first order definable modal systems

7/23/2019 First Order Definable Modal Systems

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Fi st-Order-DefiableModal

Systems

I

Motivation

Our dcvclopmcnt

of modaLlogic

has bccn ahistorical

or

cvcn anti-his-

torical:

modal deduction

systems,eachcharacterizinga

deductive

con-

scqucncc elation, wcrc introduccd and studied long bclbrc

possiblc

worlds semanticswas discovered.But the semanticalviewpoint

pro-

vidcs a supcdor

pcrspcctivc

rom

vr'hich

o cxplain

why

thcrc arc many

modal systcmsof intcrcst bcsidcs5M

(S5).

n thjs scction,wc introducc

an cxtra componcnt nto intcrprctations, whosc

paramctcrs

can bc sct

in

dilltrent

ways

o

generate

dillerent

(scmantically

delined)

systems.

In the SM semantics, f sometring is possibleat one wor1d, t is so al

ovcryworld. But this excludesan dea about

possibility

which has somc

intuitive force, he idea

hat

what is

possible at

a

world) is detcrmined

in

part

by

how

things in

fact

arc

(at

that

world).

Pcrhapsccrtain statcs

of al'lairsarc

mpossible,

given

he

way

things actually are,

but if

things

had been ditTerent, hose statesof aflairs would

have

been

possible.

n

othcr words, what is

possiblc

may vary from world to world-

Thcrc arc no uncontrovcrsialcxamplcsof this

phcnomcnon,

but hcrc

is a conlrovcrsial onc. t is hcld by sornc

philosophcrs

hat if an organ-

ism

in lact dcvclops fiom

a cc ain cnljty or cntitics,

then tftat

organ

ism

could

not have

developed

iom dif'lerent entities- jor instance, f l{

is a

human

being who actuallyoriginates

iom a

sperm

.s,and an

egg

e,,

then according o this

view

H

could not have

originated

rom different

entities. Thc idea is that anlthing devcloping

from diffcrcnt cntitics

would notbc H, but someoneelse, ven

f very

similar to

H. But thc vicw

as statcd

s imprecise:

he claim

may

be

that H cor.rld ot havc odginat-

ed

from

a difl'erent egg

dnd a

difi'erent

sperm,or tbat H could not havc

originatcd from a different

egg or

a

different sperm. Supposcwc takc



Chapter First-Order-DefinablerodalSystems

thc first rcading. Thcn it is allowcd that H could havc originatcd liom

a dillerent egg er, so long as it is sr

which

fertilizes er, and that H could

have originated tiom a ditTerent sperrn s2, so long as

it

is e1

which

s,

f?rtjlizes.

So

given

that H originates fiom s, and er, that

is,

that

in v/o

these irre I{'s

'propagules',

a world u in which I{ originates from.sr and

€, is

possiblc,

and a world u' in which H originatcs from.st and e, is

pos-

siblc,

but

a

world v in which H

originates

from

.ri and

s, is impossible.

But

the

impossibility is from

the

perspectivc

of the

world u.,* wherc H

originates from s, and €r. Assuming that the doctrine about origin

is

not itsclf sensitive to how things in fact arc, thcn if wc considcr mattcrs

from the

point

of vicw of ll, whcrc H originatcs from s, and u,, we find

that thc world vis

possiblc,

sincc onc of thc entitics from which H orig-

inates

(in

a) is retained in

v.

Thus

yis

possiblc

rclativc to u, u is

possiblc

relative

to

)a,",

but

v

is impossible relative to ]r":

relative

possibility

is

not transitivc.

2

Systems

'[o

captu-rehese deas,we introduce the relation of relative

possibiJity

inlo th€ notion

of

interpretation,

and

we w te Rav

to

mean

that

v is

possible elative o u, or as t is often read, v is acces'siblerom ui tlf,for-

mally, modal logicians oftcn rcad as Rav as u can see v. As cxplaincd

in Chapter 1.3, a binary rclation on a domain of djscourscD is a sct of

ordcrcd

pairs

of clementsof D. fhus our ncw accountof intcrprctation

is

as

bllows:

A,rr nterpretation

I

with relative

possibility

of an LSML

sequent,

a

general

interp retation, is a set t4lof

worlds

and a sct

li

of

ordered

pairs

of elements

of

I44each v € I4l s

associated

with

an assifJnmentof truth-values to the sentence-letters

which

occur n the sequent,every

4/

assitining

1

to

',\';

and one )v

€

I4l

is

designated h€ actual

world,

conventionally

written

'lv"'"

Wc usc

'stzmdard

inte.rpretation' or the kind of interpretation dis-

cusscd n Chaptcr 2.2, n which thcrc is no il" As wc leamed n Chaptcr

1"3,

a binary relation can

have

a

variety of

stluct]ral

properties,

such

as reflexivity, slrnnetry or transitivity. We can

generate

different

modal systcms scmantically by dcf.ining diffcrcnt

semantic conse-



52:

Systems 9 l

qucncc

clations tss, s,,, tc.,whcrc thc diffcrcncc lics in

thc structu-ral

requircmcnts wc imposc on R. For nstance, hcrc will bc a systcm fbr

thc cascwhcrc.l{ s rcflcxivc, a systcm or thc cascwhcrc it is tTansitivc,

one

for

the casewhere t is both transitive and reUexive,and so on, as

wcll

as a

basic

system

where

thcrc arc

no

constraints on

R

(howcvcr,

thc

fact that

onc collcction of constTaints

on R diffcrs from

anothcr

docs not

guarantcc

hat thc systcms hcy dctcrminc

will

diffcr). Wc will

use appropriate subscriptson the semanticconsequence lmbol io dis-

tinguish the various systems iom eachother^A systemwhose seman-

tic consequenceelation is defined by stipulations on R expressible n

first-order language

LFOL)

s said to be

a

first-order

definablem<>dal

systcm. fl the rest of tllis sectionwe will introducc a few of

the

best-

known first-ordcr definable systcms

(wcll

ovcr a hundrcd havc bccn

studied).

But before

going

any further,

we

have o tie in the relative

possibjlity

rclation to thc cvaluation clauscs or the modal opcrators. ntuitively,

a statcmcnt

'op'

should hold at a world w iff therc is some world u

wlic}Jis

possible

relat:ive o w w}J1crf holds. Notice that we

spcak

now

of worlds, not

possible

worlds; the

'possible'

has become a character-

ization of a relationship betlveen worlds. Irr

place

of the evaluation

clauses or 5M, we usc the following evaluationclauscs, 'ith thc oncs

for the modal operatorsbeing changcd:

(EA):

Vu' € W, wlrrl

=

r ilf w.'rr

+

r;

(Eno):

Vwe W,wlnpl

=

T iff

(Va

e W)(Rwu ulpl: a);

(E.J:

Vra/ W,wl.pl: T itf

(la

e W)(Rwu ulpl

=

r);

(tr):

W(p)

=

T iff )a/tlpl

=

r

For the rest of this section,we use

(E *)

and

(E.,.)

n

place

of

(Er)

and

(Ea). ED,r)

ays hat

p

is necessary

t

a world }v ff

p

is

true

at all worlds

lry an see,and

(EaR)

ays hat

p

is

possible

at a world lry ff

p

is tTuc at

somc

world ]a,

an see-

Our

initial

first-order definable system

s

called

Gen'

since

t

ariscs

by

placing

no

resffictions

on

R.

We

will

symbolize

he

semantic

conse

quence

elauon of

Gen by'er' to

indicate the

absence

of .restrictions.

This relation is deiined bv

p

t,...,p,

F

a 4

iff

there

is no

general

nterprctation on which all

of

p1,...,p,

re rue while

qisfalse.

Gen

s

the collection of semantically

orrect sequ€nts

p,,...,p,

e

"

q,

and.



92 Chapter r First-Order-Defina bleodalSystems

wc

will immediatcly scc that

*a

nA

-

A. Conscqucntly,Cen

and SM

(S5)

are ditlerent systems,sinceof course Fe\arA

-

A-

Example

.2.1:Sbow

s6

A

*

A-

Interpretation:

W

=

{y*},

R

:

@,w": A

-

L. Fxplanation:

The condition

'(Vu

e W\(Rw"'u ulAl

=

T)'holds since the

or

y

instance s'(Ru/*w*

-

)v{Al

-

T)', which is

true

because

R)a.,*yr'"'

is false. -Iencew.[ Al

-

T. In words, sincc )ry* an

sce

no worlds,

A'

holds at cvcry world which

\y* can sce.

It is

crucial to the counterexample

o

'uA

-

A'

that the world )ry*not

b€ able

to

see tself- If

instead of R

-

O we had set fi

:

(w*,yi

*),

tlen

rcgardlessof what assignment y* makes

o

'A'

wc

would havc

trA

-

A'

truc at )v*; n

particular,

f r4,*;A r,

thcn by

(Elr),

w.[nAl

=

-Las wcll,

since there is a world lry* can see, tself, where

'A'

is

not

truc; and i{

w"llAl

=

r, then rry*ltrA A]

:

T. But it may

seem unreasonablc o

allow

interpretations

in which there are worlds which

cannot see hem-

selves.How

can

what

happensat a world fail to be

pos$ible

relative

to

that very world? n

other words, it sccms that thc relation of relativc

possibility

should bc ref'lexive:

Vw)Rww.r

So our second first-o.rdcr

del'inablesystem s the systcm Rf, whose

semantic consequence cla-

tion is delined by:

pt,...,p,

ts*

4

iff there s no

general

nterprctation ? in which .ll

is rcflcxivc and

on

which

all of

p,,...,p,,

ate true whilc

4

is falsc.

'l'hc

systcm Rf s

the collectionof scmanticallycorrcct scquents

pr,.,.,p,

FRr

,

a collection which

contains he collection of semanticallycorrect

sequents

pv..,p,

ea

q-

For f

pb...,p"

Fi,

q,

then no

general

nterpreta-

tion makes

pr,...,p,

all truc and

4

false,so n

particula.r

no

general

nter-

prctation

with

rcflcxivc R makcs

p.,...,p,

aJ\

ruc and

4

falsc; hcnce

pr...,pn

t=nr

.

Thus any sequentwhjch belongs o Gen, belongs to Rf.

On the other hand, there are sequents which belong

to

Rf but not

to

Gen:

tso

DA

-

A but FRr A

-

A.

On

tlfs

accountwe say hat Gen

s

a

proper

subsystem f Rf and that Rf

properly

containsGen. More

gener-

ally, S is a

propcr

subsystcmof S', and S'

properly

contains S, f cvc.ry

S-sequent

s

an s'-sequcntwhilc somc Slscqucnt is not

an

S-scqucnt.

r

'lbis

does not mean that there is no intqest in systcms

of

moda.l ogic in which ti

is

not required to be reflexive. It merely means that it would be implausible

to

intcrpret

the

'n'

of such systcmsas exprcssing rcadly logicalneccssity-

Systems

withou rcncxivc R

arc

usualJy

motilated by very differcnt readings of

'D'.



52:

Systems 93

Thc

percepuve

eadermay havc noted that it sufficcs to validatc

DA

-

A'

that

we

restrict ourselves o

just

those

general

nterprctations in

which R is reflexiveat w",that is,

n which Rw"w*,

even

f R is not

'glo-

bally' reflexive, hat is, even f there is some other

world n

such tJrat

'Rau"

We could

introduce

a

semantic consequence elation for this

wider

class of

general

nterpretations, and wc would obtain a somcwhat

dift'ercnt system

from Rf. However,

global

conditions

like

'(Vw)Rww'

yield

systems

with certain convenicnt formal featurcs-if

p

is a valid

formula in

such

a system,so s ap so n this chapterwe consideronJy

conscqucncc elations defincd globally:wc say hat a conditjon is glo-

bal iff aI its terms are bound variablcs.

An altcrnativc

global

constraint on rclativc

possibility

which wc

might impose is that of slmmctry:

(Vw)(VuXRwu

Ruw).We have

already

proved

otrA FssA, and henceby

the soundness

of F* for Frr,

otrA trsM , which

is also easy o see n

its

own

right: if

'otrA'

holds

at

].r.,"n

a sta.ndard

nterpretation

I'l then

for

some

world

u

in ty, u[trA]

:

r,

and so'A'holds at every

worldin

W,

ncluding ]r". However, f n is

not

rcquired

to be s)mmeuic,

we can

easily

give

a

gcneral

ntcrprcta-

tion which relutcs this sequent-Whetheror not n is refletve

makes no

difference,but we n'ill

give

an examplewhere

t is.

Example.:].2.2: how oaA

*RrA.

LctW=

{y/",u},

R:

{(}i/*,a),

y/",}y*),

u,u)},

a/*;A r, u.'A

*

T. Wcca n

give

a dia$am of this

interpretation in which the directed arrows indi-

cate

lines of

sight,

and absenceof an arrow rurrring from onc world to

anothcr ndicatcs that the fi-rstcannot see he second-So his interprc-

tation

is

as Dicturedbelow:

Wc havc w*[A]

=

r. To see hat

]v*[o nA]

=

T, note

that by

(En,,),

we have

[ Al = T, since'A'is true at everyworld u can see Ir cannot se€ w*)^

Then since w* can see a,

u,"[onAl

=

T

by

(EoJ.

Thus .trA

FRr

A, and

ipso

acto

otrA

Fo

A.

In

Example 1.2.2he failurc of

R tobc symmc ic is crucial.Our third

tirst-order definable sysrem

s

the system

Sym,whose semanticconse-

quence

elation

tss,. s

defined

by:

u

\ ,



chapter3: First-Order-Defina bleodalSystems

p

t,...,p,

ssv.

iff there

s no

gcncral

ntcrprctation in which R is

synmetric and on

which

a]l of

pr,...,p,

arc true whilc

4

is falsr.

The

system

Sym is the collection of semantically correct sequents

p

r.

..,pr

=rnn

,

a collection

which contains he collectionof scmantically

corfectsequents

b...,p"

sa

q.For

f

p1,...,p"

=@

,Ihenno general

nter-

prctation

makcs

pr,-..,p,

all truc and

4

falsc, so n

pafiicular

no

gcncral

interpretation with slmmetric R makes

p,,...,p,

?l'l true and

q

fatse;

hence

p,,...,p,

Fsy-

4"

Thus any sequcnt which belongs

o Cen,

belongs

to Sym- On the other hand, olA Fa A while ocA Fsy.A. So Gen is a

proper

subsystemof Sym. But cven though

otrA

trRr

, it does not fol-

low that Rf is a subsystemof Sym, since here may be sequents

which

bclong to Rf but not to Sym.

And

there arc;

for instance, FRrBA

-

A,

but it is easy o chcck that

*5y.

trA

*

A

(this

is an exercise).Soneither

Sym nor Rf is a

subsystem

of

the

other"

The remaining main structural

property

of binary

relations s

that

of

transitivity:

(VwXVUXVvX(Rwu

Ruv)

*

Rwv).The exampleabout bi}

bgical

origin we

gave

n

S

1 in €xplanationof thc notion of rclative

pos-

sibilily

motivates the vicw that rclativc

possibility

is not tTansitive.For

if )v is a world in which H odginates liom s, and e, and u a

world in

which

FI originates

rom

s, and er, hen

a is

possible

elative o lv

(Rwu);

and if v is a world in whjch H originates

rom s,

and e,,

then v is

possi-

blc rclative to a (Ruv).But v is not possible elativc to Ir (-Rwv),and so

Fansitivity of R fails in this

particular

example.

However, wc will ccr-

tainly want to investigate a system

with

transitive

R,

so our

fourth first-

order definablesystem s the system

Trn,

whosesemanticconsequence

rclation Fr," s dcfincd by:

p,,...,p^

r*

q

lff

therc s no

general

nterpretation in which R is

transitive and on which all of

p,,...,p,

arc

truc whilc

4

is falsc"

'lhc

system

Trn is

the collection of semantically

co.ffect sequents

pt,...,p,

p*,4,

a collection

which,

as can

be shown by a now familiar

argumcnt, contains the collection

of semantically coffcct sequents

pt,...,p,ca 4. In addition,wc haveooA F@ A, while ooA Frr" A. So

Gen is a

proper

subsystemofTrn. In i'act,

we

also

have aoA

tssy.

oA

and

ooA

tsRr

A.



52:

Systems

Example3.2.3:ShowoaA

trsvm

A

and

ooA

*Rr

oA-

Interpretation:

ln the following interpretation, R is rcflcxivc and sym-

metric.

w:

lw*,u,v,\,

R

:

{(w*,a),

a,v), u,w*),(v,u),(w",w"l,(u,u),

(v,v)),

y".'A

*

-1,a.'A

*

f

,

y:

A

*

T, or as

pictuled:

t ,

u

l )

Hcrc wc have w*[A]

=

r, and also

wo[on]

=

1, since

u

is the only other

world w" can see and u[A]

=

r- However,

a[aA]

:

T sincc u can scc v

and v[A]

-

T. But f uloAl

:

T, thcn ]t"[ooA]

=

T since

v/"

can scc u.

In

te.rmsof oul example,suppose hat

'A'

means

H

originates

rom s, and

er'; hen this interpretation

llustrates

that

if H acflrallyor.iginatesrom

s, and e],

t

is not

possible

hat

H odginate from .ri and e2,but it could

havc bccn

possible;

t

is

possibly possible,

so

to speak.Sincenon'tran-

sitivity is crucial to the interprctation,

it

also shows

that Trn contains

a sequent hat

Rf

and Sym both

lack. However,each of thcse systems

also

contain

sequents

Trn lacks,so none of the three systemswith con-

straints on I{ is a subsystemof any other"

lhc sequentswe haveused to illustrate djffercnccs betwecn he sys-

tems havenot been chosenat random-Thcy are n a certain sensechar-

actenslrcof thcir associated cmanlic consequence

elations.

.

FRroA

-

A, but if C s any

global

constraint on

R which does

not entail that

R is reflexive, hen

tsc

DA

-

A.4

.

o trA Fsv.A, but if C s any

global

constraint on

R which does

not entail that R

is

slanrnetric,

hen atrA

*c

A.

.

oaA Fr- oA, but if C is any

gklbal

constraint on R which

does not cntail that n is ffansitivc, then

o oA

trc

oA.

Wc could continuc in thc sanc

vcin,

using

futher

structural

propcr-

ties to

define new semanticconsequenceelations.

But

the best-known

systems of

modal logic

are

obtained by combining some of tfle

proper'

ties of

R we have consideredso far. There are th,ree uch systems,

cor-

responding to the

global

constraints

that R be

(i)

reflexive

and

4

Thc

cstriction to

global

constnints

is essential, s already

ndjcatcd.

f Cis simply

thc

constraint

hat Rr4,*}v,,

he

sct ol valid formulae

s

not closed

undcr Necessiratron-

A * r A * r



Chapter

First-Order-DefinableodalSystems

sFnmetric,

(ii)

rcflcivc and

transitivc,

and

(iii)

rcflcxivc, symmctric

and transitive,

that is, an equivalencc clation

(secpage

59).The asso-

ciated scmantic

consequcncc clations arc labcled

ts*, tsR,

and

truo

respectively.They are defined by:

pv..,p,

t-as iff therc is no

gcneral

ntcrprctation in $'hich R is

refledve and symmctric and on which all of

pr,...,p,

are truc

while

a

is falsc.

pv..,p, t=w4 iff there s no general nterpretation in which R is

rcflexive and transitive and on

which all of

p,,.-.,p,

are true

while

q

is falsc.

p,...,p"

t=q

4

ifT there s no

general

nt€rpretation in which R is

an cquivalence rclation and on

which

all of

pr,..-,p.

arc truc

while

q

is

false.

The

systems

RS,RT and Eq arc thc conespondingcollectionsof seman-

tically

correct sequents.Cen s a

propcr

subsystemof all threc, neithcr

of

RS

nor RT is a subsystemof the other, and

every

sequcnt

of Rs or

RT s in Eq-But Eq s

distinct

from both RSand RT- t is an exercise o

show that oA

tsRs

oA and

oA

FRr

roA,

and

it is easy o see hat oA

tr& oA. For f 'oA' holds at wn n W hen )r* can seesome u in I/ such

that

'A'

holds at l,|.

Let v

bc

any world in I/ which ]v* can scc. Sinccwo

can see

v,

then by symmetry

v can sce rry*.Also, sincc )v" can see , by

lTansilivity

v

can see a.

But'A'holds at a; so

'oA'holds

at v. And sincc

v was

chosen

arbitraflly from the worlds w" can see, his means that

'oA'holds

at

everv

world wo can scc.Thus

'noA'holds

at

]1,*- lenceRs

and RT arc botr subsystcms [ Eq.

We have now introduccd sevcn irst-ordff

dcfinablc systcms of scn-

tcntial modal

logic,

Gen,

Rf, Sym, Trn, RS, RT, and

Eq,

and havc

explained

he

notion of one system'sbeing a subsystemof another.

To

keep

rack of

the relationshipsamong hesesystems, he

following

dia-

gram

is

useful,

in which the system at thc tail of each arrow is a sub-

system of the system

at its tip. Arows illc transitive, ]ikc thc

subsystem

elationship; that is, any two arrows

which

connect tip

to

tail

can

be

considereda single anow Thus one system

s a

proper

sub-

system of another

f

there

s

a

path

through the diagram 'rom the

first

to thc second"Where

neither of two systems s a subsystemof

the oth-



S2:Systems

er,

no

path

of a-rlows

onnects

hcm.

As the diagram suggests, hcrc is

plcnty

of'logical space' for the

inse.rtionof other

systems-

For example, herc is a

group

of frequently

studied systems which lie on the

path

from RT to Eq, each of

$'hich

contains RT and is contained

n Eq.

One of thesc

is

the

system RTc,

which has thc semanticconsequenceelation trRrc efined by:

pb...,p,

F=R.rc

itf rhere

s no

general

nterpretation n which n is

reflexive, ransitive and connectedand on

which

p,,...,p"

are all

tTucwhile

q

is false.

Thc rcquircmcnts of rcflcxivity and transitivity ensurc that every

sequcnt

of RT is a sequentof RTC,since a sequentnot

in RTC has

a

reflcxive transitive counterexample and so

is not in RT- For a character-

istic sequent or

RTC,

consider he

following-

Example 3.2.4: how

E(oA v oB)

FRr

roA v troB.

Interpretation:W

=

{w*,u,v},

R

=

1@",ul,lw",v),

w",w*),(

u,n),(v,v)},

w*. 'A r , B

*

T, u--A

T, B

*

I , v . 'A l , B

*

T.

A - T A * ' T

A - - L

B * T B - T

B * T

In this interpretation,

aA'holds

at u and

v/* and

oB'holds

at

%

so

oA

v oB'holds at everyworld which

),|,'"

an see,

and hence

(oA

v oB)'

holds at

)v*.But'troA'is falscal

r.v"

ince'oA'docs

not hold at cvcry

Eq

_t

Y.

. / \

RS RT

, / \ r / \

u/^

R"ft r)n

r----.-

t ,,-'-,

---t,/

Ge n

i

,

;



Chapler J: f irst-Order -Definable , loddl Systems

world which r4.," an scc

(considcr

v),

and

'trcB'is

falsc at w* sinc c

'oB'

docs not hold at cvcry

world which w"

can scc

(considcr

a). Howcvcr,

|hc

lack

ol' any conncction belween

u

and

v is

cruciitl

Lo I his countcrcx

ample to thc scqucnt; any

rcflexive

fansitive

counte.rexample to il will

havc

at

least

two

unconnected worlds visiblc from w".'ltrus u(oA v oB)

r=Rrc roA v troB. Sincc thc.rc is no countcrcxamplc in which li is an

cquivalcncc rclation, and

sincc oA

FRrc

r aA, wc also havc RTC a

propcr

subsystem

of Eq.

Wc end

this discussion of first-ordcr definablc systcms with thrcc

points of intercst. First, it is natu-ral to ask what relationship thcrc is, if

any, betlvcen these

systems iuld thc systcm SM

(S5)

of standard modal

logic which wc introduccd in

Chaplcr

2" Wc arlluc that SM is Eq. Rccall

th.rl an cquivalcncc relation on a domain

partitions

thc domain into

mutually

exclusive

and

jointly

exhaustive

equivalcnce classcs. So if 7 s

a

gcncral

intcrpretation in

which

R

is an equivalenre

relation, there is

a uniquc cquivalcncc class 4.I., o

which )v'-

bclongs

(in

this

spccial casc,

'Ilrv'can

bc rcad as'a and

v

can scc cxacllv thc samc

worlds'). Morc-

ovcr, thc truth-valuc of any scntencc

p

on'1, that

is,

lhe

lruth-valuc

()1'

p

at )v", dcpcnds only on thc lruth-valucs of scntcnccs at lhc othcr

wo.rlds in a-I,,.,,or if wc bcgin tbc cvaluation of

p

at w*, thcn

no mattcr

how many modal opcrators

p

contains,

we

arc

ncvcr lcd outside L/-.,

sincc no world in {/., can scc any

world

outsidc a/.,"

constqucntly, thc

truth-valucs of all scntcnccs a[

w* rcmain

thc samc

if wc

'discard'

al l

worlds

in

W

whjch

arc not

in

[.r., and all

pairs

ol

worlds in ii in which

some

member is not in L/... lhis leaves us with a new interprctation 1;

which makes cxactly the same scntenccs truc as docs 1" lut in'1l, n

is

univcrsal-(Vw)(Vu)Rwu sincc ?+'sdomain t// is

just

U*, and cvcry

wo.rld in

a/,,,can scc cvcry

othcr

(as

wcll as itscu). Howcvcr, whcn,R

js

univcrsal, thc

cvaluation

clauscs

(ElR)

and

(l1oR)

ol this chaptcr arc

equivaleft

to

the clauses

(Ur)

and

(Eo)

of SM" lhus, for any world t,t

n

ty#, hcro is a world v in llzowhich I,,can scc and at

\a'hich

a scntcncc

O

is

truc

iff

therc

is a world v in l4l+at which

tf

is truc

'which

u can scc'

is rcd[ndant. Conscqucntly,

if wc transform I into a standard intc{prc-

talion

.',f

y

removing mcntion of R, and cvaluatc scntcnccs in

.7

by thc

standard rather than thc general cvaluation clauscs, cxactly lhc same

s(:ntcnces

will

come out

flue in

J

as are true in 1#,and hcncc as

in

?.

lhis mcans that SM and

Eq are

exactly

thc samc systcms. Iror whcncvcr

wc havc a

gcncral

intcrprctation 7 which cstablishcs

pr,.,.,p,,

iirq

4,

thcn

by lbllowing thc

proccdurc

just

described

wc

gct

a stand.lrd intcrprc-

tation

J

which establishcs

pt,...,p"*su

4.

And convcrscly,

f

a standard



52:Systems

J

cstablishcs

pt,..

,p,

*su

4,

thcn by adding a univcrsal i?, wc

gct

a

gcn-

cral ? which c stablishcs

pr,...,p,

*rq

4.

n

sccond

point

ofnolc

is

that

il would bc wronlt to concludc rom thc

constnrction of thc cxtcnsions of Gen

that

any

first-order

del'inable

constraint on R dctcrmincs its own systcm,

Each

such constraint

deter-

mincs a systcm, of coursc, but

it

may bc thc samc

as

onc

dctcrmincd

by a di{Tcrcnt constrajrt. Ihc discussion of Eq in thc

prcvious par

'

graph

illustralcs

this. l}]c condjtion o{' univcrsality,

(Vw)(Vu)Rwu,

is

a

strongcr constraint on

/l

t l lan the condit ion that

R

be an equivalence

rclation (a univcrsal rclation is an cquivalcncc rclation, bul not cvcry

cquivalcncc clation is a univcrsal

rclation)-Yet

as

wc havc

us{

st:cn,

f

wc wcrc

to usc uflivcrsality to

dciinc

a scmantic

conscqucncc rclation,

thc

rcsulting

systcm

would bc Eq ovcr again. Anothcr cxamplc is irrc-

flcxivily,

(Vw)-Rww:

the

semantic consequence relation Fr,, n fact coin

cjdcs

with

Fz

(this

is an excrcis(r,

'fhird

and

lastly,

usc of

thc nomcnclatura

'first-order

d,ctinablc

sys-

tcm' sullgcsts that thcrc arc such things as second'order

dcl'irablc sys-

tcms

(somc

lamiliarity with sccond

ordcr

logic is

prcsumcd

in thc ncxt

l] vo

paragraphs).

Sincc

first-ordcr logic is

contained

in

sccond ordcr

lo,jic, first{rder

delinable

systems are ipso

facto

sccond order dclin

ablc, so by a sccond-t.rrdcr dcfinablc systcm wc mcan onc thal is eisen-

dally sccond-ordc r: at lcast onc of tbc c onstraints on R dcfining thc

scmantic conscqucncc rolation is cxprcsscd by a formula contalining

second-ordcr

quantilicrs,

and no flrst-ordcr conditions capture thc

samo

scmantic conscqucncc relation. Simply using sccond-order lan-

guagc

to impose some constraints on li is insufficicnt to

gcncratc

such

a

systcm, sincc tho

sccond

ordcr Iormulac may bc logically cquivalcnt

to lirst-ordcr oncs. It is not cvcn suificicnt to usc somc conccpt

which

we know is

nol.

first-order

cxpressible. For cxamplc, [hcrc

is no way ol'

cxpressing

'thcrc

arc at

most finitely many x

such

that' in lirsl ordcr

languagc,

but

it can bc cxprcsscd w'ith sccond ordc-r

quantificrs.

Yet il

docs

not follow tiom

this that thc

systcm F whosc scmantic consc

qucncc

rclation FF s

givcn

by

'pr,.

..,p" -t

q

rtt t 'hcrc s no

gcncral

inLcr-

prcl.alion

with linite

t'f

on which all ol

pt,...,p,

arc truc whilc

q

is

l'alsc'

is essentially second-order. ln lact, FF again coincidcs with pa"

'lo

construct an

(essentially)

second order

definable system, dcfine

an n-chain in an intcrpretation

to be a sequence of worlds wo, wt,

tv2...

such that I{}ro}vr,nv,,r}/r, ctc.

(Wc

do not -rcquirc that thc worlds in an /{-

chain be distinct, so any

world which

can

scc itscll' immcdiatcly

givcs

dsc to an intinitcly long R-chain.)

Thc

scmantic consequence

rclation



IOO Chapter : First-Order-Definable,iodalSystems

trc

dclincd as

pt,.".,p,

F=c iff

there

is no

general

nterpretation in which R is

transitive,

and

every l?-chain is finite, and on which all of

pt,...,p,are

rue while

4is

false

yiclds

a systcm nown as G which is essentidlllsecond-ordcr. n othcr

wo.rds, herc is no collection C of ftst-ordcl conditions on n

klobal

or

othcrfi/ise)such tJrat

Fr...,p,

t=c

q

ltf

p,,...,p"

Fc

4.

G contajns Trn and

hence Gen, but differs from all the systemsof tJ]is section n view of

thc scqucnt Fc D(trA

-'

A)

-

aA. Thc formula'tr(trA

--

A)

*

tlA'is

known as Ltib's Formula, and can be understood as expressing an

important corollary of Godcl's Sccond ncomplctcncssThcorcm

(G

s

so-labclled after Gitdel). At this

point,

modal logic links up with the

main results of twentieth-century mathematical logic"

u Exercises

I Show the following, explaining

your

solutions:

(1)

DA Fo oA;

(2)

oA +z aA;

( 3 )

t r A & t r B F o A - B ;

( 4 )

A

- ^

t s o 0 - A ;

(5) trA - oA tsoDA * A; *(6) trA - A Fo rlA - .A;

(7)

trA

tso

rtrA;

(8)

aaA

F6

oEA;

*(9)

oEA

tso

roA;

(10)

trA

*

A

Fo

tr(DA

'

AX

(11)

D( (A

*

A)

*

A)

tso

A;

(12)

A

*

B

Fu

DA

-

B;

( I3)

nA

-

o(B

C)

Fo

o(B

(DA

oC).

II Show the following, explaining

you,r

solutions:

(l)

A For

roA;

(2)

trA

FRr

rEA;

(3)

D(nA

'

A)

*n,

rA;

(4)

Fsv.aA

*

A;

(5)

oA

FRs

oA;

(6)

.A

FRr

r.A;

(7)

D(A

v

oB)

#Rs

A

v oB;

(8)

Fsv.

o-,r;

(9)

EoA

tsrq

EA;

(10)

tr(trA

'

A)

Feq

A;

(11) tr(tr(A

--

trA)

-

A) FRrA

(12)

FRs

r(nA

-

B)v D(nB A)

(13)

tr(o.A

-

.A)*Rs lA

-

DDA

III For each of the following, suggesta

global

first-order condition or

set of condiuons C

which makes t correct

(perhaps

different C n each

case)and explain

your

suggestion,

C may not be any of the condjtions



53:

Deducibilityn

semanticallyefined ystems 0l

or scts of conditions uscd to dcfinc

systcms

n this

scction,

and

should

be as weak a condition or sct of conditions as

possiblc:

(1)

Fc D(oA

*

A);

(2)

Fc DA

-

oA;

(3)

Fc cA.

IV Show hat RTC s a

propcr

subsystemof Eq

(this

has two

parts,

(a)

every

RTC

sequent s an Eq scquent,and

(b)

some Eq scquent s not an

RTC

scqucnt).

V Explain

why trr,,

coincideswith Fo.

lHint

for the non-trivial direc-

tion,

suppose hat some

general

ntcrprctation ? cstablishcs

pr,...,p,

so

4.

lndicate how

an

irleflexive interpretation

J

could be constructed

from

'l

which wor.rld

establish

pu...,p,

*n

4.

Considcr replacing each

world in I which can

see

tself with

two

worlds

of a certain sort.l

3 Deducibil i tyn semantically

efined ystems

We

urn

now to

the

question

of

providing

rules

of

proof

for

someof the

systems ntroduced n

the

previous

section" deally,

what we

se€k s the

following: for

each irst'order definablesystem t

with

s€manticconsc-

qucncc

rclation +, wc would likc to constTucta dcduction systcm .9'

with deductiveconsequenceelation r9 such that ry and *r coincidc.

In

other

words,

the following two conditions should bc fulfilled:

(Sndj".J:

f

pr,...,p,

-

s'

qthenpb...,p,Esqi

(Comp.t,.r):

f

p,,.

.,

n

F=

q

r}]rcn

b...,p,,

-.

q.

According

to

(Sndy"),

hc deductivc systcm S' is sound

with

respect o

(or

lbr)

the semanticallydefined systemS,which means hat each

prov-

able scquent

in

S' is scmantically corrcct on thc scmantics for S" And

according o

(CompsrJ,

he deductivesystemS' s completewith respect

to

(or

for)

the semantically dcfined systcm S,

which

mcans that for

every sequent hat

is

semanticallycorrect on the semanticsof S, herc

is a proof of it using thc rulcs of S'. Wc also spcakof thc dcductivc con-

sequence elation ry being sound and/or complete br,/with respect o

the semanticconsequenceelation

trs.

Thcrc is no

guarantce

hat for eachof the systems n

52

some sould

and completc dcductivc conscquencc clation cxists;

wc

might bc

pa.r-

ticularly doubtful of finding such a relation for an csscnrially sccond-



l02 Chapter : First-Order-Defina bleodalSystems

ordcr systcm likc G. Wc alrcady know, &ough, that thc dcductiv(: sys-

tcm 55 is sound and complclc

I'or Eq,

sincc

Eq is

lhc samc system as

SM, and wc statcd carlicr

(wilhout prool)

that 55

s

sound and

(:omplctc

for

SM-

temarkably,

every system

in

$2

can

be

provided

wilh sound

ilnd

(:omplctc

rulcs of

proof.

But thcre is a catch. Onc's lirst thoughl is

that

Lhc rulcs lor thc

othc.r

systcms will bc variations on thc 55 rulcs

thc othcr scmantic conscqucncc

rclations might

bc capturablc simply

by adjusling thc constraints on

El

and

ol- Ilowever, while it is

possiblc

to obtain thc systcm RT in this way, tbcrc is no known way of obtaininit

any of thc othe.r systems in 52 similarly. A gcncrirl approach to dcduc

tion which works for all systems is rather diffcrcnt in naturo from nat

ural

dcduction,

and it is rathcr unwicldy. For that rcason, rvc wlll

d is (uss hc mclhod

Ia i r ly

hr ic l ly .

We definc a deductive conscqucncc clation r* which is so und and

complete for Gen in the fbl[owing n'ay.

(i)

lhe deductive system K has as non-modal basis all lhc.rulcs

of

NK

for thc scntcntial

corincctivcs.

(ii)

K has a nrlc callcd Neces'sitation,which is a rcstdctcd vcrsion

of trI; whcn a formula

p

at

linc

j

in a

proof

has

been

inferred

Irom

prcvious

lincs in such a way [h:rt iL dcpcnds on no

assumptions or

prcmiscs

at

j,

or

if it

occurs at

.i

by

fl,

thcn

at a latcr linc k wc may add

'np',

also dcpcnding on no

prc-

miscs

or

assumptions.k is labcl€d

l,

Ncc'.

(iii)

K has a rulc I)fo, likc thc NK-ru lc

Df lbr'*', n'hich

allows us

lo add a new

ine

k to a

proof

by

rcplacing

any occurrcncc ol'

'-r-'

in

a

fbrmula

at

line

i

with'o', or in the ofher dircdion,

any occurrc ncc of

'o'

with

'-tr-'-

k is labehd

'j,

Dfo',

and

dcpcnds on whatcvcr k dcpcnds on. Notc ttat, lor <rrnvc-

niencc, we do not restrict applications

of Dfo

to

cascs whcrc

thc Iirst

'-'

of

thc

'-tr-'

bcing abbrcviatcd is thc main con-

ncctivc of thc formula: drll occurrcncc o1

'-r-l-'

may bc

abbrcviatcd, and any

'o'

may bc cxpiuded.

(iv)

K has a spccial cxtcnsion of Scqucnt

Introduction: wc

arc

allowcd to usc thc scqucnt n(A

-'

B) F LrA

'

nB

(Examplc

2.4.2 on

page

82) br SI.what makcs this cxtcnsion spccial

s

that wc do nol prove thc scqucnt first, belbre using it in SI

later. I{ather, making this sequcnt avallablc to oursclvcs

is

what

cts

thc systcm

K

off

thc

ground.

Equivalently, wc can

use thc scqucnt F u(A

-

B)

-

(LrA

-

lfB)

in

any application

of

TI- Thc

pr€misc,/conclusion

vcrsion is slightly morc con-

vcnicnt, but thc thc orcm vrrsion is morc common

in

prcscn-



53:

Deducibiliryn

semantically

efined

ystems

103

"

tations of this kind of approach o modal dcduction. In thc

'

thcorcm vcrsion hc formula'n(A

.-

B)

*

(nA

-

trB)'is callcd

an dxiom

to

indicatc

that

it is

simply assumed o be corrcct,

without independent

proof

(in

tJrc way Euclid assumes his

axioms of

geometry

to be conect without indepcndcnt

proof).

We will refer to our two sequentsas axiom-seque.nts,

and we use whichever

of them

leads to the

quicker proof,

labeling

the

line

L,

SI

(K)',

or'TI

(K)'as

appropriate.

These four items define the deductive system K and the

deducibility

rclation L-K, hich coincidcswith tso.Thcproofs of this and subsequent

claims about which deducibiliw rclations match which

scmanticcolrse-

qucncc

rclations a-re

ostponcd

until Chaptcr

5.

But we will illustrate

the

proof

systemswith sampledeductions,such as he follow:ing

proof

of a basic scqucnt of modal logic. In it, we use

'NK'

as

a

labcl for any

stcp

justificd

by a scqucnt of non-modal scntential logic which does

not havc ts own namc.

Example3.3.1:Show r(A & B) FK rA & nB

1

(1)

D(A& B)

( 2 ) ( A & B ) - A

(3)

tr(A

&

B)

-

A)

(4) tr(A & B)* trA

1

(s)

trA

( t ) ( A & B ) - B

(7)

tr(A & B) B)

(8)

tr(A & B)

-

uB

1

(9)

aB

1

(10)

trA & DB

Prcmisc

TI

(NK)

2

Nc c

: I S I K

4 1 - F

TI

(NK)

6 Nec

T S I K

8,r

-E

5,9&I '

Line

(4)

s by the substitlrtion-instancc f the K axiom scquentobtained

by

putting

'A

& B' or

'A'

and

'A'

for

'B'.

Proofs likc this are arrived at by some combination of luck, expcri-

enceand insight,

since

hcre are few reliable heuristics for finding

the

right non-modal hcoremsand axiom-sequent ubslitulion instanccs o

work witr. But despite its hard-to-work-with nat1ue, he

system

K as

described n

(i)-(iv)

above s in fact sound and complete for Gen, and

other

proof

systems

which

i e sound and

complete for

the

va.rious

semantic consequenceelations of

$2

can be obtained simply by aug'

menting

K with

futher axiom sequents

which

calbc

uscd in

SL



l04 Chapter : First-Order-Definableodal ystems

.

Thc systcm

KT, T

for short,

s obtaincdby adding thc axiom-sequent

r- rA

*

A, equivalently

aA F A,

to

K. So

a

proof

in KT is the same as a

proof

in K cxccpt that lines may be

ustified

by appeal

o SIusing A F

A

or

F trA

-

A;

such

a ine will be abeled

TI

(T)'

or

'j

SI

O)'.

KT s sound

and complete

with respect o Rf; that is, FKrand

trRt

oincide,

Here s a

samplc

proof

in KT; noticc that though line :l contains a

modal

opcra-

tor, its

justification

is non-modal:

wc usc a modal formula in a substi-

tution-instanceof a sequent

of non-modal ogic.

F;<ample.)-3-2:ShowA FKT A"

1

( 1 )

A

(2)

tr-A

- -A

I

(3)

-tr-A

1

(4)

.A

1

(r) A

(2)

on-A

- -A

1

(3)

-o -A

1

(4)

-^ -D*A

I

(5)

tr-tr-A

1

(6)

roA

Premise

TI

(T}

2,

sr

cMT)

3

Dfo a

The

system

KB is

obtaincd

by addjng to K the axiom-sequent

oEA F

A

(equivalently,

F o[A

-

A), which

we

call

B. As

the

reader

pcrhaps

anticipatcs, KB is sound and

complete with respect to Sym; that is, FKB

and er- coincide. In KB

we

have thc following

proof.

Example 3.:1.3:Show

A FKB roA.

Premise

n

(B)

2,1

SI

CMT)

4 Dfo

5 D N

6

Dfo a

Linc 2 uses he axiom-sequent

B, replacing

A'

with

'-A'.

The

system

K4 is obtained by adding to

K

either

the axiom-sequcnt

ooA F oA

(equivalently,

F

ooA

.*

oA), or else he sequent A F trA

(cquivalently,

F trA

-

A), known as

4.

Wc fcaturcd

thc

'o'vcrsion

as

Exuunplc .2.:l

pagc

95),but

in thc

proof

systcm or

Trn it is morc com-

mon to usc the'tr'version as the axiom sequentand derive thc

'o'vcr-

sion, which is the approach

we shall adopt.

K4 is

sound

and complete

with respect o

Trn;

that

is, FK4 nd trr," coincide-

n K4 we have he fol-

lowing

proof.



S3:

Deducibil i tyn

semanticallyefined ystems

l0 5

Example 3.3.4:Show

ooA

FK4

oA

I

( 1 )

o o A

|

(2)

- -- -A

(3)

-A

*

n -A

(4)

-A

.'

--D-A

(s)

tr(n-A

' --tr-A)

(6)

trtr-A

-

tr--tr-A

(7)

-A

*

tr-- -A

I (8)

-E-A

I

(9)

oA

Prcmisc

1 Dtlo

Tr

4)

TI

(NK)

4 Ncc

5 SI

K)

3,6 Sr

NK)

7,2 SI (tfl)

B Dfo

a

Line

ti uses

K,

putting

'D-A'

for

'A'

and

'--E-A'

for

'B'.

From this

point

it

is

straightforward to obtain sound and complcte

systemsof

proof

lbr RS,RT and Eq, simply by adding axiom-sequents

tO KT:

.

The system KTB, usually known simply as B, is KT

plus

the

.

axiom sequent

B.

(KT)B

s

sound and

complete or Rs.

.

The system KT4, usually known as 54, is KT

plus

thc axiom

sequent4. 54 is sormd and complete or

RT.

.

The system KTs,

whcre

5 is the axiom sequent

oA F troA, is

usually called 55. 55 is sound and completc for Eq.

It is

a consequence f thes€

esults that

tJIesubsystem

diagram of

thc

previous

section could be dupucatedhere,

with

the appropriate

proof

systems replacing the semantic systems

in

that diagram. We under-

stand the semantic nclusions as resu.lting

rom

a

restdction in

the

classof

interprctations:

a semanticsystem.t

s

a

subsystcm

of

a

seman-

tic system S' when the interpretations for S' are a subset of those for

S. On the derivation side, one

proof

systemS'

includes

another S

when

the

rules

of

S are available in S', and when all the axiom sequcnts and

definitions of S are availablc n .t'; for

propcr

inclusion, some axiom-

sequent or mle or definition must be available in S' and not in S.

Finalfy, therc is the

qucstion

of whether an essentiaw second-order

semantic system can have a matching proof systcm similar to thosc

just

described"The answer

s

that this

is

possible,

and the systemG is

an example-

The

proof

system

KL

s

obtainedby addjng to

K

the axiom

sequentL,which

is

tr(trA

-

A) F trA

or

equivalcntlyF tr(lA

-

A)

-

trA,

bascd on Lijb's Formula

tr(trA

-

A)

-

trA'. KL s sound and completc

for G-



106 Chapter : First-Order-Definableodal

ystems

.

Wc cnd this discussion

of dcducibility

n various

systcmsby remark-

ing that lie unwieldy nature of this kind of approach

to

prool

mcans

that thcoretically significant scquents

can

require

proofs

which

even

those on the forefront of development

of

modal logic have

some diffi-

culty discovcring.Thc

problem

of showing hat 4 is derivable n KL s

an cxampleof this

phcnomcnon,

but thc following

proof

was evcntlal

ly discovcrcd

indcpcndcntly)

by Kripkc, dc

Jongh

and S.filbin. In

it wc

assume hc sequent r&, FKL

nA

& trB)

-

tr(A & B);we have

ahcady

(ir

eilect)

proved

one half of this; the otler is

an exercise-

Example 1.:1.5:}J.ow KL

A

-

trlA

(1)

A

*

[( A

& BDA)

-

(A

& trA)] rl

(NK)

(2)

tr{A & trA)

-

( A

& nnA) Tl

(K,

r&)

(3)

A

*

[ (A

& DA)

-

(A

& nA)l 1,2 sI

(NK)

(4)

D(A

*

[D(A

& DA)

-

(A

& nA)l)

3

Ne

(s)

A

-

[tr(A & A)

'

(A

& A)l 4 sI

(K)

(6)

D(tr(A

& A)

*

(A

& rA))

-

tr(A & nA) TI

(L)

(7)

DA

*

(A & A) s,6 SI

(NK)

(8)

trA

-

(trA

& trtrA) 2,7

SI

(NK)

(9)

trA

*

nrA

8 SI

(NK)

+

Iine I is a substitution nstanceof FNK - (B & C) - (A & B)),and line

6

results

from replacing'A'in Ldb's Formula with'A & trA'. There are

oths thcorctically significant scquents whose

proofs

are even morc

obscurc.

To

end

with

a caveat,lest the dcvclopmcnt of this scction lcavc an

oversimplified

pictu.re

n the reader'smind- Not every consistent com-

bination of first or sccond-orderconditions

on

R

can be

matched

with

a rcasonable deductjon system like

the

systems

we

have

discussed

hcrc- In thc othcr dircction, thcrc arc dcduction systcms which corrc-

spond to no collection of first-order conditions, such as KL, and which

detemine

second-orderconditions which are much less manageable

than the'no infinite R-chain'condition for C. Most interestingly, here

are also deduction systems t'which do not match any semanticconsc-

quence

€lation which can be dcfincd by first or sccond-ordcr condi-

tions on l?. So f S is any scmantic system clativc to which such an .9'

is sould, there will be

some semanticallycorrect sequent

n

S

which

cannot be

proved

in

the deduction system; hat

is,

S'

must

be

incom-

plete

with respect o .S. hus the whole subject of

general

elationships



53:

Deducibilityn semanticallyefined ystems

07

likc lhcsc betweensemanticsystemsand

proof

systcms

s rathcr com-

plicatcd;

we

make a start on

it in

Chapter

5.

t-J

Exercises

I

Show the

follow.ing

(in

thcsc

problems you

may not use the natural

deducdon rules for n and o from Chapter 2):

(1)

nA

&

trB FK r(A

&

B)

Hint

use r** A

-

(B

-

(A

& B))]

(2) tr(A - B) F-Kr(-g - -A)

(3)

-oA

iF,( -A

(4)

-aA

-rFx

o-A

(5)

oA, tr(A

*

B)F

k

oB

(6)

o(A & B)FK A & cB

*(7)

oA

v oB

iF

K

o(A v B)

only

eft to right in solutions)

(8)

D(A

-

B)F* .A

-

rB

(9)

FKB nA

.

loA

*(l0)

F-Kr

DA

*

B) v

( B

-

A)

(11)

FKBr(A

'

trB)

.-

(oA

*

B)

(12)

(oA

*

B) x , A

-

cB

{l:l)

FKBa uA

-

trA

(KB4

is the system K

plus

axioms B and 4)

*(14)

t-'<u

oA

*

loA

(15)

FKB4r(tr4

-

B) v tr(trB

-

A)

(16)

(A

v

.B) F

s4

A

v oB

(17)

loA

- '

oDA Fs ao(A

*

t rA)

(18)

Fs5 trA

-

tr A

(19)

trA

-

DB Fs5D(trA

'

trB)

(20)

Fss A

*

tr lA

U We

have formulated

the system

55 in two djfferent

ways- Let us

use

S5Ax to mean the axiomatic

system of tlfs scction. Show that for any

formula

p,

if

p

is fully modalized then

F,ro,

p

-

rtp.lHtnt: distinguish

a spccial kind of fully modalized tbrmula,

from which all other fully

modalized

formulae are constructed ^Show that if

p

is

a fi.tlly

modalized

formula of this special

kind, then Fs5a,

*

p. Then show that

for

each

fully modalized

formula

p

not of the special kind, if the main subfor

mulas of

p

imply

their

olrn ncccssitations in sAx,

Fs5^,

p -

trp-l

III Lct rrs usc SsND to mean the natural deduction

syslem

givcn

in

Chapter 2,

54.

Show

dircctly

(wifhout

appeal to scmantics) that

pr,...,p,

rrr

o

qiff pr,...,p,

Fs5n"

.

[Hint:

given

a derivation of

4

fiom

Fv...,p"in

SsAx, describe

how

to

transform it into a dcdvation of

q

frtm

pt,...,p,



l08 Chapter : First-Order'Defina bleodalSystems

iJr S5ND; thcn do thc convcrsc, dcscribing how to Oansform

a dcriva

tion in 55 ND into a dcrivation in SsAx -

lhis dirccl.ion s hardcr. lsc thc

r<rsu l ts

1 'prob lcm

2.4.1T,

agc

Ul l , and o l

prob lcm

l l i rnm( :d ia tc ly

abovc. l

first order definable modal systems

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