Semantic Tableaux Methods for Modal Logics That
Include the B(rowerische)
and G(each) Axioms
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
Francis Jeffry Pelletier
Depts. of ComputingScience & Philosophy
Univ.Alberta
Edmonton,Alberta
Canada TtG 2El
Introduction: Semantic tableaux are a methodfor
determiningvalidity of argumentsin a certain class of
logics. Themethodis perhapsnot so weB-known
to the
automatedtheoremproving communityas resolution,
1but it has long beenpresent in the logic community.
As its namesuggests, the methoduses semanticfacts
about the languagein an attemptto generatea countermodelfor the argument
at hand(or to showthat there is
no such counter-model and that the argument is
therefore valid). In this, the methodbears a strong
similarity to Resolution, but it does not require any
conversion to a normal form. The method is also
different in being an analytic system:eachnon-initial
formulain any proof is a subformulaof someearlier
formulain the proof. Naturaldeductionis not analytic
in this sense, nor is anyResolutionmethodother than
UnitResolution.
Thereare various different formatsfor stating the
methodof semantictableaux. Participants in this AAAI
Symposium
are doubtless familiar with the form given
by Fitting (1988). Themethodhe uses is quite similar
to the methodI wishto employ,and there shouldbe no
difficulty in transforming
the oneinto the other. In this
note, I am interested in extending mymethodto a
certain class of modallogics. In this extensionwewill
only consider propositional logics - a fact that lends
someconsiderablesimplification to the description of
the tableaux method. (By not needing to describe
quantificationrules andtheir difficult integration into
modal logics. See Garson1988 for a survey of the
difficulties).
I will first lay out the tableauxmethod
that I prefer,
restricting myattention to classical propositionallogic.
I then showhowit is extendedto a certain class of
modallogics, the normal modallogics. The idea of
suchan extensionis not particularly new;it is already
mentionedin Fitting (1983, 1988), althoughnaturally
believe that the particular modificationsI maketo the
basic methodmakesthe extensionconsiderablyeasier to
understand.After developinga methodfor the basic one
of these logics I turn myattention to logics which
contain the B(rowerische) axiom. Of these logics,
Fitting (1983:192; 1988:203)says that the style
tableauxhe presentsdoesnot lend itself to logics where
the accessibility relation involvessymmetry.
Andit is
implied that it wouldbe very difficult to construct
reasonable tableaux methodsfor such logics. I show
that the alteration I makeavoids this difficulty and
allowsfor a semi-analytictableauxsystem.After this I
considermodificationsto the basic systemso as to yield
a tableaux methodsuitable for systemscontaining the
G(each) axiom. This axiom is unusual in that it
involves embedded modal operators in both its
antecedentand its consequent,and this in turn raises
interesting challengesfor tableauxsystems.Onceagain
wesee that a semi-analytictableauxsystemwill work.
Classical Propositional Semantic Tableaux:
Semantictableaux, whendone in the methodI prefer,
are in fact trees; andthe methodof makinga tableauis
in fact a methodof tree construction.Like Resolution,
it is a refutation method;andthereforeit starts with a
"negatedconclusion"formof an argumentto be tested.
(UnlikeResolution,there is no further pre-processingof
formulas, such as conversion to a normalform.) The
root of the tree contains all the premises and the
negationof the conclusion.Theconstructionof the rest
of the l~ee is governedby "branching rules", which
operate (by decomposition)on formulasalready in the
tree.
For the propositionallogic, formulasare definedin
the usual way,usingsentenceletters (A,B,C....) andthe
1 Themethodgoes backexplicitly to Beth(1952,1958), connectives~, &, v, -->, <->. For describingthe method,
it is convenientto divideformulasinto four classes: (i)
and has been popularized by such elementary logic
literals (sentence letters and their negations), (ii)
textbooksas Smullyan(1968), Jeffrey (1968), Bonevac formulaswhichare doublenegations (i.e., start with
(1987), Bergmann
et al (1980). As has been noted
two ~-signs), (iii) formulas whichhave somebinary
variousresearchers(e.g. Fitting 1983:4-6,Fitting 1988: connective(&,v,--->,<-->)as their mainconnective,and
191), there is a strong analogybetweenthe tableaux (iv) formulaswhichare negationsof type (iii) formulas.
methodand Gentzen’s (1934/5) consecution method. Toeach formulawhichis not a literal, exactly one of
(Although most people would wish to maintain
the followingninebranchingrules is applicable:
distinction betweenthe two methods.)
126
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
[for type(ii) formulas]:
Example1: Given(PvQ)&(P--->~Q),does it follow
that (Pv(Q&--~P))
-~A
I
A
~/ (PvQ)&(Q--+--~P)
q ~(Pv(Q&--,P))
I
q (PvQ)
q (Q--->~P)
I
~P
~(Q&--~)
[for type(iii) formulas]:
(AvB)
(A-->B)
/ \
/ \
A B
-~A B
(A&B)
I
A
B
(A<-->B)
/
\
A --~
B
~B
[for type(iv) formnl~]:
~(AvB)
~(~->B)
~A
~B
A
~B
[
t
~(A&B)
/
~A
\
~B
\
/
P
x
Q
/ \
~Q
~(A<-->B)
I
A
~B
\
~A
B
Sincethe constructionof a tableauproofin this system
is really the constructionof a tree, to define a proof
formallyrequires defining various notions relevant to
trees -- such as ’dominates’, ’branch’, ’leaf’,
’immediatelydominates’,and the like. For the present
audience it is best just to allow one’s background
knowledge
of trees comeinto play; andI will therefore
just presuppose such notions. One terminological
point: If a formulaandits negationare on a branch,we
say that the branchis closed;otherwiseit is open.
Onebegins the tree construction with the set of
premisesand negated conclusionat the root. Onemay
close a branchas soonas one notices that it contains
complementary
formulas(i.e., anyformulaand also its
negation on the samebranch). Otherwiseone chooses
complex
formula(of type (ii), (iii), or (iv) above)
somenode on an open branchand applies the relevant
branchingrule to it, placingthe result at the bottomof
all open branches dominatedby that node. (Done by
extendingeachsuch openbranchin the mannerindicated
by the rule.) Whenthis is done, the initial formulais
"checkedoff" (by placing a ’~/’ marknext to it),
indicatingthat it needsto be considered
no further. This
lends itself to the followingalgorithm: Continuethe
followingtwosteps until either all branchesare closed
or else there are no un-checkedcomplex
formulasin the
tree. (i) determinewhetherany just-altered branches
close; if so, close the branchby placing an ’x’ at the
bottom of the branch. (ii) choose an un-checked
complex formula from an open branch, apply the
relevantbranchingrule to it, andcheckoff the formula.
Thetwo halting conditions in the abovealgorithm
correspondto whetherthe initial argument
is valid or is
invalid, respectively. Aproof of the completenessand
soundnessof the algorithmdependson K0nig’sLemma,
and can be found in Jeffrey (1968:56). Here are two
simpleexamples
of a proof/disproofin this system.
127
x
x
Example2: Given~(A<--->B),does ~(B--->A)follow?
q -~(A<-->B)
~/ ~(A<--->B)
I
q (B-~A)
l
-~B
/ \
A -~A
~B B
X
\
A
/
\
A -~A
--i]3 B
x
The first exampleis valid, as evidencedby all the
branchesclosing, while the secondexampleis invalid,
as evidenced by branches remaining open yet all
complex
formulasbeingcheckedoff. It is pretty clear in
these examples which branches were the result of
working with which previously-appearing formulas;
other examples maynot be so clear, and so some
textbooks employa bookkeepingmethodto record this
information.It should be noted that a branchingrule
operates on a formulaat a node in the tree, and the
result of applyingthe rule is to extend every (open)
branchin the tree that the node dominatesby adding
newnode(s)as called for by the particular rule being
used. Theformulato whichthe rule has beenapplied is
then checkedoff, to indicate that it never need be
consideredagain. (Propositional tableaux systemsare
therefore decision procedures: they alwaysterminate
whenall complexformulas have been checkedoff or
whenthe tree is closed). In the classical propositional
calculusit doesnot matter, logically speaking,in which
order formulas are chosen in the proof construction
algorithm(althoughit certainly mattersfor efficiencyof
tree construction).This aspect will changein the modal
propositionalcalculus. It shouldalso be notedthat the
this methodof proof is analytic: no formulaappearsin
the tree unless it is a subformulaof somepreviouslyappearingformula. Thusall formulasin any proof are
subformulasof the initial set of negated-conclusion
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
formula. In this, classical semantictableaux stand in
sharp contrast with Natural Deductionproof methods
and with Resolution proof methods, which allow
formulasto be generated that are not subformulasof
previously-generated
formulas.This aspect of classical
tableaux systems will also change with the modal
logicsmentioned
in the title of this article.
Intuitively speaking,branchinga formulaindicates
that the formulais true if and only if at least oneof
these newly-createdsub-branchesis true (i.e., all the
formulas on the newnode of that branch are true).
Hence,the initial root nodeis true exactlyif at least one
branchremainsopen-- andthe initial argumentis valid
just in case no branchremainsopen. Onecan evenread
off a counterexample
to an invalid argument(in terms
of assignmentsto atomic sentences) by traversing an
open branchand assigning True to (atomic) unchecked
sentences whichappear unnegatedand False to those
whichappearnegated.
Normal Modal Propositional Logics: Modal
propositional logics are formed from classical
propositional logic by addingtwonew(interdef’mable)
sentence operators: L ("necessarily")
and M
("possibly"). In this section, 21 modalpropositional
systemsare described. Theyform a subset of the socalled normalmodallogics, and include most of the
well-knownmodalsystems. The techniqueused here to
describe these systems is that of Chellas (1980).
Starting with a fundamentalsystem, K, various axioms
are introduced(called suchnamesas ’D’, ’T’, "B’, ’G’,
’4’, ’5’) and more complex systems are namedin
accordancewith whichaxiomsare added to K. Thus,
the modalsystem KD45results from adding axiomsD,
4, and 5 to system K. As it turns out, someaxioms
imply others, some combinations of axioms are
equivalent to each other, and somecombinationsare
knownby other names. Attention will be called to
these casesas theyarise.
So, syntactically speaking, the various modal
systemsof interest are seen as being built uponK.
SystemKcan be axiomatizedas follows:
1. ClassicalPropositionalLogic(formulatedwith
modusponensas a rule of inference)
2.
3. ~--k L(~-~F)--~ (L~--~LVF)
4. if ~k¯ then ~---kL~
Typical theoremsof systemK are
M(p--u:I)~-~Lp--->Mq)
tLt~q~--~pSa.q)
Butsuch familiar formulasas
Lp~p
Lp--->LLp
Lp--->Mp
are not theoremsof K. To build uponK, we consider
the followingsix axioms:
D. L~--->M~
T. L~--->~
128
G. ML~--->LM~
B. ~--->LM~
4. L~---->LL~
5. M~--->LM~
Starting with K(whichadds 0 of these axioms),there
are 64 different combinationsof these six axioms,and
so one might expect 64 different modal systems.
However,there are certain implications betweenaxioms
and equivalences amongst groups of axioms. The
relevantimpficationsare:
T~--~>D
B~--~>G
5 ~> G
andthere are the followingequivalences
KIM <~> KB5
KDB4<~>KTB4<~>KT45<~-.m->KT5<~>KTB5
(and anyother implicationsandequivalencesentailed by
these). This leaves us with the 21 modal systems
pictured in Figure 1 (adaptedfromChellas 1980p.132)
presentedat the endof this article. Thelines between
systemsindicate properinclusion: all theoremsof the
weakersystemare also theoremsof the stronger system,
but not conversely.
Oneinteresting fact about these systemsis that
they have a semantics whichcan be described by a
binaryrelation (calledthe "accessibilityrelation", R)
"normal possible worlds". The intuition is that a
sentenceis (not just simplytrue but rather) true at a
possible world, and whenthe sentence has one of the
modaloperatorsas a mainconnectivethen it is true at a
particular worldjust in case the demodalized
formulais
true at every (or some-- depending on the modal
operator involved) accessible possible world. For
example,L~is true at possible worldwjust in case
is true at every possibleworldthat is accessibleto w.
In a normal modallogic, to say that a sentence is
semantically valid is to say that it is true at every
possible worldin every modelin whichR obeyscertain
constraints. Obviouslythese definitions put a lot of
weighton just whatthe relation R is.
It is well-known
whatthe relevant requirementson
R are for the modalsystemsunder consideration here.
In systemK, R has no special requirements;therefore, a
~)
sentence ¯ is valid in K (whichwerepresent as ~k
just in case for any relation R, ¯ is true at every
possibleworld(underthat accessibility relation, R). (It
is becausethere are no special requirementsplacedupon
R that systemK is "the minimalnormalmodallogic").
Otherlogics addconditionsuponR, for examplethat it
mustbe reflexive or that it mustbe connected.Eachof
the axiomsto be considered gives rise to its own
particular requirementon R, and thus a logic described
as KD45(for example) will impose upon R the
requirements relevant to D and to 4 and to 5. Anda
formula ¯ is valid in KD45( ~kd45~) iff for any
relation R whichobeysthe restrictions relevant to D, 4,
and 5, ¯ is true at every possible world (under that
accessibilityrelation, R).
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
Fromthe semanticpoint of view, a standard model
Mis a triple <W,R,P>
whereWis a (non-empty)set
indices ("possibleworlds"),R is a binaryrelation on
and P is a total function on WXLA
(whereLAis the set
of atomicsentencesof the language)into {trne,false }.
(Thatis, P says whichatomicsentencesare true/false at
whichpossible worlds). A truth-functionally compound
sentence¯ is true in the modelat possible worlda in
just the "normalway",viz.:
ff ¯ = -,W,then~ is true at a iff ~Fis false at o~
if q~= OF&F),
then ¯ is true at a iff either ~Fis
trne at ¢¢or F is true at ~
if ¢, =OF->r),
then¯ is true at a iff, if ~Fis true
at a thenr" is true at ¢
andso on for any truth functional connectivewehappen
to have. Modallycompound
sentences are true in the
modelat a possible worlda in a waythat makesuse of
the "accessibilityrelation", R:
ifO = L~, then ¯ is true at a iff~ is true atall
[3 suchthatRa[3
if ¯ = M~F,
then¯ is true at a iff ~Fis true at
some~ such that Rctl3
Aformulais true in the modelMiff it is true at every
possible worldin M. A formulais logically true in M
iff it is true at every possible world in each model
<W,R,P’>,whereP" might differ from P as to which
atomicsentences are true at whichpossible worlds. A
formulais valid iff it is logically true in everymodel
<W,R’,P>.That is, a sentence is valid iff for all R"
andfor all P" the sentenceis true at everyworldin each
<W,R’,P’>.
Thejust-dcfinod notion characterizes validity in
system K. Each other system defines validity with
respect to somerestricted set of R’-models. For
example,these R" relations might be required to be
reflexive, or they mightbe requiredto be serial, etc. As
it turns out, each of the axioms which we have
considered adding to system K gives rise to its own
requirementon what kind of relation R must be. The
roquimmonts
amstated:
(d) [serialityl (Vx)(xeW
--+ (3y)(yeW
(d), (b) implies (g), and(5) implies (g). Furthermore,
the combination (b) and (4) is equivalent the
combination (b) and (5). And the following
combinations
are all equivalentto one another: of (d),
(b), (4); of (d), (b), (5); of (t), (4), (5); of
andof (t), (b),
Thematerial of the preceding paragraphs can be
summarized
bythe chart in Fig. 2 (adaptedfromChellas
1980,p.164) given at the end of this article. Let us
abbreviatethe notionof ¯ beingvalid, with restrictions
X placed upon R, as ~xO. Then Fig. 2 reports
completenessand soundnesstheoremsfor all the modal
logics wehavediscussed,saying
where X is comecollection of ’K’ plus any of the
letters ’D’, ’G’, ’B’, ’T’, "4’, ’5’ -- thus formingoneof
the syntactic (axiomatic)systemsmentionedearlier
and x is the corresponding collection chosen from
amongst’d’, ’g’, "b’, ’t’, ’4’, "5" given immediately
above. These soundness and completeness theorems
assureus that the results of anycorrect proofprocedure
for a normal modalsystem X can be mirrored by the
results of any correct semanticevaluation procedure.
Andsince each of these systemshas the finite model
property,it followsthat there is a decisionprocedure
for
eachof the logics.
Tableaux Methods for Propositional Modal
Logics: In this section wegive tableaux methodsfor
any normal modal system containing B and G. Our
strategy will be to give a methodsuitable for systemK,
and then to explain whatnewrules or modificationsare
called for by each of these axioms.Henceto producea
tableaux argumentfor KBG,for example,one proceeds
as with systemK but adds on the modificationsrelevant
to (b) and (g). (The samemethodcan be used to
tableauxmethodsfor all the 21 systems, but that more
generalmethod
is savedfor a different occasion).
Therules consideredabovefor classical logic are
just that: classical. This meansthat theyare relevantto
any one world at a time, but are inapplicable when
(0 [reflexivity]CCx)(xe
w-->Rxx)
considering more than one world. For example,were
(b) [symmetry]
(Vx)fVy)(x,yeW--+ (Rxy--+Ryx))
formulaAto be Irue in one world and formula~Ato be
(g) [incestuality] (Yx)fYy)(Vz)(x,y,ze
true in another(evenif it be an accessibleworld),that
(Rxy&Rxz--+(3w)(weW&Ryw&Rzw)))
doesnot allow us to assert that any branchis or isn’t
(4) [transitivity] (Vx)(Yy)(Vz)(x,y,zeW--+
closed. This suggests that wedivide our rules into
(Rxy&Ryz
----> Rxz))
differentsorts:
(5) [euclidean]
(Vx)(Vy)(Vz)(x,y,ze
Type 1 rules ("World-Bound
Rules’9: This sort of rule
(Rxy&Rxz
--,Ryz))
containsthe classical rules wehavealreadyintroducedInthemodal
system
KD,forexample,
a formula
¯ is rules that operate only within a specified possible
valid
just
incase
itislogically
true
inevery
model
in world. In addition to the (classical) worldboundrules
which
R obeys
(d),
that
is,inwhich
R isserial
(ffi
con- we have already comeacross, wehave two more new
nected).
O isvalid
inKIM5
just
incase
itislogicallyones that involve the modaloperators. ThesenewType
trueinevery
model
inwhich
R obeys
(d),(4),
and
1 rules are common
to all normalmodalsystems. After
that
is,inwhich
R isserial,
transitive,
andeuclidean. applyingthe rule, wecheckoff the initial formula.
Itcanbenoted
that,
justasinthecaseofthe
--LA
~MA
axioms,
certain
conditions
imply
others
andvarious
I
I
combinations
areequivalent.
Inparticular,
(t)implies
M~A
L~A
129
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
Type 2 Rules ("World-CreatingRules"): This type of
rule is distinctive and newto modal logics. When
evaluating a modalizedformula, the semantic rules
given abovedirect us to determinewhethera certain
subformulaof the modalizedformula is true in some
other possible world. Type2 rules detect whenthis
situation occurs and directs us howto extend the
semantictree so as to encompassnewworlds, and how
these newworldsare to be initialized. This conception
is so that wewill use a doubleline (rather than a single
line) to indicate howthe proof tree is extendedwhen
these rules are applied.Therule canbe put like this:
with anybelowthe double-linebranch,exceptinsofar as
permittedto do so by Type3 InterworldCopyRules.
Aside effect of the our decree that no formulain
one worldcan interact with a formulaof another world
(except via Type3 Rules) is that all Type1 Rules
must be applied within a given world before
any Type 2 Rule can be applied in that
world. If we do not have this requirement then it
wouldbe possible for a contradictionto be presentin a
world but to go unnoticedbecausethe relevant Type1
rule was not applied to the complexformula which
wouldmakethe contradiction manifest. Our algorithm
for modallogic therefore says that weapply only Type
MA
1 rules in whateverworldwehappento be investigating
1
I
until either (a) all branchesare closedin that world(and
thus that worldhas beendeclaredto be impossible),or
/via
2
Co) there is an openbranchon whichall formulasare
I
either literals, or are checkedoff, or else havea modal
operator as their mainconnective.In case (b), if there
are anyof the last-mentionedsort of formulas,then we
I
can applyType2 rules.
MAn
To apply the Type2 rule: each M-formulawithin a
givenworldcreates a newworldunderat the endof each
openbranchthat it dominates,andinitializes that world
A1 A2 A3 ...
An
with the de-mod~liTed
formula(i.e., the formulawithout
Thewayto understandthis rule is: givena groupof M- the mainM-connective).Havingdone this with a given
M-formula,it can be checkedoff (althoughfor ease of
formulas(formulaswith ’M’as their mainoperator) all
use of someof the Type 3 rules there should be a
on the same(open)branch,then simultaneouslystart
collection of newbranchesin whicheach newdouble- different style of check-mark,since the Type3 rules
sometimes
needto makereferenceto these formulasthat
line branch is initiated by a different one of these
have been decomposed
by Type2 rules). Onecontinues
formulas(withouttheir ’M’operator).
to
apply
the
Type
2
rules
to every M-formulaon an
Type 3 Rules ("Interworld CopyRules"): Given an
open
branch
in
the
tree
at
the
given world. Onlyafter
openbranchin a world whichalready has beenextended
there
are
no
more
applications
of Type 2
bya doubleline (so that there are newworldscreatedoff
that branch),then if this branchcontainsthe L-formulas rules in a given world can one proceed to
apply the Type3 rules. (Each of the new worlds
LB
copyB1, B2.... Brn into each of the
1, LB2,...LBm,
introduced by an application of the WorldCreating
newworldsthat the L-formuladominates.
Rulesis independentof the other, and each of themis
(as it were)self-contained, oncethe Type3 Interworld
Applicationof the Type1, 2, and3 rules:
Theorderof applicationof the rules in the classical CopyRules are applied. This meansthat all of the
logic madenological difference--anyorderof application branchesbelowthe double-finescan be investigated in
wouldyield the samefinal answer(valid/’mvalid)as any paralleL)
After the Type 2 Rules have been applied in a
other. This is no longer the case, because of the
given
possible world, one can apply the Type3 rules,
introductionof differing possibleworldswithinthe tree
copying
the de-modalizedformula into each of the
structure. Intuitively, the addition of a double-line
already-existing
possible worlds dominatedby the Lbranch to an already-existing branch amounts to
formula. Whena L-formulahas had each of the relevant
constructing a newpossible world accessible from the
Type3 rules applied to it [in someof the morecomplex
world in whichwe are currently working. Giventhis
logics a formulacan satisfy the preconditionsof more
understanding
of the double-linebranch,it is clear that
than one Type3 rule, and each of them needs to be
we need somerestriction on howformulas above the
applied] then it can be checkedoff. (Althoughagain we
doubleline can be interact with those belowthe double
line. For instance, werethe formula--,A to be found employa different type of checkmarkthan with the
Type1 rules). After each of the L-formulashas been
abovethe double-line branch(that is, in the earlier
possible world), the appearanceof A belowthe double- dealt withby all the Type3 rules that are relevantto it,
one is in a position to moveto a different possible
line branch (in the newpossible world) should not
worldanddeal with the formnl~sthat are in that world
allowus to cancelthat branch.Ourruling will be that
by constructing another tree--this time within the new
no formulaabovethe double-line branchcan interact
possibleworld. In this newworld,oneagain starts with
130
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
the Type1 rules, then moveson to the Type2 rules,
This requirementcan be satisfied in the following
manner3:After applyingall the rules as indicated for
system K (thereby constructing newworlds by Type
Theclosingof trees, andof worlds:
rules and copying formulas into them by the Type3
Within each world, one uses the World-Bound rules) also copyover the followingformulainto these
Rules. If all branchesof the tree that represents a
newworldsw:
possible world close, then the double-line into that
(B-formula): M(A1& A2 & .... &
world (and hence out of the previous world) closes
wherethe Ai are all the formulason the branchleading
indicatingthat it wasnot a possibleworldafter all. If
to worldwthat do not havethe check-offmarksrelevant
any one of the double-line branches leading outward to the Type1 rules. (So they mayhavebeencheckedoff
from a branchwithin somepossible world closes, then by Type2 or 3 rules, but not by Type1 rules). This
that leaf is markedas closed. (Notethat this requires new rule will be applied to each of the worlds
only that one of the many double-line branches
constructedin the earlier world.Theeffect of this will
emanatingfroma leaf needs to close in order to close be to place the B-formulainto each world j that is
the leaf. Intuitively, from the point of viewof the
constructedfromworldi. But then it followsthat when
initial possibleworld,the branchasserts the existence branchesin worldj are expanded,it is guaranteedthat
of such-and-sopossibleworlds.If it turns out that one one of the worldsthat can be reachedwill be the one
of themdoes not exist, then that initial branch is
initialiT~d by the de-modalized
B-formula.However,
that
impossible- regardless of what occurs in the other is just worldi - and thus wehavesimulatedsymmetry.
possible worlds.)
As an example we will show that the B axiom is
The goal remains as in the classical logic:
validatedby this rule:
determinewhetherthe tree that represents the initial
world contains an open branch. If it does, then the
~(p---->LMp)
argument
is invalid; otherwiseit is valid.
I
P
TableauxRules for B and G: The preceding set of
--,LMp
rules describes system K. All and only the valid
I
arguments of K will be declared so by the above
M~Mp
tableaux method.Andsince the methodoperates only
II
by decompositionof formulasinto subformulas(it is
II
analytic),
the algorithm
will detect this in a finite time.
from Type2 rule
-~Mp
So it is a decision procedure. As mentionedat the
I
outset, the point of this paperis to describehowto add
M(p&M--dVlp)from B rule
tableaux rules for complexrules such as B and G.2 We
I
will see that the resulting methodis not analytic. The
L~p
tableauxrules for both B andGare newType3 rules.
II
II
TableauxRule for B:
p&M~Mp fromType2 rule
Thesyntactic axiomfor B says that if a formulais
~p
from Type3 rule
true in a world,thenit is necessarythat it be possible.
P
Thatis, if it is actually
true thenit is possiblytrue in
M~Mp
every accessible world. The semantic condition on
x
accessibility for B is symmetry.But in a tableaux
setting this raises seriousdifficulties since it apparently Thecontradiction of p and ~p in the third world makes
implies that whenwecreate a newworld froma Type2
that world impossible, whichgets "pushedback" into
rule, and then in that newworld wish to create yet
the secondworld, into the only branchof that world.
another newworld, that the newlycreated worldhas to
This showsthat the second world is impossible, and
either be or somehowinclude the initial starting
this gets "pushedback"into the onlybranchof the first
world...otherwise it would not be a symmetric
world. Thus the fh-st world is impossible and the
accessibilityrelation.
etc.
2 Afull versionof this papercontainstableauxrules for
eachof the six axioms,and hencefor the 21 different
systemsindicatedin Figs. 1 and2.
131
3 Thegeneral theorembeing appealedto here is known
as the Bulldozer Theorem: any formula that is
unsatisf’mble in a (possible worlds) modelwherethe
accessibility relation is reflexive or symmetricor
transitive is also unsatisfiable in an h-reflexive,
asymmetric,and non-transitive model. See Segerberg
(1971:80ff).
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
argumentis valid: (p-~LMp)is validated by the
tableaux method. Note that the B-formula is not
necessarily a subformulaof any formulain the tree up
to the point of its introduction.So this tableauxmethod
for systemscontainingthe B axiomis not analytic. But
since the B-formulacan be algorithmeticallyconstructed
fromthe formulasin the tree, wemightcall the system
semi-analytic.
TableauxRule for G:
Thesyntactic axiomfor Gsays that if a formulais
possiblynecessary,then it is necessarilyposs~le.That
is, if it is necessaryin any accessible world,then it
must be possible in every accessible world. The
semanticrule for Ois incestuality: if world1 leads to
worlds2 and3, then there is a world4 that both 2 and3
lead into. Theeffect of this relation canbe gottenby the
followingmethod.Weconsider any tree whichhas been
constructed so as to have newworlds Wl, w2.... Wn
alreadyin existenceat the end of somebranch.In these
newworldsweapply all the relevant Type1 rules. But
nowbefore applyingwhateverType2 rules mightapply
in these worlds,weapplythis newsort of Type3 rule:
If LAi, LBi... are formulasin an openbranch of
wi, and LAj, LBj... are formul~in an openbranch
of wj, then add M(Ai&Bi&...&Aj&Bj&...)
to the
end of those open branchesin each of wi and wj.
Dothis for all openbranchesin eachof the pairs of
worldsof Wl,w2.... Wn.
Notethat each pair of worldsin Wl, w2.... Wnare now
guaranteedto havea common
descendant,and that this
descendantwill havetrue all the formulasthat were
necessarily true in its parents. So wehavesimulated
incest,~lity. A simple example is to prove the G
axiom:
~(MLp~LMp)
I
MLp
--~LMp
M~Mp
//
type2 rule:
the Grule:
type2 rule:
\\
Lp
--drip :type 2 rule
I
L~p
I
I
M(p&~p)M(p&~p):G
II
II
p&~p:type 2
p&~p
I
I
P
P
--,p
~p
X
tiffs is the only branchin eachof these worlds,they too
are each impossible, and this "pushes back" the
impossibility into the initial world. (Asmentioned,it
is only necessary that one of the two double-line
branchesexiting fromthe initial worldbe impossible.)
This showsthat the only branchin the initial world
closes, and thus that the argument is valid...the
unnegatedformulais a theoremin G.
Onceagain wenote that this methodis not analytic
becausethe formulasconstructedby the G rule is not
necessarily subformnh.~of anything in the tree up to
that point. Yet since the relevant formulas can be
algorithmicallyconstructedfromthe tree, wecan call it
a semi-analyticsystemof proof.
Acknowledgements:
The research reported here was
supported in part by the Canadian NSERC
grant OPG
5525. Theauthor is also a member
of the Institute for
Robotics and Intelligent Systems and wishes to
acknowledge
the support of the Networksof Centres of
Excellence Program by NSERC,the Governmentof
Cawd~and the participation of PRECARN
Associates.
References:
Bergmann,M., J. Moor,J. Nelson (1980) The Logic
Book (McGraw-Hill).
Beth, F. (1952)Symbolic
Logic. (North-Holland).
Beth, F. (1958) "OnMachineswhichProve Theorems",
in Siekmann& Wrightsonpp. 79-90.
Bonevae,D. (1987) Deduction(MayfieldPub. Co.).
Chellas, B. (1980) ModalLogic (CambridgeUniv.
Press).
Fitting, M. (1983) Proof Methods for Modal and
lntuitionistic Logics(Reidel).
Fitting, M. (1988) "First Order ModalTableaux"
Journal of AutomatedReasoningpp. 191-213.
Garson, M. (1988) "Quantified ModalLogic" in
Gabbay& F. GuentlmerHandbookof Philosophical
Logic Vol 2.
Gentzen, G. (1934/5) "Investigations into Logical
Deduction". Translation printed in M. SzaboThe
Collected Papers of Gerhard Gentzen (NorthHolland)1969,pp. 68-131.
Jeffrey, R. (1968) FormalLogic: Its Scopeand Limits
(McGraw-HtU).
Segerberg, K. (1971) An Essay on Classical Modal
Logic.(Uppsala:FilosofiskaStudier).
Siekmann,J. & Wrightson,G. (1983) The Automation
of Reasoning,Vol. 1. Springer-Verlag.
Smullyan, R. (1968) First Order Logic. (Springer-
vcr~g).
X
The closure of at least one of the two worldsat the
bottomof the tree will "pushback"the impossibilityof
that worldinto the precedingworld. Thuseach of the
previousworlds(the oneswhichare initialized with Lp
andwith -drip) will get an x at their bottom.Andsince
132
From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
Figure I: 21 normalmodalsystems. Arrowsindicate propexinclusion.
conditions on R
(.~
¯
(.)
¯
t’~
(-)
¯
(:,~
(./
¢.~
¯
t.~
¯
¯
¯
(.)
(q
C-)
(.)
(-)
¯
(’)
(-)
..
(q
¢.~
fq
¯
.
.
¯
¯
(-)
(-)
¯
TABLE
1: The semantic requirements on R for 21 normal modal
systems.¯ = a required property, (-) = property entailed
required properties. KB4and KT5are shownwith different
(equivalent) combinations
of required vs. entailed properties.
133