some theorem schemas of normal modal logic

some theorem schemas of 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

.

Based on some of the

meta-theorems

of

Λ

(see here (

http://

planetmath

.org/SyntacticPropertiesOfANormalModalLogic

)), we can easily derive the following theorem schemas:

1.

□

⁢

(

A

∧

B

)

→

□

⁢

A

∧

□

⁢

B

2.

□

⁢

A

∧

□

⁢

B

→

□

⁢

(

A

∧

B

)

3.

□

⁢

¬

⟂

4.

□

⁢

A

↔

¬

⋄

¬

⁢

A

5.

□

(

A

→

B

)

→

(

⋄

A

→

⋄

B

)

6.

⋄

(

A

→

B

)

→

(

□

A

→

⋄

B

)

7.

⋄

A

∧

□

⁢

B

→

⋄

(

A

∧

B

)

8.

□

⁢

A

∨

□

⁢

B

→

□

⁢

(

A

∨

B

)

9.

⋄

(

A

∧

B

)

→

⋄

A

∧

⋄

B

10.

□

(

A

∨

B

)

→

□

A

∨

⋄

B

11.

⋄

(

A

∨

B

)

↔

⋄

A

∨

⋄

B

Proof.

1.

From tautologies

A

∧

B

→

A

and

A

∧

B

→

B

and meta-theorem 1, we get

⊢

□

⁢

(

A

∧

B

)

→

□

⁢

A

and

⊢

□

⁢

(

A

∧

B

)

→

□

⁢

B

. So

⊢

□

⁢

(

A

∧

B

)

→

□

⁢

A

∧

□

⁢

B

.

2.

From tautology

A

→

(

B

→

(

A

∧

B

)

)

and meta-theorem 1, we get

⊢

□

A

→

□

(

B

→

(

A

∧

B

)

)

.

From the K

instance

□

(

B

→

(

A

∧

B

)

)

→

(

□

B

→

□

(

A

∧

B

)

)

and the tautology

(

p

→

q

)

→

(

(

q

→

(

r

→

s

)

)

→

(

(

p

∧

r

)

→

s

)

)

substituting

p

for

□

⁢

A

,

q

for

□

(

B

→

(

A

∧

B

)

)

,

r

for

□

⁢

B

, and

s

for

□

⁢

(

A

∧

B

)

, and applying modus ponens twice, we get the result.

3.

From the tautology

¬

⟂

, we have the result by necessitation.

4.

All we need is

⊢

□

⁢

A

→

¬

⋄

¬

⁢

A

. From tautology

A

→

¬

⁢

¬

⁢

A

, we get

⊢

□

⁢

A

→

□

⁢

¬

⁢

¬

⁢

A

by meta-theorem 1. From tautology

□

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

¬

⁢

□

⁢

¬

⁢

¬

⁢

A

and the definition of

⋄

, we get

⊢

□

⁢

A

→

¬

⋄

¬

⁢

A

.

5.

To show

□

(

A

→

B

)

→

(

⋄

A

→

⋄

B

)

, it is enough to show

□

(

A

→

B

)

→

(

□

¬

A

∨

⋄

B

)

, which is enough to show

□

(

A

→

B

)

→

(

□

¬

B

→

□

¬

A

)

, which is enough to show

□

(

¬

B

→

¬

A

)

→

(

□

¬

B

→

□

¬

A

)

, which is just an instance of K.

6.

To show

⋄

(

A

→

B

)

→

(

□

A

→

⋄

B

)

, it is enough to show

¬

□

(

A

∧

¬

B

)

→

(

□

A

→

⋄

B

)

, which is enough to show

¬

(

□

A

∧

□

¬

B

)

→

(

□

A

→

⋄

B

)

by 1 and 2, which is enough to show

¬

□

A

∨

⋄

B

→

(

□

A

→

⋄

B

)

, which is just

(

□

A

→

⋄

B

)

→

(

□

A

→

⋄

B

)

.

7.

To show

⋄

A

∧

□

⁢

B

→

⋄

(

A

∧

B

)

, it is enough to show

¬

⁢

□

⁢

¬

⁢

A

∧

□

⁢

B

→

⋄

(

A

∧

B

)

, which is enough to show

¬

⁡

(

¬

⁢

□

⁢

B

∨

□

⁢

¬

⁢

A

)

→

⋄

(

A

∧

B

)

, which is enough to show

¬

⁡

(

□

⁢

B

→

□

⁢

¬

⁢

A

)

→

¬

⁢

□

⁢

¬

⁡

(

A

∧

B

)

, which is enough to show

□

¬

(

A

∧

B

)

→

(

□

B

→

□

¬

A

)

, which is enough to show

□

(

¬

A

∨

¬

B

)

→

(

□

B

→

□

¬

A

)

, which is enough to show

□

(

B

→

¬

A

)

→

(

□

B

→

□

¬

A

)

, which is an instance of K.

8.

Since

⊢

A

→

A

∨

B

and

B

⊢

A

∨

B

,

□

⁢

A

→

□

⁢

(

A

∨

B

)

and

□

⁢

B

→

□

⁢

(

A

∨

B

)

, and therefore

□

⁢

A

∨

□

⁢

B

→

□

⁢

(

A

∨

B

)

.

9.

By 8,

□

⁢

¬

⁢

A

∨

□

⁢

¬

⁢

B

→

□

⁢

(

¬

⁢

A

∨

¬

⁢

B

)

, so

¬

⁢

□

⁢

(

¬

⁢

A

∨

¬

⁢

B

)

→

¬

⁡

(

□

⁢

¬

⁢

A

∨

□

⁢

¬

⁢

B

)

, whence

⋄

(

A

∧

B

)

→

⋄

A

∧

⋄

B

.

10.

To show

□

(

A

∨

B

)

→

□

A

∨

⋄

B

, it is enough to show

□

(

¬

B

→

A

)

→

□

A

∨

⋄

B

, which is enough to show

□

(

¬

B

→

A

)

→

¬

□

¬

B

∨

□

A

, or

□

(

¬

B

→

A

)

→

□

¬

B

→

□

A

, an instance of K.

11.

From

A

→

A

∨

B

and

B

→

A

∨

B

, we get

⋄

A

→

⋄

(

A

∨

B

)

and

⋄

B

→

⋄

(

A

∨

B

)

, so that

⋄

A

∨

⋄

B

→

⋄

(

A

∨

B

)

. On the other hand, from

□

⁢

¬

⁢

A

∧

□

⁢

¬

⁢

B

→

□

⁢

(

¬

⁢

A

∧

¬

⁢

B

)

, we get

¬

⁢

□

⁢

(

¬

⁢

A

∧

¬

⁢

B

)

→

¬

⁡

(

□

⁢

¬

⁢

A

∧

□

⁢

¬

⁢

B

)

, or

⋄

(

A

∨

B

)

→

⋄

A

∨

⋄

B

, and the result follows.

∎

Remark

. The proofs are condensed for the sake of space. Properly, a formal proof should lay out the

sequence

of wff’s and their

derivations

. For example, the proof for

#

⁢

5

is

an instance of K

□

(

¬

B

→

¬

A

)

→

(

□

¬

B

→

□

¬

A

)

(1)

tautology

(

p

→

q

)

→

(

¬

q

→

¬

p

)

(

A

→

B

)

→

(

¬

B

→

¬

A

)

(2)

meta-theorem 1 applied to (2)

□

(

A

→

B

)

→

□

(

¬

B

→

¬

A

)

(3)

law of syllogism

on (3) to (1)

□

(

A

→

B

)

→

(

□

¬

B

→

□

¬

A

)

(4)

definition of

∨

□

(

A

→

B

)

→

(

¬

□

¬

B

∨

□

¬

A

)

(5)

definition of

⋄

□

(

A

→

B

)

→

(

⋄

B

∨

□

¬

A

)

(6)

a tautology

⁢

p

∧

q

→

q

∧

p

⋄

B

∨

□

¬

A

→

□

¬

A

∨

⋄

B

(7)

law of syllogism on (7) to (6)

□

(

A

→

B

)

→

(

□

¬

A

∨

⋄

B

)

(8)

a tautology

⁢

p

↔

¬

⁢

¬

⁢

p

□

⁢

¬

⁢

A

↔

¬

⁢

¬

⁢

□

⁢

¬

⁢

A

(9)

substitution theorem

on (8) by (9)

□

(

A

→

B

)

→

(

¬

¬

□

¬

A

∨

⋄

B

)

(10)

definition of

⋄

□

(

A

→

B

)

→

(

¬

⋄

A

∨

⋄

B

)

(11)

definition of

∨

□

(

A

→

B

)

→

(

⋄

A

→

⋄

B

)

(12)

Title

some theorem schemas of normal modal logic

Canonical name

SomeTheoremSchemasOfNormalModalLogic

Date of creation

2013-03-22 19:34:26

Last modified on

2013-03-22 19:34:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Generated on Thu Feb 8 19:51:56 2018 by

LaTeXML