modal logic

modal logic

Introduction

Modal logic

is the logic of “

necessity

” and “possibility”. It is an

extension

of the

classical logic

, with two additional unary

logical connectives

□

and

◇

. The

well-formed formulas

of modal logic is then the smallest set satisfying the following:

1.

the

propositional variables

are well-formed formulas,

2.

if

A

,

B

are (well-formed)

formulas

, then so are

A

∨

B

,

A

∧

B

,

¬

⁢

A

,

A

→

B

,

3.

if

A

is a formula, so are

□

⁢

A

and

◇

⁢

A

,

4.

and if the logic is first-order, then for any formula

A

, so are

∃

x

⁢

A

and

∀

x

⁢

A

.

Let

L

be the set of all well-formed formulas described above, and

L

0

the set of all formulas formed with condition 3 removed. Then

L

0

⊆

L

is just the set of all formulas of the classical logic (the underlying classical logic), and is part of the reason why we call modal logic an extension of the classical logic. The other reasons have to do with

semantics

and

deductions

, to be discussed below.

□

⁢

A

is usually interpreted as “it is necessarily true that

A

”, and

◇

⁢

A

as “it is possibly true that

A

”.

In order to study the logical truth and valid deductions in modal logic, we need to know how to interpret

□

and

◇

. So just what are precisely the meanings of “necessity” and “possibility”? Because of the lack of exact mathematical

interpretations

of these vague terms, and

□

and

◇

more generally, there are a great many kinds of modal logics, than the other well-known

logical systems

, such as classical logic and

intuitionistic logic

.

In fact, one can take

□

and

◇

to be qualifiers attachable to any

propositions

(or

sentences

). So

□

could mean “in South Africa” and

◇

“in the U.S.”. Then, if

A

is the sentence “July is a summer month”, then

□

⁢

A

and

◇

⁢

A

are read “July is a summer month in South Africa” and “July is a summer month in the U.S.” respectively. Some of the common interpretations of

□

⁢

A

are:

•

(

alethic

logic)

A

is necessarily true

•

(

provability logic

)

A

is provable in PA

•

(deontic logic) it ought to be that

A

•

(epistemic logic) it is known that

A

•

(doxastic logic) it is believed that

A

•

(temporal logic)

A

will always be true

Additionally, for most of the modal logics,

◇

is ‘defined” in terms of

□

:

◇

⁢

A

:=

¬

⁢

□

⁢

¬

⁢

A

.

(1)

In the next two subsections, we restrict our attention to only propositional modal logic.

Semantics

As mentioned previously, a modal logic is an extension of the classical logic. As such,

tautologies

and

tautological consequences

of valid formulas should be valid in any semantical model of modal logic. More precisely, call a formula in a modal logic

L

prime

or

quasi-atomic

if it is

•

either a propositional variable, or

•

of the form

□

⁢

A

, where

A

is in

L

.

It is easy to see that any formula in

L

can be built up from quasi-atoms using just

→

(or other

connectives

, depending on the choice of

axiom system for classical propositional logic

). In this sense,

L

can be thought of as a classical logic where the propositional variables range over quasi-atoms. Now, let

M

be any semantic model of the

L

, we write

M

⊧

A

to mean

A

is true in the model

M

. Then,

•

if

A

is an

instance

of a tautology (viewing

L

as the classical

propositional logic

here), then

M

⊧

A

,

•

if

A

1

,

…

,

A

n

tautologically implies

A

, and if

M

⊧

A

i

for each

i

, then

M

⊧

A

.

We call a formula in a modal logic

L

a

tautology

if it is an instance of a tautology of

L

viewed as the classical propositional logic over its quasi-atoms. Therefore,

□

⁢

A

→

□

⁢

A

is a tautology, since it is an instance of the tautology

X

→

X

. Also, if

□

⁢

A

and

□

⁢

A

→

□

⁢

B

are both valid, so is

□

⁢

B

, since

□

⁢

B

may be deduced by

modus ponens

.

For the majority of the modal logics (known as

normal modal logics

), the principal semantic model is the

Kripke semantics

, or possible-world semantics. It is a

generalization

of the truth-value semantics for the classical propositional logic. A Kripke model

M

consists of a triple

(

W

,

R

,

T

)

where

W

is a non-empty set of

elements

called possible worlds,

R

is a

binary relation

on

W

(called accessibility relation), and

T

is a

function

from the set of atomic propositions (propositional variables) to

P

⁢

(

W

)

, the

powerset

of

W

. Semantic truth condition of a proposition

A

is written

M

⊧

α

A

to mean “

A

is true in possible world

α

in model

M

”, or

⊧

α

A

if

M

is clear from the

context

.

The

relation

⊧

is defined recursively as follows:

1.

if

A

is atomic, then

⊧

α

A

iff

α

∈

T

⁢

(

A

)

2.

⊧

α

⟂

for no

α

∈

W

3.

⊧

α

A

→

B

iff if

⊧

α

A

then

⊧

α

B

4.

⊧

α

□

⁢

A

iff for all

β

where

α

⁢

R

⁢

β

,

⊧

β

A

The truth conditions of

¬

⁢

A

,

A

∧

B

,

A

∨

B

, and

⋄

A

can be derived. For example,

⊧

α

◇

⁢

A

iff there is some

β

where

α

⁢

R

⁢

β

, such that

⊧

β

A

.

If

A

is true in every possible world in

M

, we say that

A

is true in

M

, and write

M

⊧

A

. When

A

is true in every model, then we say that

A

is valid, and write

⊧

A

. The formula schema K, for example, is a set of valid formulas. Also, if

A

is valid, it is easy to see that so is

□

⁢

A

valid.

In

addition

to Kripke semantics, there are also a number of semantic tools, some are lattices with

operators

, some are systems of open neighborhoods in

topological spaces

, while others are just

variations

or generalizations of the Kripke models, such as including more than one binary relation on the possible worlds, or having a subset of the possible worlds called “impossible worlds”.

Axiom Systems

As explained in the previous subsection, tautologies and tautological consequences of valid formulas are valid, in any semantic interpretation of a modal logic. Similarly, in any axiom system for a modal logic:

•

tautologies are to be considered axioms (or

deducible

from axioms), and

•

modus ponens as an

inference rule

.

By including additional schemas of formulas as axioms, and imposing additional inference rules, one arrives at axiom systems for the different kinds of modal logics. Some of the common

axioms schemas

are listed above: K, 4, 5, D, T, B, and W. Some common inference rules (and their abbreviations) are

•

RK:

(

A

1

∧

⋯

∧

A

n

)

→

A

(

□

⁢

A

1

∧

⋯

∧

□

⁢

A

n

)

→

□

⁢

A

,

n

≥

0

•

RR:

(

A

∧

B

)

→

C

(

□

⁢

A

∧

□

⁢

B

)

→

□

⁢

C

•

RM:

A

→

B

□

⁢

A

→

□

⁢

B

•

RN (necessitation rule):

A

□

⁢

A

For example, a widely studied class of modal logics is the class of normal modal logics. A modal logic is normal if its axiom system consists of the axiom schema K and the rule RN.

Remarks

.

•

Like other logical systems, issues of soundness and completeness of a model with respect to an axiom system, as well as various decidability questions are also studied in modal logic. In addition, questions regarding on the equivalence of axiom systems are also very interesting.

•

Extending modal logic, one has the notion of

multi-modal logic

, where the

language

of the logical system contains several, sometimes infinitely many, modal connectives. From the discussion above, a modal logic where connectives

□

and

◇

are not inter-definable (and therefore must be non-normal) is really an example of a bimodal logic.

References

1

B. F. Chellas,

Modal Logic, An

Introduction

, Cambridge University Press (1980)

2

R. Goldblatt,

Logic of Time and

Computation

, 2nd Edition, CSLI (1992)

3

G. Priest,

An Introduction to Non-Classical Logic

, Cambridge University Press (2001)

Title

modal logic

Canonical

name

ModalLogic

Date of creation

2013-03-22 19:13:54

Last modified on

2013-03-22 19:13:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

34

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03B45

Related topic

KripkeSemanticsForModalPropositionalLogic

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