactic properties of a normal modal logic

syntactic properties of a normal modal logic

Recall that a

normal modal logic

is a logic containing all

tautologies

, the schema K

□

(

A

→

B

)

→

(

□

A

→

□

B

)

,

and

closed under

modus ponens

and

necessitation

rules. Also, the modal operator

diamond

⋄

is defined as

⋄

A

:=

¬

⁢

□

⁢

¬

⁢

A

.

Let

Λ

be any normal modal logic. We write

⊢

A

to mean

Λ

⊢

A

, or wff

A

∈

Λ

, or

A

is a

theorem

of

Λ

. In

addition

, for any set

Δ

,

Δ

⊢

A

means there is a

finite sequence

of wff’s such that each wff is either a theorem, a member of

Δ

, or obtained either by modus ponens or necessitation from earlier wff’s in the sequence, and

A

is the last wff in the sequence. The sequence is called a

deduction

(of

A

) from

Δ

.

Below are some useful

meta-theorems

of

Λ

:

1.

(RM)

⊢

A

→

B

implies

⊢

□

⁢

A

→

□

⁢

B

Proof.

By assumption and by necessitation,

⊢

□

(

A

→

B

)

, by schema K and by modus ponens, we have the result.
∎

2.

As a result,

⊢

A

↔

B

implies

⊢

□

A

↔

□

B

.

3.

(

substitution theorem

). If

⊢

B

i

↔

C

i

for

i

=

1

,

…

,

m

, then

⊢

A

[

B

¯

/

p

¯

]

↔

A

[

C

¯

/

p

¯

]

,

where

p

¯

:=

(

p

1

,

…

,

p

m

)

is the tuple of all the

propositional variables

in

A

listed in order.

Proof.

For most of the proof, consult this entry (

http://planetmath.org/SubstitutionTheoremForPropositionalLogic

) for more detail. What remains is the case when

A

has the form

□

⁢

D

. We do

induction

on the number

n

of

□

’s in

A

. The case when

n

=

0

means that

A

is a wff of PL

c

, and has already been proved. Now suppose

A

has

n

+

1

□

’s. Then

D

has

n

□

’s, and so by induction,

⊢

D

[

B

/

p

]

↔

D

[

C

/

p

]

, and therefore

⊢

□

D

[

B

/

p

]

↔

□

D

[

C

/

p

]

by 2. This means that

⊢

A

[

B

/

p

]

↔

A

[

C

/

p

]

.
∎

4.

⊢

A

→

B

implies

⊢

⋄

A

→

⋄

B

Proof.

By assumption, tautology

⊢

(

A

→

B

)

→

(

¬

B

→

¬

A

)

, and modus ponens, we get

⊢

¬

⁢

B

→

¬

⁢

A

. By 1,

⊢

□

⁢

¬

⁢

B

→

□

⁢

¬

⁢

A

. By another

instance

of the above tautology and modus ponens, and the definition of

⋄

, we get the result.
∎

5.

⊢

A

∨

B

implies

⊢

⋄

A

∨

□

⁢

B

Proof.

Since

⊢

A

∨

B

↔

(

¬

A

→

B

)

, we have

⊢

¬

⁢

A

→

B

, so

⊢

□

⁢

¬

⁢

A

→

□

⁢

B

. By the tautology

C

↔

¬

⁢

¬

⁢

C

, we have

⊢

¬

⁢

¬

⁢

□

⁢

¬

⁢

A

→

□

⁢

B

, or

⊢

¬

⋄

A

→

□

⁢

B

, and therefore

⊢

⋄

A

∨

□

⁢

B

.
∎

6.

(RR)

⊢

A

∧

B

→

C

implies

⊢

□

⁢

A

∧

□

⁢

B

→

□

⁢

C

Proof.

By assumption and 1,

⊢

□

⁢

(

A

∧

B

)

→

□

⁢

C

. Since

□

⁢

A

∧

□

⁢

B

→

□

⁢

(

A

∧

B

)

is a theorem (see here (

http://planetmath.org/SomeTheoremSchemasOfNormalModalLogic

)), we get

⊢

□

⁢

A

∧

□

⁢

B

→

□

⁢

C

by the

law of syllogism

.
∎

7.

(RK) More generally,

⊢

A

1

∧

⋯

∧

A

n

→

A

implies

⊢

□

⁢

A

1

∧

⋯

⁢

□

⁢

A

n

→

□

⁢

A

, where the case

n

=

0

is the necessitation rule.

Proof.

Cases

n

=

1

,

2

are meta-theorems 1 and 6. If

⊢

A

1

∧

⋯

∧

A

n

∧

A

n

+

1

→

A

, or

⊢

(

A

1

∧

⋯

∧

A

n

)

∧

A

n

+

1

→

A

, then

⊢

□

⁢

(

A

1

∧

⋯

∧

A

n

)

∧

□

⁢

A

n

+

1

→

□

⁢

A

by 6. But

⊢

□

(

A

1

∧

⋯

∧

A

n

)

↔

□

A

1

∧

⋯

∧

□

A

n

, the result follows.
∎

8.

Define a

function

s

on

{

¬

,

λ

}

, where

λ

is the

empty word

, such that

s

⁢

(

¬

)

=

λ

, the empty word, and

s

⁢

(

λ

)

=

¬

. Then for any wff

A

, and

ϵ

1

,

ϵ

2

∈

{

¬

,

λ

}

:

⊢

ϵ

1

□

n

ϵ

2

A

↔

s

(

ϵ

1

)

⋄

n

s

(

ϵ

2

)

A

.

Technically speaking, this is really an

infinite

collection

of theorem schemas.

Proof.

We will check the case when

ϵ

1

=

¬

and

ϵ

2

=

λ

and leave the rest to the reader. We do induction on

n

. If

n

=

0

, then we have the tautology

¬

⁢

A

↔

¬

⁢

A

. Suppose

⊢

¬

□

n

A

↔

⋄

n

¬

A

. Then

⊢

¬

□

n

+

1

A

↔

⋄

n

¬

□

A

, by applying the induction case on wff

□

⁢

A

. Since

A

↔

¬

⁢

¬

⁢

A

is a tautology,

⊢

¬

□

n

+

1

A

↔

⋄

n

¬

□

¬

¬

A

by the substitution theorem. By the definition of

⋄

, we have

⊢

¬

□

n

+

1

A

↔

⋄

n

+

1

¬

A

.
∎

9.

Let

□

⁢

Δ

:=

{

□

⁢

A

∣

A

∈

Δ

}

. Then

Δ

⊢

A

implies

□

⁢

Δ

⊢

□

⁢

A

.

Proof.

Induct on the length

n

of deduction of

A

from

Δ

. If

n

=

0

, then either

⊢

A

, in which case

⊢

□

⁢

A

by necessitation, or

A

∈

Δ

, in which case

□

⁢

A

∈

□

⁢

Δ

. In either case,

□

⁢

Δ

⊢

□

⁢

A

. Next suppose the property holds for all deductions of length

n

, and there is a deduction

ℰ

of

A

of length

n

+

1

. If

A

is obtained from

Δ

by necessitation, say

A

is

□

⁢

B

, where

B

is in

ℰ

, then a subsequence of

ℰ

is a deduction of

B

of length

≤

n

, from

Δ

. So by induction,

□

⁢

Δ

⊢

□

⁢

B

, or

□

⁢

Δ

⊢

A

. By necessitation,

□

⁢

Δ

⊢

□

⁢

A

. Finally, if

A

is obtained by modus ponens, then there is a wff

B

such that

B

,

B

→

A

are both in

ℰ

. By induction,

□

⁢

Δ

⊢

□

⁢

B

and

□

Δ

⊢

□

(

B

→

A

)

, which, by K and modus ponens,

□

⁢

Δ

⊢

□

⁢

B

→

□

⁢

A

, and as a result,

□

⁢

Δ

⊢

□

⁢

A

by modus ponens.
∎

Noticeably absent is the

deduction theorem

, for the necessitation rule says

A

⊢

□

⁢

A

, but this does not imply

⊢

A

→

□

⁢

A

. In fact, the wff

A

→

□

⁢

A

is not a theorem in general, unless of course the logic includes the entire schema. All we can say is the following:

10.

(deduction theorem) If

Δ

,

A

⊢

B

and

B

is not of the form

□

⁢

C

, then

Δ

⊢

A

→

B

.

Remark

. It can be shown that

conversely

, if a

modal logic

obeys meta-theorem 7 above as an

inference rule

, then it is normal. For more detail, see here (

http://planetmath.org/EquivalentFormulationsOfNormality

).

Title

syntactic properties of a normal modal logic

Canonical name

SyntacticPropertiesOfANormalModalLogic

Date of creation

2013-03-22 19:34:18

Last modified on

2013-03-22 19:34:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

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