Grafting

Modalities

onto

substructural

implication

systems

Marcello

D'Agostino

Dov

M.

Gabbay

Alessandra

Russo

Department

of

Computing

Imperial

College

Queen

's

Gate

London

-LRB-

UK

-RRB-

June

Abstract

We

investigate

the

semantics

of

the

logical

systems

obtained

by

introducing

the

modalities

and

into

the

family

of

substructural

implication

logics

-LRB-

including

relevant

linear

and

intuitionistic

implication

-RRB-

Then

in

the

spirit

of

the

LDS

-LRB-

Labelled

Deductive

Systems

-RRB-

methodology

we

import

this

semantics

into

the

classical

proof

system

KE

This

leads

to

the

formulation

of

a

uniform

labelled

refutation

system

for

the

new

logics

which

is

a

natural

extension

of

a

system

for

substructural

implication

developed

by

the

first

two

authors

in

a

previous

paper

Introduction

The

notion

of

modality

is

central

in

pure

and

applied

logic

Many

systems

presented

to

formalise

some

application

area

require

the

addition

of

modality

to

the

language

for

a

variety

of

reasons

to

cater

for

changes

of

the

system

in

time

or

perhaps

for

the

dependency

of

the

system

on

the

context

or

even

to

bring

metalevel

notions

into

the

object

level

Be

the

reasons

and

motivations

what

they

may

we

must

develop

the

logical

capability

of

adding

We

wish

to

thank

Krysia

Broda

Marco

Mondadori

Guido

Governatori

and

Sara

Negri

for

stimulating

discussions

and

helpful

suggestions

This

research

was

supported

by

EPSRC

grant

-LRB-

Labelled

Tableaux

for

Algorithmic

Deduction

-RRB-

modalities

into

a

system

and

have

an

understanding

of

the

landscape

of

options

available

This

paper

addressed

the

problem

of

grafting

modalities

onto

substructural

implication

logics

The

term

substructural

logics

is

now

of

widespread

use

to

indicate

a

family

of

logics

weaker

than

classical

logic

-LRB-

including

intuitionistic

relevant

and

linear

logic

-RRB-

which

result

from

dropping

some

or

all

of

the

traditional

structural

rules

of

the

classical

sequent

calculus

-LRB-

see

-LSB-

D

-RSB-

for

an

introduction

-RRB-

and

are

proving

useful

in

several

applications

of

logic

to

Computer

Science

and

AI

-LRB-

see

for

instance

-LSB-

-RSB-

for

an

application

of

relevance

logic

to

modular

reasoning

systems

and

-LSB-

-RSB-

for

an

overview

of

computational

applications

of

Linear

Logic

-RRB-

The

introduction

of

modalities

into

such

logics

adds

a

new

dimension

which

can

explicitly

and

naturally

account

for

accessibility

relations

involved

in

the

processes

that

are

being

modelled

Formally

we

have

a

system

with

the

conditional

operator

and

we

want

to

add

unary

operators

and

which

behave

like

modalities

We

would

like

to

to

this

in

a

way

which

is

compatible

with

the

spirit

of

the

system

i.e.

to

define

modalities

from

within

the

system

in

some

natural

way

Substructural

logics

can

be

characterized

by

structures

of

objects

which

can

be

understood

as

pieces

of

information

or

sometimes

as

resources

A

natural

way

of

defining

modalities

within

such

semantics

consists

therefore

in

adding

an

accessibility

relation

between

pieces

of

information

This

is

the

approach

we

shall

adopt

in

this

paper

By

analogy

with

classical

modal

logics

the

intuitive

idea

is

that

the

verification

of

a

proposition

of

the

form

or

by

means

of

a

given

piece

of

information

-LRB-

or

resource

-RRB-

x

depends

on

what

is

verified

by

other

pieces

of

information

-LRB-

or

resources

-RRB-

accessibile

from

x.

Unlike

classical

modal

logics

however

the

verification

of

non-modal

formulae

such

as

a

conditional

formula

by

a

given

piece

of

information

x

also

depends

on

what

is

or

is

not

verified

elsewhere

in

the

structure

-LRB-

as

with

the

intuitionistic

conditional

-RRB-

We

shall

not

pursue

any

particular

intuitive

interpretation

because

it

may

change

from

one

logic

to

the

other

and

shall

leave

it

to

the

reader

to

envisage

his

or

her

favourite

application

contexts

Previous

work

on

this

topic

has

been

concentrating

on

intuitionistic

modal

logics

-LRB-

-LSB-

-RSB-

see

in

particular

-LSB-

-RSB-

for

a

comprehensive

overview

-RRB-

Here

we

start

developing

a

more

general

approach

which

fits

naturally

into

the

framework

for

substructural

logics

presented

in

-LSB-

-RSB-

The

latter

consists

of

a

labelled

generalization

of

the

classical

refutation

system

KE

-LSB-

-RSB-

where

the

labels

can

be

interpreted

as

pieces

of

information

that

may

or

may

not

verify

a

given

formula

In

this

paper

we

shall

restrict

our

attention

to

the

fragment

containing

the

operators

and

The

interplay

between

modalities

and

the

other

operators

will

be

discussed

in

a

subsequent

work

Owing

to

space

constraints

proper

comparisons

with

related

work

will

also

have

to

be

postponed

to

another

occasion

The

classical

KE

system

The

system

KE

like

the

tableau

method

and

resolution

is

a

refutation

system

for

classical

logic

Unlike

resolution

however

KE

is

not

restricted

to

clausal

form

and

unlike

the

tableau

method

it

includes

a

cut

rule

which

can

not

in

general

be

eliminated

This

classical

cut

rule

is

called

PB

-LRB-

from

Principle

of

Bivalence

-RRB-

and

has

the

following

forms

depending

on

whether

we

deal

with

signed

or

unsigned

formulae

TA

FA

A

A

The

formula

A

introduced

by

an

application

of

this

rule

is

called

PB-formula

or

cut

Once

such

a

cut

rule

has

been

allowed

the

branching

elimination

rules

typical

of

the

tableau

method

become

unnecessarily

strong

and

can

be

replaced

by

weaker

non-branching

rules

-LRB-

with

two

premises

-RRB-

The

rules

of

the

KE

system

-LRB-

for

unsigned

formulae

-RRB-

are

illustrated

in

Table

The

two-premiss

elimination

rules

correspond

to

familiar

principles

of

inference

modus

ponens

modus

tollens

disjunctive

syllogism

its

dual

The

one-premiss

elimination

rules

are

the

same

as

the

tableau

rules

A

KE-refutation

of

a

set

of

formulae

is

as

usual

a

closed

tree

of

formulae

constructed

according

to

the

rules

of

KE

starting

from

formulae

in

A

crucial

property

of

KE

is

the

analytic

cut

property

the

applications

of

the

cut

rule

can

be

restricted

to

subformulae

of

the

formulae

occurring

above

in

the

branch

without

loss

of

completeness

This

property

allows

for

systematic

and

efficient

refutation

procedures

Indeed

results

in

-LSB-

-RSB-

imply

that

any

refutation

procedure

which

can

be

formulated

in

terms

of

the

tableau

rules

can

be

efficiently

-LRB-

linearly

-RRB-

simulated

by

means

of

the

KE

rules

but

there

are

efficient

and

systematic

KE-procedures

which

can

not

be

polynomially

simulated

by

means

of

the

tableau

rules

One

of

these

procedures

is

the

canonical

procedure

consisting

in

giving

priority

to

the

see

that

PB

plays

the

same

role

as

the

cut

rule

in

the

classical

sequent

calculus

think

of

it

as

a

rule

that

allows

one

to

construct

a

closed

KE-tree

for

given

a

closed

KE-tree

for

A

and

a

closed

KE-tree

for

A

for

a

discussion

of

this

point

see

-LSB-

-RSB-

linear

elimination

rules

over

the

cut

rule

so

that

the

cut

rule

is

applied

only

when

no

elimination

rule

is

further

applicable

and

the

choice

of

the

cut

formulae

is

restricted

to

pairs

A

A

such

that

A

is

an

immediate

subformula

of

a

formula

of

type

-LRB-

in

the

Smullyan

notation

-RRB-

occurring

above

in

the

branch

Disjunction

Rules

A

B

A

B

A

B

B

A

-LRB-

A

B

-RRB-

A

B

E

Conjunction

Rules

-LRB-

A

B

-RRB-

A

B

E

-LRB-

A

B

-RRB-

B

A

E

A

B

A

B

E

Implication

Rules

A

B

A

B

E!

A

B

B

A

E!

-LRB-

A

B

-RRB-

A

B

E

Negation

Rule

A

A

E

Principle

of

Bivalence

A

A

PB

Table

KE-rules

for

unsigned

formulae

In

-LSB-

-RSB-

the

system

KE

is

extended

into

a

labelled

deductive

system

-LRB-

in

the

sense

of

-LSB-

-RSB-

-RRB-

which

provides

a

unifying

proof

framework

for

-LRB-

the

multiplicative

fragment

of

-RRB-

classical

and

intuitionistic

substructural

logics

Information

Frames

In

this

section

we

briefly

review

the

treatment

of

substructural

implication

systems

in

-LSB-

-RSB-

to

which

we

refer

the

reader

for

further

details

-LRB-

See

Section

for

related

work

on

this

topic

-RRB-

In

the

context

of

this

paper

by

a

consequence

relation

over

a

given

logical

language

L

we

intend

a

relation

between

sequences

of

L-formulae

and

L-formulae

satisfying

the

following

two

conditions

-LRB-

Identity

-RRB-

A

A

-LRB-

Surgical

Cut

-RRB-

A

and

A

B

=-RRB-

B

Let

L

be

a

propositional

language

containing

two

binary

operators

and

By

an

implication

logic

we

mean

a

consequence

relation

over

L

characterised

by

the

following

universal

conditions

on

the

operators

and

C

A

B

-LRB-

-RRB-

A

B

C

A

B

-LRB-

-RRB-

A

B.

Let

us

now

consider

the

following

structural

rules

A

B

C

Exchange

B

A

C

A

A

C

Contraction

A

C

A

C

Expansion

A

A

C

C

Weakening

A

C

Then

we

can

consider

the

family

of

implication

logics

closed

under

different

collections

of

these

structural

rules

We

shall

call

the

logical

systems

belonging

to

this

family

substructural

implication

logics

Notice

that

in

all

the

systems

allowing

Exchange

the

two

implications

and

collapse

In

this

context

we

shall

use

the

symbol

without

subscripts

Notice

also

that

every

system

closed

under

Weakening

is

closed

also

under

Expansion

Substructural

logics

in

general

and

implication

systems

in

particular

can

be

characterized

in

terms

of

information

structures

that

we

call

quantale

frames

or

information

frames

Definition

An

information

frame

or

quantale

frame

is

a

structure

F

-LRB-

Q

ffi

v

-RRB-

such

that

Q

is

a

non-empty

set

of

elements

called

pieces

of

information

or

information

tokens

v

is

a

partial

ordering

which

makes

Q

into

a

complete

lattice

x

v

y

can

be

interpreted

as

y

contains

at

least

the

same

information

as

x

ffi

is

a

binary

operation

on

Q

which

is

-LRB-

a

-RRB-

associative

x

ffi

-LRB-

y

ffi

z

-RRB-

-LRB-

x

ffi

y

-RRB-

ffi

z

-LRB-

b

-RRB-

distributive

over

F

for

every

non-empty

family

fzig

Q

Ffzi

ffi

xg

Ffzig

ffi

x

and

Ffx

ffi

zig

x

ffi

Ffzig

Q

and

for

every

x

Q

x

ffi

ffi

x

x

Observe

that

the

properties

of

ffi

imply

that

this

operation

is

order-preserving

i.e.

v

=-RRB-

ffi

y

v

ffi

y

and

y

ffi

v

y

ffi

We

can

define

classes

of

quantale

frames

which

satisfy

additional

conditions

on

the

ordering

v

Definition

We

say

that

a

quantale

frame

is

commutative

if

x

ffi

y

v

y

ffi

x

contractive

if

x

ffi

x

v

x

expansive

if

x

v

x

ffi

x

monotonic

if

x

v

x

ffi

y

We

shall

call

these

conditions

structural

constraints

Information

tokens

may

verify

formulae

of

the

given

language

L

This

verification

is

governed

by

the

notion

of

valuation

Definition

Given

a

quantale

frame

F

let

V

be

a

function

F

Q

fT

Fg

where

-LRB-

i

-RRB-

F

is

the

set

of

formulae

of

the

language

-LRB-

ii

-RRB-

Q

is

the

set

of

information

tokens

in

the

frame

F

-LRB-

iii

-RRB-

T

F

are

truth

values

on

which

we

assume

the

usual

order

F

T

we

say

that

V

is

a

valuation

over

F

when

it

satisfies

the

following

conditions

-LRB-

Heredity

-RRB-

For

all

formulae

A

if

V

-LRB-

A

x

-RRB-

T

and

x

v

y

then

V

-LRB-

A

y

-RRB-

T

-LRB-

Continuity

-RRB-

For

each

given

A

and

every

non-empty

S

Q

ffl

V

-LRB-

A

FS

-RRB-

FfV

-LRB-

A

x

-RRB-

jx

Sg

and

ffl

V

-LRB-

A

dS

-RRB-

dfV

-LRB-

A

x

-RRB-

jx

Sg

As

usual

when

V

-LRB-

A

x

-RRB-

T

we

shall

also

say

A

is

true

at

x

or

A

is

verified

at

a

or

a

verifies

A

and

sometimes

we

shall

write

xk?A

Similarly

when

V

-LRB-

A

x

-RRB-

F

we

shall

also

say

A

is

false

at

x

or

A

is

not

verified

at

a

or

a

does

not

verify

A

and

sometimes

write

Definition

An

implication

structure

is

a

pair

-LRB-

F

V

-RRB-

where

F

is

a

quantale

frame

and

V

is

a

valuation

over

F

satisfying

the

following

condition

V

-LRB-

A

B

x

-RRB-

T

-LRB-

-RRB-

V

-LRB-

A

y

-RRB-

F

or

V

-LRB-

B

x

ffi

y

-RRB-

T

-LRB-

-RRB-

The

valuation

clause

for

is

clearly

a

generalization

of

Urquhart

's

semantics

of

relevant

implication

-LSB-

-RSB-

The

only

difference

is

that

the

underlying

algebraic

structure

is

a

quantale

rather

than

a

semilattice

We

say

that

an

implication

formula

is

verified

in

an

implication

structure

if

it

is

true

at

its

identity

point

Given

a

set

S

of

structural

constraints

we

call

S-frame

a

frame

closed

under

all

the

constraints

in

S

An

implication

formula

is

verified

in

an

S-frame

if

its

verified

in

all

the

implication

structures

based

on

it

and

is

valid

for

S

if

it

is

verified

in

all

S-frames

We

also

say

that

a

finite

sequence

An

of

formulae

implies

a

formula

A

for

S

if

the

implication

formula

-LRB-

-LRB-

An

A

-RRB-

-RRB-

is

valid

for

S

In

-LSB-

-RSB-

different

substructural

logics

are

seen

to

correspond

to

different

classes

of

quantale

frames

in

the

expected

way

each

structural

rule

being

associated

with

a

structural

constraint

on

v.

Indeed

one

deals

only

with

the

implication

fragments

full

quantale

frames

are

not

necessary

and

semilattice

ordered

monoids

are

sufficient

However

as

we

shall

see

in

the

sequel

the

extra

structure

of

quantale

frames

allows

us

to

represent

modalities

in

a

very

natural

way

The

correspondence

between

structural

rules

and

structural

constraints

is

summarized

in

Table

Some

of

the

resulting

logics

are

well-known

implication

systems

For

instance

if

ffi

satisfies

only

the

commutativity

condition

the

resulting

logic

is

the

implication

fragment

of

Girard

's

Linear

Logic

-LSB-

-RSB-

if

it

satisfies

both

commutativity

and

contraction

then

the

system

of

relevant

Commutativity

x

ffi

y

v

y

ffi

x

A

B

C

B

A

C

Contraction

x

ffi

x

v

x

A

A

C

A

C

Expansion

x

v

x

ffi

x

A

C

A

A

C

Monotonicity

x

v

x

ffi

y

C

A

C

Table

Correspondence

between

structural

constraints

and

structural

rules

x

ffi

y

v

y

ffi

x

x

ffi

x

v

x

x

v

x

ffi

x

x

v

x

ffi

y

Lambek

's

implications

Linear

implication

ffl

Relevant

implication

ffl

ffl

Mingle

ffl

ffl

ffl

BCK

implication

ffl

ffl

Intuitionistic

implication

ffl

ffl

ffl

ffl

Table

Correspondence

between

implication

systems

and

sets

of

structural

constraints

TA

B

x

TA

y

TB

x

ffi

y

TA

B

x

FB

x

ffi

y

FA

y

FA

B

x

TA

a

FB

x

ffi

a

where

a

is

a

new

atomic

label

TA

x

FA

x

TA

TA

xn

FA

y

if

u

u

xn

v

y

Table

The

LKE

rules

for

substructural

implication

implication

-LSB-

-RSB-

is

obtained

finally

if

ffi

satisfies

all

the

structural

conditions

on

ffi

the

resulting

logic

is

intuitionistic

implication

See

Table

for

an

overview

Implicational

LKE

An

inferential

characterization

of

substructural

implication

logics

can

be

obtained

by

turning

the

semantics

described

above

into

the

rules

of

a

labelled

deductive

system

-LRB-

in

the

sense

of

-LSB-

-RSB-

-RRB-

In

-LSB-

-RSB-

D'Agostino

and

Gabbay

presented

a

labelled

refutation

system

consisting

of

a

generalization

of

the

classical

system

KE

investigated

in

-LSB-

-RSB-

The

rules

of

this

labelled

refutation

system

that

we

call

LKE

are

tree-expansion

rules

which

are

immediately

justified

by

-LRB-

and

are

indeed

equivalent

to

-RRB-

our

previous

definitions

-LRB-

see

-LSB-

-RSB-

for

a

detailed

discussion

of

the

route

which

leads

from

sequent-based

presentation

of

substructural

consequence

relations

to

labelled

refutation

systems

via

algebraic

and

Kripke-style

semantics

-RRB-

The

rules

for

the

implication

fragment

are

listed

in

Table

Notice

that

the

only

branching

rule

can

be

seen

as

a

labelled

generalization

of

the

classical

principle

of

bivalence

Observe

also

that

a

simple

special

case

of

the

closure

rule

given

in

the

table

is

the

simple

rule

TA

x

FA

y

provided

x

v

y

Indeed

this

special

case

is

sufficient

for

completeness

except

for

the

implication

systems

which

do

not

allow

weakening

but

do

allow

expansion

-LRB-

see

-LSB-

-RSB-

for

the

details

-RRB-

In

these

rules

the

declarative

units

are

not

just

signed

formulae

as

in

the

classical

KE

system

-LRB-

or

in

the

system

of

analytic

tableaux

-RRB-

but

labelled

signed

formulae

or

LS-formulae

for

The

points

of

the

quantale

frames

are

turned

into

labels

while

signs

play

the

usual

role

so

that

TA

x

is

interpreted

as

A

is

true

at

point

x

and

FA

x

is

interpreted

as

A

is

not

true

at

point

x

Different

classes

of

frames

correspond

to

different

labelling

algebras

i.e.

different

sets

of

rules

that

can

be

used

in

manipulating

the

labelling

terms

to

verify

whether

or

not

the

condition

for

the

application

of

the

closure

rule

is

satisfied

The

labelling

language

consists

of

a

denumerable

set

of

atomic

labels

denoted

by

a

b

c

etc.

-LRB-

possibly

with

subscripts

-RRB-

a

distinguished

atomic

label

and

two

binary

operators

ffi

and

Complex

labels

are

built

up

from

the

atomic

ones

by

means

of

the

two

binary

operators

The

basic

labelling

algebra

which

applies

to

all

frames

consists

of

the

following

elements

ffl

the

usual

axioms

expressing

the

fact

that

ffi

is

a

monoid

operation

with

as

identity

ffl

the

usual

axioms

expressing

the

fact

that

u

is

a

semilattice

meet

The

partial

ordering

v

can

be

defined

as

usual

by

putting

x

v

y

if

and

only

if

x

u

y

x

This

basic

labelling

algebra

augmented

with

a

class

of

structural

constraints

S

provides

a

specific

labelling

algebra

sufficient

to

charaterize

the

notion

of

validity

in

S-frames

We

shall

denote

by

LKE

-LRB-

S

-RRB-

where

S

is

an

arbitrary

set

of

structural

constraints

the

LKE

system

equipped

with

similar

approach

is

used

by

Fitting

in

his

prefixed

tableaux

for

classical

modal

logics

-LSB-

-RSB-

operator

is

used

only

in

connection

with

the

general

closure

rule

given

in

the

table

and

is

not

required

whenever

the

simpler

special

case

described

above

is

sufficient

for

completeness

namely

for

all

classes

of

frames

which

either

are

monotonic

or

nonexpansive

the

labelling

algebra

obtained

by

adding

the

constraints

in

S

to

the

basic

labelling

algebra

A

proof

of

the

validity

of

a

formula

A

for

the

class

of

S-frames

consists

in

a

refutation

of

the

assumption

that

A

is

false

at

the

identity

element

of

some

implication

structure

based

on

an

S-frame

Such

a

refutation

is

represented

by

a

closed

LKE

-LRB-

S

-RRB-

tree

starting

with

the

LS-formula

FA

where

the

constraints

in

S

may

be

used

together

with

the

basic

labelling

algebra

in

order

to

close

a

branch

Whenever

such

a

closed

tree

can

be

constructed

we

say

that

A

is

a

LKE

-LRB-

S

-RRB-

theorem

We

shall

also

denote

by

LKE

-LRB-

S

-RRB-

the

-LRB-

finitary

-RRB-

deducibility

relation

of

LKE

-LRB-

S

-RRB-

defined

as

follows

-LRB-

where

-LSB-

-RSB-

represents

the

empty

list

of

formulae

-RRB-

-LSB-

-RSB-

LKE

-LRB-

S

-RRB-

A

iff

A

is

an

LKE

-LRB-

S

-RRB-

theorem

An

LKE

-LRB-

S

-RRB-

B

iff

-LSB-

-RSB-

LKE

-LRB-

S

-RRB-

-LRB-

-LRB-

An!B

-RRB-

-RRB-

We

shall

also

write

simply`LKE

-LRB-

S

-RRB-

instead

of

-LSB-

-RSB-

LKE

-LRB-

S

-RRB-

A

This

definition

implies

that

An

LKE

-LRB-

S

-RRB-

B

if

and

only

if

there

is

a

closed

LKE

-LRB-

S

-RRB-

tree

starting

with

the

sequence

of

LS-formulae

TAn

an

FB

ffi

ffi

an

where

an

are

all

distinct

atomic

labels

In

this

approach

the

whole

family

of

substructural

implication

logics

is

therefore

characterised

by

the

same

tree-expansion

rules

and

different

members

of

the

family

are

identified

by

the

different

labelling

algebras

that

can

be

employed

to

check

branch-closure

This

is

what

we

call

the

separation-byclosure

property

of

the

LKE

system

Another

remarkable

property

of

LKE

is

the

atomic

closure

property

if

there

is

a

closed

tree

starting

from

a

given

set

of

initial

LS-formulae

then

there

is

also

one

starting

from

the

same

set

of

LS-formulae

such

that

the

closure

rule

is

applied

only

with

atomic

LS

formulae

i.e.

LS-formulae

of

the

form

sP

x

-LRB-

where

s

T

or

F

-RRB-

whith

P

atomic

An

example

of

an

LKE-proof

is

given

in

Table

This

example

does

not

make

any

use

of

the

branching

rule

of

generalized

bivalence

Indeed

this

rule

introduces

a

good

deal

of

non-determinism

into

the

system

in

that

it

allows

for

the

use

of

-LRB-

a

-RRB-

arbitrary

formulae

and

-LRB-

b

-RRB-

arbitrary

labels

in

each

rule

application

However

this

non-determinism

can

be

tamed

to

some

extent

As

for

-LRB-

a

-RRB-

it

can

be

shown

that

the

applications

of

PB

can

be

restricted

to

analytic

ones

i.e.

involving

only

subformulae

of

formulae

previously

occurring

in

the

branch

-LRB-

and

even

further

to

canonical

applications

F

-LRB-

A

-LRB-

A

B

-RRB-

-RRB-

-LRB-

A

B

-RRB-

TA

-LRB-

A

B

-RRB-

a

FA

B

ffi

a

-LRB-

a

-RRB-

TA

b

FB

a

ffi

b

TA

B

a

ffi

b

TB

a

ffi

b

ffi

b

Table

The

contraction

axiom

is

valid

for

the

class

of

contractive

frames

-LRB-

i.e.

those

satisfying

the

condition

x

v

x

ffi

x

for

all

x

-RRB-

For

b

ffi

b

v

b

and

since

ffi

is

order-preserving

a

ffi

b

ffi

b

v

a

ffi

b.

Therefore

this

one-branch

tree

is

closed

The

axiom

is

not

valid

in

Linear

Logic

which

is

characterised

by

a

class

of

frames

which

are

not

contractive

as

in

the

canonical

procedure

for

the

classical

system

outlined

in

Section

-RRB-

As

for

-LRB-

b

-RRB-

it

can

be

shown

that

similar

restrictions

hold

for

the

labels

and

the

best

strategy

is

to

apply

the

PB

rule

with

a

variable

as

label

In

this

way

the

closure

of

a

branch

depends

on

the

solution

of

an

inequation

in

the

given

algebra

of

the

labels

and

the

closure

of

the

whole

tree

on

the

simultaneous

solution

of

a

system

of

inequations

This

a

well

defined

algebraic

problem

which

can

be

addressed

via

unification-like

techniques

For

a

detailed

and

systematic

presentation

of

the

LKE

system

-LRB-

including

the

other

multiplicative

operators

different

from

-RRB-

and

more

examples

of

LKE-proofs

the

reader

is

referred

to

-LSB-

-RSB-

The

modal

operators

We

introduce

in

our

quantale

frames

a

binary

relation

R

between

their

points

called

the

accessibility

relation

We

assume

that

this

relation

is

closed

under

arbitrary

F

and

d

namely

-LRB-

S

-RRB-

xRy

=-RRB-

xRGS

and

-LRB-

S

-RRB-

xRy

=-RRB-

xR

l

S

-LRB-

-RRB-

We

then

introduce

into

our

language

the

two

unary

operators

and

intended

as

the

usual

modalities

Valuations

are

extended

to

these

modal

operators

in

the

obvious

way

For

we

have

the

following

clause

V

-LRB-

x

-RRB-

T

iff

V

-LRB-

A

y

-RRB-

T

for

all

y

such

that

xRy

-LRB-

-RRB-

and

for

V

-LRB-

x

-RRB-

T

iff

V

-LRB-

A

y

-RRB-

T

for

some

y

such

that

xRy

-LRB-

-RRB-

In

order

to

preserve

the

hereditary

property

of

valuations

-LRB-

see

point

of

Definition

-RRB-

R

must

also

satisfy

the

following

conditions

-LRB-

see

-LSB-

-RSB-

and

-LSB-

-RSB-

-RRB-

x

v

y

and

xRz

=-RRB-

-LRB-

-RRB-

-LRB-

and

z

v

-RRB-

-LRB-

-RRB-

and

x

v

y

and

yRz

=-RRB-

-LRB-

-RRB-

-LRB-

and

v

z

-RRB-

-LRB-

-RRB-

Definition

A

modal

quantale

frame

is

a

pair

-LRB-

F

R

-RRB-

where

F

is

a

quantale

frame

and

R

is

a

binary

accessibility

relation

defined

on

the

domain

of

F

and

satisfying

conditions

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

If

R

is

defined

as

above

any

atomic

valuation

V

over

a

quantale

frame

can

be

extended

over

a

modal

quantale

frame

by

means

of

the

valuation

clauses

for

and

preserving

the

hereditary

property

of

valuations

Definition

A

modal

implication

structure

is

a

triple

-LRB-

F

R

V

-RRB-

where

-LRB-

F

R

-RRB-

is

a

modal

quantale

frame

and

V

is

a

valuation

satisfying

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

Let

M

be

a

modal

implication

structure

-LRB-

F

R

V

-RRB-

and

let

be

the

relation

that

holds

between

two

points

x

and

y

when

they

verify

exactly

the

same

formulae

i.e.

x

y

means

that

xk?A

if

and

only

if

yk?A

We

can

consider

the

quotient

structure

F

and

define

a

valuation

V

on

it

as

follows

V

-LRB-

A

-LSB-

x

-RSB-

-RRB-

T

iff

V

-LRB-

A

x

-RRB-

T

where

-LSB-

x

-RSB-

denotes

the

equivalence

class

of

x

under

Similary

we

can

define

-LSB-

x

-RSB-

-LSB-

y

-RSB-

iff

xRy

Clearly

the

structure

-LRB-

F

V

-RRB-

is

a

modal

implication

structure

and

we

can

restrict

our

attention

without

loss

of

generality

to

modal

implication

structures

with

the

additional

property

that

any

two

points

verifying

exactly

the

same

formulae

are

identical

Definition

We

say

that

a

formula

A

is

verified

in

a

modal

implication

structure

M

if

it

is

verified

at

the

identity

point

of

M

We

also

say

that

A

is

verified

in

a

frame

F

if

is

verified

in

all

modal

implication

structures

based

on

F

Observe

that

these

definitions

are

essentially

different

from

the

classical

definitions

In

classical

modal

logics

a

formula

is

verified

in

a

modal

structure

-LRB-

or

model

as

structures

are

often

called

in

the

literature

-RRB-

if

it

is

verified

in

all

the

worlds

beloging

to

that

structure

Our

definition

is

equivalent

to

the

classical

one

only

in

the

case

of

monotonic

information

frames

For

suppose

a

formula

A

is

verified

at

the

identity

point

of

a

structure

based

on

a

monotonic

frame

i.e.

V

-LRB-

A

-RRB-

T

Then

by

the

monotonicity

of

the

frame

v

ffi

a

-LRB-

a

-RRB-

for

every

point

a

in

the

frame

So

by

the

hereditary

property

of

valuations

V

-LRB-

A

a

-RRB-

T

for

all

the

points

in

the

frame

Our

final

aim

is

to

incorporate

the

semantics

of

the

modal

operators

into

the

proof-theoretical

framework

of

Section

For

this

purpose

we

shall

simplify

this

semantics

by

replacing

the

binary

relation

R

with

a

pair

of

unary

operators

corresponding

to

the

modalities

-LRB-

These

operators

are

reminiscent

of

the

ones

used

in

the

algebraic

semantics

of

classical

modal

logics

for

which

see

-LSB-

-RSB-

-RRB-

Such

a

reformulation

will

allow

us

in

section

to

formulate

an

algebra

of

the

labels

which

is

a

simple

extension

of

the

one

characterizing

substructural

implication

systems

obtained

by

adding

to

the

structural

constraints

a

-LRB-

possibly

empty

-RRB-

set

of

modal

constraints

i.e.

inequations

of

the

form

ff

v

where

ff

and

are

terms

of

the

extended

algebra

including

the

new

unary

operators

Let

us

denote

by

S

-LRB-

x

-RRB-

the

sphere

of

x

i.e.

the

set

of

all

pieces

of

information

accessible

from

x

We

shall

also

use

F

and

d

to

denote

the

usual

lattice

join

and

meet

We

now

define

the

two

unary

operators

and

as

follows

x

def

l

S

-LRB-

x

-RRB-

-LRB-

-RRB-

and

x

def

GS

-LRB-

x

-RRB-

-LRB-

-RRB-

Now

observe

that

-LRB-

-RRB-

implies

that

the

following

property

of

R

is

satisfied

in

every

frame

-LRB-

-RRB-

-LRB-

-RRB-

xRy

-LRB-

-RRB-

In

other

words

for

all

x

S

-LRB-

x

-RRB-

In

fact

for

every

point

x

both

x

and

x

are

accessible

from

x

that

is

-LRB-

-RRB-

xR!x

and

xR?x

-LRB-

-RRB-

In

the

theory

of

classical

modal

logics

a

property

like

-LRB-

-RRB-

is

known

as

seriality

So

in

our

approach

all

frames

are

serial

Using

the

and

operators

this

assumption

is

expressed

by

the

following

inequation

which

holds

for

all

modal

quantale

frames

x

v?x

-LRB-

-RRB-

So

in

our

approach

the

points

x

and

x

have

a

special

status

x

is

a

point

in

S

-LRB-

x

-RRB-

that

verifies

all

and

only

the

formulae

which

are

true

at

all

points

in

S

-LRB-

x

-RRB-

while

x

is

a

point

in

S

-LRB-

x

-RRB-

that

verifies

all

and

only

the

formulae

which

are

true

at

at

least

one

point

in

S

-LRB-

x

-RRB-

It

is

not

difficult

to

show

that

-LRB-

-RRB-

and

-LRB-

-RRB-

above

are

respectively

equivalent

to

the

following

conditions

on

and

expressing

the

fact

that

the

operators

and

are

both

order-preserving

x

v

y

=-RRB-

x

v?y

-LRB-

-RRB-

x

v

y

=-RRB-

x

v!y

-LRB-

-RRB-

Proposition

In

every

modal

quantale

frame

-LRB-

-RRB-

holds

if

and

only

if

-LRB-

-RRB-

holds

and

-LRB-

-RRB-

holds

if

and

only

if

-LRB-

-RRB-

holds

Proof

We

show

only

that

-LRB-

-RRB-

is

equivalent

to

-LRB-

-RRB-

and

leave

it

to

the

reader

to

show

the

other

equivalence

First

assume

-LRB-

-RRB-

and

suppose

x

v

y.

By

-LRB-

-RRB-

we

know

that

yR!y

By

-LRB-

-RRB-

-LRB-

-RRB-

v!y

and

Then

by

definition

of

x

v

and

therefore

x

v!y

Now

assume

-LRB-

-RRB-

and

suppose

-LRB-

i

-RRB-

x

v

y

-LRB-

ii

-RRB-

yRz

Then

y

v

z

and

by

-LRB-

-RRB-

x

v

z

Since

xR!x

there

exists

such

that

and

v

z.

Given

our

definitions

of

the

operators

and

the

valuation

clauses

for

and

can

be

reformulated

more

concisely

as

below

V

-LRB-

x

-RRB-

T

iff

V

-LRB-

A

x

-RRB-

T

-LRB-

-RRB-

and

V

-LRB-

x

-RRB-

T

iff

V

-LRB-

A

x

-RRB-

T

-LRB-

-RRB-

Let

us

now

show

the

equivalence

of

the

valuation

clauses

-LRB-

-RRB-

and

-LRB-

-RRB-

with

-LRB-

-RRB-

and

-LRB-

-RRB-

respectively

First

we

assume

that

the

clause

in

-LRB-

-RRB-

holds

and

show

that

-LRB-

-RRB-

holds

too

For

the

only-if

direction

let

us

assume

V

-LRB-

x

-RRB-

T

then

by

-LRB-

-RRB-

V

-LRB-

A

y

-RRB-

T

for

all

y

S

-LRB-

x

-RRB-

Now

by

the

definition

of

valuation

A

remains

true

at

the

greatest

lower

bound

of

S

-LRB-

x

-RRB-

that

is

at

x.

For

if-direction

let

us

assume

V

-LRB-

x

-RRB-

F

It

follows

that

there

exists

a

point

y

S

-LRB-

x

-RRB-

such

that

A

is

false

at

y.

Since

x

v

y

A

must

be

false

at

x

Now

we

show

that

-LRB-

-RRB-

implies

-LRB-

-RRB-

For

the

only-if

direction

suppose

V

-LRB-

x

-RRB-

T

Then

by

-LRB-

-RRB-

V

-LRB-

A

x

-RRB-

T

So

A

is

true

at

every

point

y

such

that

x

v

y

and

therefore

at

every

point

in

S

-LRB-

x

-RRB-

For

the

if-direction

assume

that

A

is

true

at

every

point

in

S

-LRB-

x

-RRB-

Suppose

that

is

false

at

x.

Then

by

-LRB-

-RRB-

V

-LRB-

A

x

-RRB-

F

So

the

assumption

that

A

is

true

at

every

point

in

S

-LRB-

x

-RRB-

is

contradicted

Hence

must

be

true

at

x

A

similar

argument

shows

the

equivalence

of

-LRB-

-RRB-

and

-LRB-

-RRB-

We

can

now

exploit

the

new

operators

to

express

complex

statements

about

the

accessibility

relation

R

concisely

in

the

form

of

simple

inequalities

of

the

form

ff

v

where

ff

and

are

expressions

built

up

from

atomic

terms

by

means

of

the

operators

ffi

and

Let

us

consider

some

of

the

most

familiar

properties

of

the

accessibility

relation

R.

Seriality

A

frame

is

serial

if

for

every

point

x

S

-LRB-

x

-RRB-

that

is

for

every

point

x

there

exists

a

y

such

that

xRy

Observe

that

a

frame

is

serial

if

and

only

if

it

verifies

the

formula

As

mentioned

above

in

our

approach

all

frames

are

serial

as

a

consequence

of

-LRB-

-RRB-

and

seriality

corresponds

to

the

assumption

that

x

v?x

for

all

x.

Reflexivity

A

frame

is

said

to

be

reflexive

if

xRx

for

all

x.

Observe

that

a

frame

is

reflexive

if

and

only

if

it

verifies

the

following

formula

A

In

our

notation

reflexivity

can

be

expressed

by

the

following

condition

-LRB-

Reflexivity

-RRB-

-LRB-

-RRB-

-LRB-

x

v

x

-RRB-

The

equivalence

between

the

two

formulations

is

immediately

seen

as

follows

First

assume

that

x

v

x

Then

every

formula

true

at

x

is

true

also

at

x.

Suppose

xRx

does

not

hold

for

some

x

This

means

that

there

exists

a

point

a

such

that

aRa

Now

consider

a

valuation

V

and

a

formula

A

such

that

V

-LRB-

A

x

-RRB-

T

for

all

x

S

-LRB-

a

-RRB-

i.e.

V

-LRB-

a

-RRB-

T

but

V

-LRB-

A

a

-RRB-

F

-LRB-

which

is

possible

since

a

S

-LRB-

a

-RRB-

-RRB-

For

this

V

we

have

that

V

-LRB-

A

a

-RRB-

T

and

since

a

v

a

V

-LRB-

A

a

-RRB-

T

This

is

a

contradiction

Hence

if

x

v

x

for

all

x

then

xRx

for

all

x.

Conversely

if

xRx

for

all

x

every

formula

that

is

true

at

all

points

accessible

from

x

is

true

also

at

x

i.e.

for

all

x

x

v

x.

Transitivity

A

frame

is

said

to

be

transitive

if

the

following

condition

holds

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

yRz

=-RRB-

xRz

-RRB-

A

frame

is

transitive

if

and

only

if

it

verifies

the

following

formula

In

our

notation

transitivity

is

expressed

by

the

following

condition

-LRB-

Transitivity

-RRB-

-LRB-

-RRB-

-LRB-

x

v

x

-RRB-

Assume

that

x

v

x

for

all

x.

Now

suppose

that

the

frame

is

not

transitive

Then

there

exists

a

point

a

a

valuation

V

and

a

formula

A

such

that

V

-LRB-

a

-RRB-

T

but

V

-LRB-

a

-RRB-

F

This

implies

by

-LRB-

-RRB-

that

V

-LRB-

A

a

-RRB-

T

and

V

-LRB-

A

a

-RRB-

F

But

since

a

v

a

this

is

impossible

Hence

if

x

v

x

for

all

x

the

frame

is

transitive

Conversely

suppose

the

frame

is

transitive

We

show

that

x

v

x.

Let

A

be

any

formula

which

is

true

at

x

Then

A

is

true

at

all

points

in

S

-LRB-

x

-RRB-

Moreover

x

is

accessible

from

x

Since

the

frame

is

transitive

if

a

point

y

is

accessible

from

x

it

is

also

accessible

from

x.

So

if

A

is

true

at

all

the

points

accessible

from

x

it

is

also

true

at

all

the

points

accessible

from

x

that

is

x

v

x.

Symmetry

A

frame

is

said

to

be

symmetric

if

the

following

condition

holds

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

=-RRB-

yRx

-RRB-

A

frame

is

symmetric

if

and

only

if

it

verifies

the

following

formula

A

In

our

notation

symmetry

is

expressed

by

the

following

condition

-LRB-

Symmetry

-RRB-

-LRB-

-RRB-

-LRB-

x

v

x

-RRB-

For

this

property

as

well

as

for

the

next

two

We

leave

it

to

the

reader

to

verify

the

equivalence

between

the

correspoding

formulations

Euclideanism

A

frame

is

said

to

be

Euclidean

if

the

following

condition

holds

true

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

xRz

=-RRB-

yRz

-RRB-

A

frame

is

Euclidean

if

and

only

if

it

verifies

the

following

formula

Modal

formulae

Conditions

on

R

Modal

constraints

A

-LRB-

-RRB-

xRx

x

v

x

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

yRz

=-RRB-

xRz

-RRB-

x

v

x

A

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

=-RRB-

yRx

-RRB-

x

v

x

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

xRz

=-RRB-

yRz

-RRB-

x

v

x

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

xRz

=-RRB-

-LRB-

-RRB-

-LRB-

yRu

and

zRu

-RRB-

-RRB-

x

v

x

Table

Correspondence

between

modal

formulae

conditions

on

R

and

constraints

on

the

modal

quantale

frames

The

condition

corresponding

to

this

property

is

the

one

stated

below

-LRB-

Euclideanism

-RRB-

-LRB-

-RRB-

-LRB-

x

v

x

-RRB-

Directedness

A

frame

is

directed

if

the

following

condition

holds

true

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xRy

and

xRz

=-RRB-

-LRB-

-RRB-

-LRB-

yRu

and

zRu

-RRB-

-RRB-

which

corresponds

to

-LRB-

Directedness

-RRB-

-LRB-

-RRB-

-LRB-

x

v

x

-RRB-

Observe

that

a

frame

is

directed

if

and

only

if

it

verifies

the

following

formula

Our

discussion

so

far

amounts

to

a

reformulation

of

the

well-known

correspondence

theory

in

our

new

setting

This

correspondence

is

summarized

in

table

We

shall

now

state

a

crucial

lemma

Let

and

be

-LRB-

possibly

empty

-RRB-

strings

of

and

The

dual

of

oei

denoted

by

is

the

string

obtained

by

interchanging

and

in

oei

Moreover

let

M

M

where

M

is

the

set

of

modal

conditions

listed

above

We

have

the

following

duality

principle

Duality

Lemma

Assume

that

v

holds

for

all

x

in

all

M

frames

i.e.

there

is

a

chain

of

inequalities

v

v

OEn

such

that

-LRB-

i

-RRB-

-LRB-

ii

-RRB-

OEn

and

-LRB-

iii

-RRB-

each

inequality

OEi

v

OEi

with

i

n

is

one

of

of

the

primitive

inequalities

in

M

Then

v

also

holds

for

all

x

in

all

M

frames

Proof

The

proof

is

an

easy

induction

on

the

length

of

the

chain

v

v

OEn

Base

n

Then

the

inequality

v

is

an

element

of

M

The

reader

can

easily

verify

that

the

lemma

holds

for

all

primitive

inequalities

in

M.

Step

n

k

Let

ae

be

the

prefix

of

OEk

i.e.

OEk

aex

By

inductive

hypothesis

v

Moreover

since

aex

v

again

by

inductive

hypothesis

we

have

that

v

Hence

v

Let

now

be

a

propositional

language

containing

the

two

binary

operators

and

plus

the

the

two

modal

operators

and

-LRB-

Recall

that

the

two

conditionals

collapse

in

all

the

systems

allowing

the

structural

rule

of

Exchange

in

which

case

we

use

the

symbol

without

subscripts

-RRB-

By

a

modal

implication

logic

or

MIL

for

short

we

mean

a

-LRB-

substructural

-RRB-

implication

logic

over

satisfying

the

following

conditions

on

the

operators

and

A

B

-LRB-

-RRB-

and

A

B

-LRB-

-RRB-

In

addition

to

-LRB-

-RRB-

and

-LRB-

-RRB-

above

different

MIL

's

may

also

satisfy

different

subsets

of

the

following

modal

axioms

A

A

In

this

paper

we

restrict

our

attention

to

serial

MIL

's

i.e.

those

satisfying

the

modal

axiom

In

the

next

section

we

shall

describe

a

concrete

example

of

a

modal

implication

structure

defined

in

terms

of

the

deducibility

relation

associated

with

any

given

-LRB-

serial

-RRB-

MIL

We

call

this

concrete

model

the

canonical

model

The

canonical

model

In

the

canonical

model

the

points

of

the

underlying

quantale

frame

are

decreasing

sets

of

formulae

i.e.

if

A

belongs

to

any

such

set

and

A

B

then

B

also

belongs

to

the

set

where

is

the

deducibility

relation

of

an

arbitrary

MIL

-LRB-

we

call

these

sets

decreasing

because

A

B

can

be

interpreted

as

the

information

carried

by

A

is

less

than

or

equal

to

the

information

carried

by

B

-RRB-

For

every

MIL

the

canonical

frame

generated

by

is

then

a

structure

hM

ffi

Ri

such

that

ffl

M

is

the

set

of

all

decreasing

sets

of

formulae

with

respect

to

the

given

ffl

ffi

is

a

binary

operation

defined

as

follows

x

ffi

y

def

fC

j

-LRB-

x

-RRB-

-LRB-

y

-RRB-

A

B

Cg

ffl

is

the

set

fA

j

-LSB-

-RSB-

Ag

where

-LSB-

-RSB-

denotes

the

empty

list

ffl

the

relation

is

set-inclusion

Clearly

M

is

a

complete

lattice

under

with

standard

union

and

intersection

as

join

and

meet

Moreover

it

is

not

difficult

to

verify

that

ffi

is

a

monoid

operation

with

as

identity

Furthermore

it

is

easy

to

show

that

ffi

distributes

over

arbitrary

joins

Therefore

the

structure

hM

ffi

i

is

a

quantale

frame

Let

us

now

define

two

functions

on

the

points

of

the

canonical

model

h

-LRB-

x

-RRB-

def

xg

and

g

-LRB-

x

-RRB-

def

xg

Notice

that

for

every

x

h

-LRB-

x

-RRB-

and

g

-LRB-

x

-RRB-

are

themselves

decreasing

sets

of

formulae

because

of

the

monotonicity

properties

of

and

-LRB-

-RRB-

and

-LRB-

-RRB-

Therefore

they

also

belong

to

the

canonical

model

Notice

also

that

our

seriality

assumption

implies

that

for

all

x

h

-LRB-

x

-RRB-

g

-LRB-

y

-RRB-

Accessibility

between

points

of

this

canonical

model

is

then

defined

as

follows

xRy

iff

h

-LRB-

x

-RRB-

y

g

-LRB-

x

-RRB-

It

follows

from

our

definitions

that

R

satisfies

the

continuity

property

-LRB-

-RRB-

x

y

implies

h

-LRB-

x

-RRB-

h

-LRB-

y

-RRB-

x

y

implies

g

-LRB-

x

-RRB-

g

-LRB-

y

-RRB-

x

y

and

xRz

implies

and

z

to

see

this

take

g

-LRB-

y

-RRB-

we

have

that

because

h

-LRB-

y

-RRB-

g

-LRB-

y

-RRB-

g

-LRB-

y

-RRB-

moreover

since

xRz

we

have

that

z

g

-LRB-

x

-RRB-

and

since

g

-LRB-

x

-RRB-

g

-LRB-

y

-RRB-

we

can

conclude

z

g

-LRB-

y

-RRB-

Therefore

z

x

y

and

yRz

implies

and

z

to

see

this

take

h

-LRB-

x

-RRB-

notice

that

h

-LRB-

x

-RRB-

h

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

so

moreover

since

h

-LRB-

x

-RRB-

h

-LRB-

y

-RRB-

and

h

-LRB-

y

-RRB-

z

-LRB-

because

yRz

-RRB-

we

have

that

h

-LRB-

x

-RRB-

h

-LRB-

y

-RRB-

z.

TS

-LRB-

x

-RRB-

h

-LRB-

x

-RRB-

SS

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

x

iff

A

h

-LRB-

x

-RRB-

x

iff

A

g

-LRB-

x

-RRB-

Properties

and

above

show

that

R

satisfies

the

conditions

-LRB-

-RRB-

and

-LRB-

-RRB-

and

therefore

the

canononical

frame

is

a

modal

quantale

frame

A

valuation

V

over

M

can

be

defined

simply

as

follows

V

-LRB-

A

x

-RRB-

T

-LRB-

-RRB-

A

x

Such

a

function

V

satisfies

all

the

conditions

on

the

definition

of

a

valuation

-LRB-

Definition

-RRB-

As

for

the

heredity

property

-LRB-

condition

-RRB-

observe

that

V

-LRB-

A

x

-RRB-

T

implies

by

definition

of

V

that

A

x

which

together

with

the

condition

x

y

gives

that

A

y

and

therefore

V

-LRB-

A

y

-RRB-

T

As

for

the

continuity

property

-LRB-

condition

-RRB-

notice

that

since

the

relation

is

set-inclusion

the

F

and

d

are

the

set

operations

S

and

T

respectively

So

suppose

that

for

a

non-empty

family

S

M

V

-LRB-

A

SS

-RRB-

T

By

definition

of

V

A

SS

Then

there

exists

a

set

x

S

such

that

A

x

so

for

such

an

x

V

-LRB-

A

x

-RRB-

T

Hence

FfV

-LRB-

A

x

-RRB-

j

x

Sg

T

For

the

converse

suppose

that

FfV

-LRB-

A

x

-RRB-

j

x

Sg

T

Then

there

exists

an

x

S

such

that

V

-LRB-

A

x

-RRB-

T

that

is

A

x.

Therefore

A

Sfx

j

x

Sg

Hence

V

-LRB-

A

SS

-RRB-

T

An

analogous

argument

shows

that

V

also

satisfies

the

other

continuity

property

V

-LRB-

A

d

-RRB-

dfV

-LRB-

A

x

-RRB-

j

x

Sg

We

now

show

that

the

valuation

V

defined

as

above

satisfies

also

clauses

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

and

therefore

the

canonical

frame

together

with

this

valuation

is

a

modal

implication

structure

We

shall

call

this

structure

the

canonical

model

-LRB-

generated

by

the

given

MIL

-RRB-

Clause

-LRB-

-RRB-

For

the

only-if

direction

we

assume

that

V

-LRB-

A

B

x

-RRB-

T

and

that

for

an

arbitrary

y

M

V

-LRB-

A

y

-RRB-

T

Then

A

B

x

and

A

y.

By

the

identity

property

of

and

by

the

deduction

theorem

we

can

easily

derive

that

A

BA

B.

Hence

by

definition

of

ffi

B

x

ffi

y

which

means

that

V

-LRB-

B

y

-RRB-

T

For

the

if

direction

assume

that

for

all

y

V

-LRB-

A

y

-RRB-

T

=-RRB-

V

-LRB-

B

x

ffi

y

-RRB-

T

Let

y

fC

j

A

Cg

Then

for

such

a

y

A

y

-LRB-

since

A

A

-RRB-

and

therefore

V

-LRB-

A

y

-RRB-

T

It

then

follows

that

V

-LRB-

B

x

ffi

y

-RRB-

T

This

means

that

B

x

ffi

y.

By

definition

of

ffi

there

exist

C

x

and

D

y

such

that

C

D

B

Since

every

formula

in

y

is

proved

by

A

by

surgical

cut

we

obtain

C

A

B

and

therefore

C

A

B

Since

C

x

we

can

conclude

that

A

B

x

and

hence

V

-LRB-

A

B

x

-RRB-

T

Clause

-LRB-

-RRB-

For

the

only-if

direction

let

V

-LRB-

x

-RRB-

T

and

let

y

be

an

arbitrary

point

such

that

xRy

By

definition

of

V

x

which

implies

that

A

h

-LRB-

x

-RRB-

Since

xRy

h

-LRB-

x

-RRB-

y

then

A

y.

Hence

V

-LRB-

A

y

-RRB-

T

For

the

converse

we

assume

that

for

all

y

such

that

xRy

V

-LRB-

A

y

-RRB-

T

This

implies

that

dfV

-LRB-

A

y

-RRB-

j

y

S

-LRB-

x

-RRB-

g

T

Moreover

by

definition

dS

-LRB-

x

-RRB-

h

-LRB-

x

-RRB-

Then

by

the

continuity

property

of

V

V

-LRB-

A

h

-LRB-

x

-RRB-

-RRB-

T

that

is

A

h

-LRB-

x

-RRB-

Hence

x

which

gives

by

definition

of

V

that

V

-LRB-

x

-RRB-

T

Clause

-LRB-

-RRB-

For

the

only-if

direction

we

assume

that

V

-LRB-

x

-RRB-

T

By

definition

of

V

x

which

implies

that

A

g

-LRB-

x

-RRB-

Notice

that

xRg

-LRB-

x

-RRB-

since

h

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

Hence

there

exists

a

y

namely

g

-LRB-

x

-RRB-

such

that

xRy

and

V

-LRB-

A

y

-RRB-

T

For

the

converse

we

assume

that

V

-LRB-

A

y

-RRB-

T

for

some

y

in

S

-LRB-

x

-RRB-

It

follows

that

A

y

and

a

fortiori

A

g

-LRB-

x

-RRB-

Hence

x

and

V

-LRB-

x

-RRB-

T

Let

C

S

-LSB-

M

where

S

is

an

arbitrary

subset

of

the

structural

rules

in

Table

and

let

M

is

an

arbitrary

subset

of

the

modal

axioms

in

Table

We

denote

by

C

the

smallest

MIL

satisfying

all

the

rules

and

axioms

in

C.

Moreover

let

be

the

set

of

constraints

corresponding

to

the

rules

and

axioms

in

C

as

outlined

in

Tables

and

By

a

we

mean

a

frame

satisfying

all

the

constraints

in

Observe

that

the

canonical

frame

generated

by

C

is

a

-LRB-

the

proof

is

left

to

the

reader

-RRB-

Then

by

considering

the

canonical

model

it

is

easy

to

show

the

following

Proposition

If

A

is

verified

in

all

then

C

A.

Proof

Suppose

C

A

Then

the

canonical

model

generated

by

C

de

scribes

a

modal

implication

structure

-LRB-

based

on

a

frame

-RRB-

which

does

not

verify

A.

Modal

LKE

rules

Let

us

enrich

our

labelling

algebra

with

the

two

unary

operators

and

satisfying

conditions

and

the

duality

principle

-LRB-

see

the

Duality

Lemma

above

-RRB-

The

valuation

clauses

-LRB-

-RRB-

and

-LRB-

-RRB-

immediately

imply

simple

LKE-style

rules

for

the

modal

operators

Elimination

rules

for

x

TA

x

x

FA

x

Elimination

rules

for

x

TA

x

x

FA

x

The

rules

have

a

clear

intuitive

interpretation

if

is

true

at

a

point

x

then

it

is

true

at

all

points

accessible

from

it

and

therefore

it

is

true

at

the

special

point

x

which

verifies

all

and

only

the

formulae

which

are

true

at

all

points

in

S

-LRB-

x

-RRB-

Conversely

if

is

false

at

x

then

it

must

be

false

at

x.

Similarly

if

is

true

at

a

point

x

then

it

is

true

at

some

point

accessible

from

it

and

therefore

at

the

special

point

x

which

verifies

all

and

only

the

formulae

which

are

true

at

at

least

one

point

in

S

-LRB-

x

-RRB-

Conversely

if

is

false

at

x

then

it

must

be

false

at

x.

By

adding

these

rules

to

the

implication

rules

we

obtain

an

LKE

system

for

modal

substructural

implication

Different

systems

will

be

characterized

by

adding

to

the

labelling

algebra

different

sets

of

structural

and

modal

constraints

to

be

used

in

checking

for

branch-closure

So

this

extended

method

preserves

the

separation-by-closure

property

of

the

original

method

Moreover

it

also

preserves

the

atomic

closure

property

-LRB-

see

Section

above

-RRB-

Notice

that

by

virtue

of

the

duality

principle

whenever

the

constraint

v

is

derivable

in

the

algebra

of

the

labels

so

is

its

dual

v

In

the

next

section

we

shall

see

some

examples

of

proofs

in

this

modal

extension

of

the

LKE

system

Examples

Example

The

axiom

T

A

is

valid

in

all

reflexive

frames

A

a

FA

ffi

a

-LRB-

a

-RRB-

TA

a

This

one-branch

tree

is

closed

in

all

reflexive

frames

since

a

v

a.

Observe

that

by

the

duality

principle

the

constraint

x

v?x

also

holds

in

all

reflexive

frames

and

by

means

of

this

the

dual

of

T

A

can

be

proved

Example

The

axiom

D

is

valid

in

all

serial

frames

a

a

TA

a

FA

a

This

tree

is

closed

in

all

serial

frames

because

a

v?a

Example

The

axiom

is

valid

in

all

Euclidean

frames

a

ffi

a

-LRB-

a

-RRB-

TA

a

a

FA

a

A

similar

tree

making

use

of

the

dual

constraint

x

v!x

shows

the

validity

of

the

dual

axiom

Example

The

formula

-LRB-

A

-LRB-

A

B

-RRB-

-RRB-

-LRB-

B

-RRB-

is

valid

in

all

contractive

and

reflexive

frames

F

-LRB-

A

-LRB-

A

B

-RRB-

-RRB-

-LRB-

B

-RRB-

TA

-LRB-

A

B

-RRB-

a

B

a

b

FB

a

ffi

b

TA

b

TA

B

affi!b

TB

affi!bffi!b

In

all

contractive

frames

bffi!b

v!b

so

affi!bffi!b

v

affi!b

Moreover

in

all

reflexive

frames

b

v

b.

So

in

all

contractive

and

reflexive

frames

affi!bffi!b

v

a

ffi

b.

Example

The

formula

-LRB-

B

-RRB-

is

valid

in

all

frames

which

are

both

monotonic

and

transitive

-LRB-

B

-RRB-

a

FB

a

TB

b

a

ffi

b

TA

a

-LRB-

a

ffi

b

-RRB-

FA

-LRB-

a

ffi

b

-RRB-

This

one-branch

tree

is

closed

for

all

monotonic-transitive

frames

For

a

v

affib

by

monotonicity

therefore

a

v

-LRB-

affib

-RRB-

by

-LRB-

-RRB-

moreover

by

transitivity

-LRB-

a

ffi

b

-RRB-

v

-LRB-

a

ffi

b

-RRB-

so

a

v

-LRB-

a

ffi

b

-RRB-

and

the

branch

is

closed

Example

The

formula

A

-LRB-

A

-RRB-

is

valid

in

all

frames

which

are

expansive

and

symmetric

FA

-LRB-

A

-RRB-

TA

a

FA

a

a

ffi

a

-LRB-

a

ffi

a

-RRB-

FA

-LRB-

a

ffi

a

-RRB-

This

one-branch

tree

is

closed

in

all

expansive-symmetric

frames

For

a

v

a

ffi

a

by

expansion

-LRB-

a

ffi

a

-RRB-

v

-LRB-

a

ffi

a

-RRB-

by

symmetry

These

two

inequalities

imply

that

a

v

-LRB-

a

ffi

a

-RRB-

Example

The

formula

-LRB-

A

A

-RRB-

is

valid

in

all

monotonic

frames

-LRB-

A

A

-RRB-

FA

A

TA

a

FA

ffi

a

In

all

monotonic

frames

a

v

ffi

a

and

the

branch

is

closed

Example

The

proof

rule

A

B

is

valid

in

all

modal

quantale

frames

This

is

shown

by

the

following

closed

LKE-tree

a

a

TA

a

FB

a

TA

B

TB

a

bb

FA

B

T

where

T

is

a

closed

LKE-tree

for

FA

B

which

exists

by

hypothesis

Example

The

proof

rule

A

is

valid

in

all

monotonic

frames

This

is

shown

by

the

following

closed

LKE-tree

FA

TA

FA

T

where

T

is

a

closed

LKE-tree

for

FA

which

exists

by

hypothesis

Here

the

lefthand

brach

is

closed

because

v

by

the

monotonicity

of

ffi

Example

-LSB-

Regular

modal

implication

systems

-RSB-

We

can

define

the

class

of

regular

MIL

's

as

the

class

of

all

MIL

's

closed

under

the

additional

axiom

-LRB-

A

B

-RRB-

-LRB-

-RRB-

-LRB-

Recall

that

all

MIL

's

considered

in

this

paper

satisfy

the

other

condition

that

in

the

context

of

classical

modal

logic

is

used

to

characterized

regular

modal

logics

namely

Condition

-RRB-

The

class

of

regular

frames

corresponding

to

regular

MIL

's

is

characterized

by

the

following

constraint

xffi!y

v

-LRB-

x

ffi

y

-RRB-

The

reader

can

easily

check

that

the

above

axiom

becomes

provable

if

this

constraint

is

added

to

the

algebra

of

the

labels

The

tree

in

Table

shows

that

the

following

formula

-LRB-

-RRB-

-LRB-

-RRB-

C

-RRB-

C

is

verified

in

all

frames

which

are

both

regular

and

direct

-LRB-

we

use

x

y

z

as

label-variables

and

skip

the

first

few

nodes

in

the

construction

of

the

tree

-RRB-

Since

this

example

involves

an

application

of

the

branching

rule

of

generalized

bivalence

it

can

serve

also

the

purpose

of

illustrating

the

use

of

variables

in

the

labels

hinted

at

the

end

of

Section

Since

the

lefthand

branch

of

this

tree

contains

the

pair

of

LS-formulae

TC

x

ffi

b

and

FC

a

ffi

b

it

is

closed

if

x

ffi

b

v

a

ffi

b.

On

the

other

hand

-LRB-

-RRB-

a

T

-LRB-

-RRB-

C

b

FC

a

ffi

b

a

x

TC

x

ffi

b

HHH

x

c

x

ffi

c

c

-LRB-

x

ffi

c

-RRB-

FB

-LRB-

x

ffi

c

-RRB-

affi!c

TB

-LRB-

affi!c

-RRB-

Table

Example

this

tree

is

closed

for

all

regular

and

directed

frames

under

the

substitution

x

a.

the

righthand

branch

is

closed

if

-LRB-

affi!c

-RRB-

v

-LRB-

x

ffi

c

-RRB-

because

of

the

pair

of

LS-formulae

TB

-LRB-

affi!c

-RRB-

and

FB

-LRB-

x

ffi

c

-RRB-

Hence

this

tree

is

closed

for

a

given

class

of

frames

if

the

system

consisting

of

the

two

inequations

x

ffi

b

v

a

ffi

b

-LRB-

affi!c

-RRB-

v

-LRB-

x

ffi

c

-RRB-

has

a

solution

in

the

corresponding

algebra

of

the

labels

In

this

simple

case

it

is

easy

to

work

out

the

solution

x

a

for

the

algebra

of

the

labels

corresponding

to

directed

regular

frames

To

verify

this

is

a

solution

of

the

inequation

generated

by

the

righthand

branch

it

is

sufficient

to

go

through

the

following

steps

affi!c

v

-LRB-

a

ffi

c

-RRB-

by

regularity

-LRB-

affi!c

-RRB-

v

-LRB-

a

ffi

c

-RRB-

by

the

monotonicity

of

-LRB-

a

ffi

c

-RRB-

v

-LRB-

a

ffi

c

-RRB-

by

directedness

and

duality

-LRB-

affi!c

-RRB-

v

-LRB-

a

ffi

c

-RRB-

by

transitivity

of

v

A

general

technique

for

solving

systems

of

inequations

generated

by

LKE

trees

-LRB-

with

variables

-RRB-

within

a

given

algebra

of

the

labels

is

crucial

for

developing

decision

algorithms

based

on

our

method

This

problem

will

be

discussed

in

a

subsequent

paper

Here

we

only

observe

that

our

approach

separates

each

proof

in

two

components

-LRB-

i

-RRB-

a

logical

component

which

depends

only

on

the

universal

meaning

of

the

logical

operators

as

defined

by

the

inference

rules

which

are

a

straightforward

generalization

of

the

classical

ones

and

-LRB-

ii

-RRB-

an

algebraic

component

which

consists

in

solving

a

system

of

inequations

within

a

given

labelling

algebra

that

mirrors

the

specific

properties

of

the

information

system

in

which

the

inferential

process

takes

place

While

algorithms

for

the

first

component

can

be

devised

as

straightforward

extensions

of

known

algorithms

for

classical

logic

-LRB-

simply

by

shifting

from

formulae

to

labelled

formulae

-RRB-

algorithms

for

the

second

component

can

be

devised

by

generalizing

existing

algebraic

methods

for

solving

similar

problems

in

related

areas

-LRB-

e.g.

unification

term-rewriting

constraint

programming

etc.

-RRB-

Completeness

of

LKE

The

following

table

summarizes

the

constraints

on

the

algebra

of

the

labels

we

have

been

considering

so

far

Structural

Constraints

Modal

Constraints

commutativity

x

ffi

y

v

y

ffi

x

seriality

x

v?x

contraction

x

ffi

x

v

x

reflexivity

x

v

x

expansion

x

v

x

ffi

x

transitivity

x

v

x

monotonicity

x

v

x

ffi

y

symmetry

x

v

x

Euclideanism

x

v

x

directedness

x

v

x

Let

C

be

as

above

a

set

of

structural

rules

and/or

modal

axioms

and

let

be

the

corresponding

constraints

-LRB-

according

to

the

correspondence

outilined

in

Table

and

Table

-RRB-

We

call

LKE

-LRB-

-RRB-

the

basic

LKE

system

obtained

by

augmenting

the

basic

labelling

algebra

-LRB-

see

page

-RRB-

with

the

additional

constraints

in

and

we

denote

by

LKE

-LRB-

-RRB-

the

deducibility

relation

associated

with

this

system

Then

we

can

prove

the

following

Proposition

If

C

A

then

LKE

-LRB-

-RRB-

A.

Proof

We

need

to

show

that

LKE

-LRB-

-RRB-

intended

as

a

-LRB-

finitary

-RRB-

consequence

relation

is

a

MIL

and

is

closed

under

all

the

structural

rules

and

modal

axioms

in

C.

First

of

all

we

have

to

show

that

LKE

-LRB-

-RRB-

is

a

MIL

namely

that

it

is

closed

under

the

basic

conditions

for

a

consequence

relation

-LRB-

identity

and

surgical

cut

-RRB-

under

conditions

C!i

and

under

conditions

-LRB-

-RRB-

and

-LRB-

-RRB-

We

shall

show

only

that

LKE

-LRB-

-RRB-

is

closed

under

surgical

cut

and

leave

it

to

the

reader

to

verify

that

the

other

conditions

are

also

satisfied

Suppose

LKE

A

and

A

LKE

B

We

wish

to

show

that

LKE

B.

Let

us

write

T

to

denote

a

sequence

of

LS-formulae

obtained

by

the

sequence

of

formulae

by

-LRB-

i

-RRB-

signing

them

with

T

and

-LRB-

ii

-RRB-

attaching

to

each

of

them

a

distinct

atomic

label

We

shall

also

use

ffi

to

denote

the

ffi-concatenation

of

these

atomic

labels

Similarly

we

shall

use

the

abbreviations

T

ffi

ffi

ffi

T

and

ffi

By

hypothesis

there

is

a

closed

LKE-tree

with

initial

LS-formulae

T

FA

ffi

and

there

is

also

a

closed

LKE-tree

with

initial

LS-formulae

T

ffi

TA

a

T

FB

ffi

ffi

ffi

a

ffi

ffi

We

can

assume

without

loss

of

generality

that

the

sets

of

the

atomic

labels

occuring

in

and

are

disjoint

Then

the

following

tree

is

closed

T

ffi

T

T

FB

ffi

ffi

ffi

ffi

ffi

ffi

TA

ffi

-LSB-

a

fl

ffi

-RSB-

FA

ffi

where

-LSB-

a

fl

ffi

-RSB-

denotes

the

result

of

substituting

every

occurrence

of

the

atomic

label

a

in

with

the

label

ffi

-LRB-

The

reader

can

check

that

such

a

substitution

does

not

affect

correcteness

-RRB-

The

above

closed

tree

shows

that

LKE

B.

To

complete

the

proof

we

need

to

show

that

for

any

given

set

C

of

structural

rules

and

modal

axioms

LKE

-LRB-

-RRB-

where

is

the

set

of

constraints

corresponding

to

the

elements

of

C

is

closed

under

C

As

for

the

modal

axioms

some

of

them

are

dealt

with

in

the

examples

of

Section

above

and

the

others

can

be

proved

in

a

similar

way

For

the

structural

rules

we

show

only

closure

under

contraction

and

leave

the

rest

to

the

reader

To

show

closure

under

contraction

assume

A

A

LKE

-LRB-

C

-RRB-

B

We

wish

to

show

A

LKE

-LRB-

C

-RRB-

B

whenever

the

constraint

corresponding

to

contraction

i.e.

a

ffi

a

v

a

belongs

to

C.

By

hypothesis

there

is

a

closed

LKE

-LRB-

C

-RRB-

tree

with

initial

LS-formulae

T

TA

a

TA

b

T

ffi

FB

ffi

ffi

a

ffi

b

ffi

ffi

ffi

We

can

replace

without

affecting

correctness

every

occurrence

of

b

in

this

tree

with

an

occurrence

of

a

and

the

result

will

still

be

a

closed

LKE

-LRB-

C

-RRB-

tree

with

initial

LS-formulae

T

TA

a

T

ffi

FB

ffi

ffi

a

ffi

a

ffi

ffi

ffi

So

if

a

ffi

a

v

a

belongs

to

C

the

following

tree

will

also

be

closed

T

TA

a

T

ffi

FB

ffi

ffi

a

ffi

ffi

ffi

TB

ffi

ffi

a

ffi

a

ffi

ffi

ffi

HHH

FB

ffi

ffi

a

ffi

a

ffi

ffi

ffi

-LSB-

b

a

-RSB-

Propositions

and

imply

that

LKE

is

complete

with

respect

to

modal

implication

structures

To

close

the

circle

we

only

need

to

prove

that

LKE

is

sound

with

respect

to

this

semantics

Proposition

If

LKE

-LRB-

-RRB-

A

then

A

is

verified

in

all

The

proof

is

routine

and

is

omitted

It

immediately

follows

from

Propositions

and

that

Corollary

C

A

if

and

only

if

LKE

-LRB-

-RRB-

A

if

and

only

if

A

is

verified

in

all

The

subformula

property

In

the

classical

KE

system

the

application

of

the

bivalence

rule

PB

-LRB-

which

is

equivalent

to

classical

cut

-RRB-

can

be

restricted

without

loss

of

completeness

so

as

to

preserve

the

subformula

property

Is

a

similar

restriction

possible

also

in

the

LKE

systems

presented

here

As

remarked

in

Section

the

labelled

version

of

PB

introduces

a

new

type

of

non-determinism

in

the

system

since

this

rule

allows

for

the

introduction

not

only

of

arbitrary

formulae

but

also

of

arbitrary

labels

In

this

section

we

show

how

the

first

type

of

non-determinism

concerning

the

formulae

can

be

removed

without

side-effects

on

the

labels

This

is

a

crucial

step

towards

the

development

of

systematic

refutation

procedures

based

on

our

method

However

this

step

is

by

no

means

sufficient

In

order

to

obtain

a

fully

mechanical

refutation

procedure

we

need

also

to

tame

the

second

type

of

non-determinism

related

to

the

use

of

labels

in

the

generalized

PB

rule

i.e.

we

have

to

develop

a

procedure

which

terminates

either

with

a

closed

tree

or

with

a

tree

containgin

enough

information

to

construct

a

countermodel

The

solution

of

this

problem

crucially

depends

on

what

structural

constraints

are

allowed

in

the

algebra

of

the

labels

and

on

the

development

of

suitable

algorithms

for

solving

the

systems

of

inequations

associated

with

LKE

-LRB-

C

-RRB-

trees

-LRB-

as

illustrated

in

Example

above

-RRB-

It

seems

plausible

that

efficient

decision

algorithms

based

on

our

method

will

have

to

be

logic-specific

exploiting

the

computational

properties

of

each

given

labelling

algebra

Let

us

say

that

an

LKE

-LRB-

C

-RRB-

tree

enjoys

the

subformula

property

if

for

every

LS-formula

sA

x

occuring

in

it

where

s

is

one

of

the

two

signs

T

or

F

the

formula

A

is

a

subformula

of

some

formula

B

such

that

sB

y

for

some

s

and

some

y

occurs

above

in

the

same

branch

In

the

context

of

the

next

proposition

by

saying

that

two

closed

LKE

-LRB-

C

-RRB-

trees

are

equivalent

we

mean

that

they

are

constructed

starting

from

the

same

set

of

initial

LS-formulae

Proposition

There

is

a

procedure

to

turn

every

closed

LKE

-LRB-

C

-RRB-

tree

into

an

equivalent

one

which

enjoys

the

subformula

property

Proof

First

notice

that

all

the

elimination

rules

of

our

systems

preserve

the

subformula

property

So

to

prove

the

above

proposition

we

only

need

to

prove

that

all

the

applications

of

the

labelled

PB

rule

which

violate

the

subformula

property

can

be

eliminated

without

loss

of

deductive

power

If

an

LKE

-LRB-

C

-RRB-

tree

violates

the

subformula

property

some

of

the

formulae

introduced

by

means

of

the

labelled

PB

rule

-LRB-

let

us

call

them

PB-formulae

-RRB-

are

not

subformulae

of

any

formula

occurring

above

in

the

same

branch

We

shall

call

such

PB-formulae

non-analytic

Now

we

associate

with

every

LKE

-LRB-

C

-RRB-

tree

T

the

pair

-LRB-

c

n

-RRB-

where

c

is

the

maximal

of

a

non-analytic

PB-formula

in

T

while

n

is

the

number

of

non-analytic

PB

formulae

of

maximal

complexity

We

call

this

pair

degree

of

non-analyticity

of

T

Obviously

for

every

tree

enjoying

the

subformula

property

both

c

and

n

are

equal

to

zero

We

can

define

an

ordering

on

degrees

of

non-analyticity

as

follows

-LRB-

c

n

-RRB-

-LRB-

-RRB-

if

either

c

or

c

and

n

Now

suppose

there

is

a

procedure

to

transform

every

closed

tree

with

degree

of

non-analyticity

d

-LRB-

c

n

-RRB-

-LRB-

-RRB-

into

an

equivalent

tree

with

degree

of

analyticity

such

that

d

Then

by

iterating

this

procedure

we

shall

eventually

obtain

a

closed

tree

equivalent

to

the

initial

one

but

with

the

subformula

property

It

is

not

difficult

to

define

such

a

procedure

Let

T

with

degree

of

non-analyticity

d

and

let

C

be

a

non-analytic

PB-formula

of

maximal

complexity

We

distinguish

three

cases

-LRB-

-RRB-

C

A

B

-LRB-

-RRB-

C

and

-LRB-

-RRB-

C

Let

us

consider

case

-LRB-

-RRB-

Then

T

has

the

following

form

X

TA

B

x

aeae

ZZ

FA

B

x

Where

X

is

the

parent

node

of

the

two

sibling

nodes

containing

the

PB

formulae

Now

notice

that

in

the

LS-formula

TA

B

x

can

be

used

complexity

of

a

formula

is

measured

in

one

of

the

usual

ways

e.g.

by

the

total

number

of

occurrences

of

propositional

letters

and

logical

operators

only

as

major

premiss

of

an

application

of

the

ET

rule

-LRB-

since

there

is

no

formula

in

the

tree

with

complexity

greater

than

A

B

-RRB-

with

some

-LRB-

possibly

several

-RRB-

LS-formulae

of

the

form

TA

y

as

minor

premisses

to

obtain

LS-formulae

of

the

form

TB

x

ffi

y.

Notice

also

that

in

the

LS-formula

FA

B

x

can

be

used

only

to

generate

the

two

LS-formulae

TA

a

for

some

new

atomic

label

a

and

FB

x

ffi

a.

Let

OE

be

a

brach

containing

an

LS-formula

TB

x

ffi

y

obtained

by

using

the

PB-formula

TA

B

x

as

major

premiss

with

TA

y

as

minor

premiss

Then

OE

will

have

the

following

form

TA

y

TB

x

ffi

y

Let

T

be

the

result

of

removing

every

occurrence

of

TA

a

from

Now

replace

OE

with

the

following

tree

TA

y

TB

x

ffi

y

ll

FB

x

ffi

y

T

-LSB-

a

y

-RSB-

where

T

-LSB-

a

y

-RSB-

is

the

result

of

replacing

every

occurrence

of

the

atomic

label

a

with

y

in

T

It

is

not

difficult

to

see

that

the

resulting

tree

is

a

closed

tree

equivalent

to

T

Moreover

the

LS-formula

TB

x

ffi

y

no

longer

depends

on

the

PB

formula

TA

B

x.

By

applying

the

same

procedure

for

every

LS-formula

resulting

from

an

application

of

ET

with

the

PB-formula

TA

B

x

as

major

premiss

we

obtain

a

tree

T

of

the

following

form

X

TA

B

x

T

aeae

ZZ

FA

B

x

where

the

PB-formula

TA

B

x

is

never

used

in

T

It

follows

that

T

is

equivalent

to

the

following

tree

T

X

T

Finally

notice

that

the

degree

of

non-analyticity

of

the

new

tree

T

is

strictly

less

than

that

of

T

This

completes

the

proof

of

case

Let

us

now

consider

case

Then

T

has

the

following

form

X

x

ll

x

Observe

that

being

a

formula

of

maximal

complexity

x

can

be

used

in

only

as

premiss

of

the

rule

with

TA

x

as

conclusion

Moreover

x

can

be

used

in

only

as

premiss

of

the

rule

with

FA

x

as

conclusion

Let

T

and

T

be

the

results

of

removing

respectively

all

the

occurrences

of

TA

x

in

and

all

the

occurrences

of

FA

x

in

Then

it

is

easy

to

see

that

the

following

tree

T

is

a

closed

tree

equivalent

to

T

X

TA

x

T

FA

x

T

Notice

again

that

the

degree

of

non-analyticity

of

the

new

tree

T

is

strictly

less

than

that

of

T

This

completes

the

proof

of

case

The

proof

of

case

is

similar

to

that

of

case

and

is

omitted

Discussion

and

related

work

Related

work

on

the

semantics

of

substructural

logics

sketched

in

Section

-LRB-

following

-LSB-

-RSB-

-RRB-

can

be

found

in

a

wide

number

of

papers

We

mention

in

particular

-LSB-

D

D

-RSB-

but

also

-LSB-

-RSB-

The

Kripke-style

semantics

of

implication

described

below

and

which

can

be

found

in

a

number

of

papers

is

nothing

but

a

generalization

of

Urquhart

's

semantics

for

relevant

implication

described

in

-LSB-

-RSB-

For

more

semantic

investigations

into

substructural

logics

which

can

be

related

to

the

contents

of

Section

see

also

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

Previous

work

on

the

specific

topic

of

this

paper

-LRB-

combining

modal

and

substructural

logics

-RRB-

has

been

concentrating

on

the

intuitionistic

modal

logics

The

implication

fragments

of

these

logics

belong

to

the

wide

family

of

substructural

modal

systems

presented

in

this

paper

-LRB-

i.e.

those

satisfying

all

the

structural

rules

of

Table

and

are

captured

by

the

corresponding

LKE

systems

i.e.

the

LKEC

systems

obtained

by

including

in

the

set

C

all

the

structural

constraints

We

consider

in

particular

the

works

of

Bo

zi?c

and

Do

sen

-LSB-

-RSB-

and

Plotkin

and

Stirling

-LSB-

-RSB-

-LRB-

see

also

-LSB-

-RSB-

for

an

overview

-RRB-

The

main

problem

in

defining

modal

connectives

on

Kripke-style

intuitionistic

structures

is

to

guarantee

the

intuitionistic

hereditary

property

for

modal

formulae

This

can

be

achieved

either

by

defining

approriate

satisfiaction

clauses

for

the

modal

operators

which

embed

some

sort

of

monotonicity

-LRB-

or

hereditary

property

-RRB-

feature

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

or

by

imposing

conditions

between

the

accessibility

intuitionistic

relations

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

Bo

zi?c

and

Do

sen

propose

-LSB-

-RSB-

three

different

intuitionistic

modal

logics

one

with

only

the

necessity

operators

-LRB-

i.e

-RRB-

one

with

only

the

possibility

operator

-LRB-

i.e.

-RRB-

and

the

third

with

both

modalities

-LRB-

i.e.

-RRB-

In

the

system

a

frame

is

a

possible

world

structure

-LRB-

W

R

-RRB-

where

the

partial

ordering

and

the

binary

relation

R

satisfy

the

following

condition

If

x

y

and

yRz

then

such

that

and

z

-LRB-

-RRB-

A

class

of

models

is

obtained

by

adding

on

a

frame

a

valuation

functions

V

which

satisfies

the

hereditary

property

for

the

propositional

variables

Standard

intuitionistic

and

modal

satisfaction

clauses

are

then

considered

In

the

the

following

condition

is

added

to

the

same

type

of

frame

If

x

y

and

xRz

then

such

that

and

z

-LRB-

-RRB-

These

are

Conditions

-LRB-

-RRB-

and

-LRB-

-RRB-

in

this

paper

Plotkin

and

Stirling

-LRB-

-LSB-

-RSB-

-RRB-

combine

the

two

approaches

described

above

in

that

they

consider

the

condition

of

Bo

zi?c

and

Do

sen

which

allows

to

preserve

the

traditional

-LRB-

classical

-RRB-

satisfaction

clause

for

but

they

define

an

alternative

satisfaction

clause

for

the

x

j

def

x

y

if

yRz

then

z

j

A

-LRB-

-RRB-

In

this

way

Plotkin

and

Stirling

generate

intuitionistic

modal

logics

in

which

modal

operators

are

interpreted

similarly

to

the

intuitionistic

quantifiers

and

and

the

two

modal

operators

are

not

interdefinable

An

extension

of

the

work

presented

in

this

paper

to

full

substructural

modal

logics

-LRB-

including

negation

-RRB-

will

allow

better

comparisons

with

-LSB-

-RSB-

and

others

In

future

papers

we

also

plan

to

extend

our

research

in

the

following

directions

ffl

accommodating

Girard

's

modalities

-LRB-

the

exponentials

of

Linear

Logic

-RRB-

into

the

framework

outlined

in

the

present

paper

ffl

investigating

the

decision

problem

for

modal

substructural

logics

on

the

basis

of

the

LKE

approach

this

involves

introducing

a

suitable

notion

of

completed

or

saturated

LKE

tree

and

adapting

existing

constraint

satisfaction

methods

for

solving

the

systems

of

inequalities

generated

when

the

applications

of

the

labelled

PB

rule

are

carried

out

with

variable

labels

-LRB-

see

Example

in

Section

-RRB-

ffl

extending

the

present

approach

to

cover

modal

many-valued

logics

ffl

investigating

the

relation

between

our

methods

and

the

methods

based

on

Belnap

's

Display

Logic

-LRB-

-LSB-

-RSB-

-RRB-

References

-LSB-

-RSB-

A.

R.

Anderson

and

N.D.

Belnap

Jr.

Entailment

the

Logic

of

Relevance

and

Necessity

volume

Princeton

University

Press

Princeton

-LSB-

-RSB-

V.M.

Abrusci

Phase

semantics

and

sequent

calculus

for

pure

noncommutative

classical

linear

propositional

logic

Journal

of

Symbolic

Logic

-LCB-

-LSB-

-RSB-

G.

Allwein

and

J.M.

Dunn

Kripke

models

for

linear

logic

the

Journal

of

Symbolic

Logic

-LCB-

-LSB-

-RSB-

V.

Alexiev

Applications

of

linear

logic

to

computation

An

overview

Bulletin

of

the

IGPL

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

Giambattista

Amati

and

Fiora

Pirri

A

uniform

tableau

method

for

intuitionistic

modal

logics

I.

Manuscript

-LSB-

-RSB-

A.

Avron

The

semantics

and

proof

theory

of

linear

logic

Theoretical

Computer

Science

-LCB-

-LSB-

-RSB-

M.

Bo

zi?c

and

K.

Do

sen.

Models

for

normal

intuitionistic

modal

logics

Studia

Logica

-LCB-

-LSB-

-RSB-

Krysia

Broda

Marcello

D'Agostino

and

Alessandra

Russo

Transformation

methods

in

LDS

In

Logic

Language

and

Reasoning

An

Essay

in

Honor

of

Dov

Gabbay

Kluwer

Academic

Publishers

To

appear

-LSB-

-RSB-

Nuel

Belnap

Display

logic

Journal

of

Philosophical

Logic

-LCB-

-LSB-

-RSB-

Nuel

Belnap

Linear

logic

displayed

Notre

Dame

Journal

of

Formal

Logic

-LCB-

-LSB-

-RSB-

Robert

Bull

and

Krister

Segerberg

Basic

modal

logic

In

D.

Gabbay

and

F.

Guenthner

editors

Handbook

of

Philosophical

Logic

volume

II

chapter

II.I

pages

-LCB-

Reidel

Dordrecht

-LSB-

-RSB-

Marcello

D'Agostino

and

Dov

M.

Gabbay

A

generalization

of

analytic

deduction

via

labelled

deductive

systems.Part

I

Basic

substructural

logics

Journal

of

Automated

Reasoning

-LCB-

-LSB-

-RSB-

Marcello

D'Agostino

and

Marco

Mondadori

The

taming

of

the

cut

Journal

of

Logic

and

Computation

-LCB-

-LSB-

D

-RSB-

Kosta

D

osen

Sequent

systems

and

groupoid

models

I.

Studia

Logica

-LCB-

-LSB-

D

-RSB-

Kosta

D

osen

Sequent

systems

and

groupoid

models

II

Studia

Logica

-LCB-

-LSB-

D

-RSB-

Kosta

D

osen

A

historical

introduction

to

substructural

logics

In

P.

Schroeder

Heister

and

Kosta

D

osen

editors

Substructural

Logics

pages

-LCB-

Oxford

University

Press

-LSB-

-RSB-

J.

M.

Dunn

Relevant

logic

and

entailment

In

D.

Gabbay

and

F.

Guenthner

editors

Handbook

of

Philosophical

Logic

volume

III

pages

-LCB-

Reidel

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-LSB-

-RSB-

M.

Fitting

Proof

Methods

for

Modal

and

Intuitionistic

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Reidel

Dordrecht

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D.M.

Gabbay

Labelled

deductive

systems

volume

foundations

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Report

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Preliminary

partial

draft

of

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intended

for

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University

Press

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Garson

Modularity

and

relevant

logic

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Dame

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of

Formal

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-LCB-

-LSB-

-RSB-

J.-Y

Girard

Linear

logic

Theoretical

Computer

Science

-LCB-

-LSB-

-RSB-

Wendy

MacCaull

Relational

semantics

and

a

relational

proof

system

for

full

Lambek

calculus

Technical

report

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Mathematics

and

Computing

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Xavier

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for

substructural

logics

In

Peter

SchroederHeister

editor

Substructural

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pages

-LCB-

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-RSB-

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C.P.

Stirling

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framework

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intuitionistic

modal

logic

In

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Y.

Halpern

editor

Theoretical

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pages

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-LSB-

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Sambin

The

semantics

of

pretopologies

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SchroederHeister

editor

Substructural

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pages

-LCB-

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University

Press

-LSB-

-RSB-

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Fischer

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modal

logic

with

an

intuitionistic

base

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-LCB-

-LSB-

-RSB-

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K.

Sympson

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Proof

Theory

and

Semantics

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Intuitionistic

Modal

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thesis

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of

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-LSB-

-RSB-

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Urquhart

Semantics

for

relevant

logic

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Journal

of

Symbolic

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-LCB-

-LSB-

-RSB-

Heinrich

Wansing

The

logic

of

information

structures

In

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editor

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Notes

in

Artificial

Intelligence

volume

-LSB-

-RSB-

Heinrich

Wansing

Sequent

calculi

for

normal

modal

propositional

logi

Journal

of

Logic

and

Computation

-LRB-

-RRB-

-LCB-