From: AAAI-83 Proceedings. Copyright ©1983, AAAI (www.aaai.org). All rights reserved.
DEFAULT REASONING USING MONOTONIC LOGIC:
A Modest Proposal
Jane Terry Nutter
Department of Computer Science, Tulane University
This paper presents, a simple extension of
rodicatEflog+c to include a.default
first order
inference overning the
Piules
operator.
operator are s ecified, and a mo%el theory for
interpretin sen? ences involvin default operators
is develo e% based on standard 4 arskian semantics.
The resus ting system is trivial1 sound. It is
es an adequate
argued that (a) this logic proviar
basis for default reasoning in A.I. s stems, and
is purpose,
(b) unlike most lo its proposed for t i!
retains the v1 rtues of standard first order
Egic , including both monotonicity and simplicity.
Reasoning from incomplete information and from
which
atterns
default, generalizations follows
;s?Fctd
f+rst order predicate Po ic does not
devia,ion
i;;;lv;t
!
l?he most sttz;;ing
conclusions
making ' inferences
counterindicated by further information which does
previously
contradict anything
nc&explicitly
.
In standard logics, if a set of premises
entails a conclusion, aneY,tt;;r set containing all
?
that conclusion.
also
those premises
roperty are called monotonic.
Logics with this
The above departure Prom standard logic's reasoning
patterns has led researchers to adopt and dev;t;zg
non-monotonic logics for use in A.I. systems
Mcf;gtott and Doyle,a;9!JOigaMgCDermott,
1982;
; Duda et al.,
Aronson et
i%Tier
19713;&nd others).
Critics of non-monotonic logics have p;ignStd;d
out technical weaknesses (see e.g. Davis,
that itd
and Israel (19UO) argues persuasive1
sup orters confuse logic with complex ju5 gments of
Dut unless some
kin% s logic cannot deal with.
with default
for dealing
alternative appears
reasoning, non-monotonic logic must continue to
appeal.
I have argued elsewhere that (1) default
reasonin only appears non-monotonic if we fail to
distin u ?sh warranted assertions from warranted
assump9 ions (gi;y~, 19821, and (2) an log;zs;hi;;
reasonYng
default
adequately
monotonic (Nutter, 1983).
If this is accepted, a conservative approa:;
promises more than do radical ones, not because
1s
"safer", bu;apurusa it simultaneously preserves
its
what is of
in standard lo ic -simplicity, clarity, and ade uacy 4n domains of
complete information -- and aP lows distinguishing
warranted assumptio2;omfrom warranted assertions
universals,
(generalizations
def;E
consequences from genuines inferences, etc.).
pa er presents an al~~~iEtrivha~chBxtenslon
of first
distinguishes
or%er
predicate
generalizatii;; and the resulting resumptions from
and permits
their
universals
reasoning from default
resultin proposition is- obviously related to its
componen8 in an interesting way, but its truth
value is not a function of the component's.
This paper describes how to extend standard
first order predicate logic to a logic appro ;;;te
for default reasoning. This involves four P
t:e
(1) We must characterize the grammar of
extended langua e. That is, we must give rules for
f
determining whe her a particular string of symbols
re resents a pro osition in the extended lan uage.
2
(L
4 We must expP ain how to extend the dtn&;;e
system of the original logic so that all
r
desirable inferences will be legal.
descruEtthe seyh;ycs for the lan uage:'3'th%? m?:
of
th;!
9
the it:;rpre ations
say
the
values
how
are,
Enguage
inter retations are determined, and ho?logical
(4)
we
should
Finally
entaiPment is defined.
investi at; ;;a,least,brief;i the me;;;: ;c of the
;I
.
That
extende%
ourselves tK at the new exte6deiesystem is %%:
conclusions
to
be derived
it never permits false
from true premises. We now take up these tasks.
III
w
The first component of a formal logic ;; its
the
rammar, which determines the langua e
be defined as either e he set of
4 ogic, which ma
formulas or tEe set of propositions which are
considered well formed. Formulas are distinguished
variable
from propositFhyt by containin open
occurrences,
lS, placehol% ers which could
either be replaced b specific individual constants
ound by an existential or a
-- names -- or be 1:
universal quantifier -- turned into an 8 uivalent
of either "something" or "everything" -- % ut which
297
at present
present the
our logic.
In this section, we
occur unbound.
rules of grammar for the language of
,, written "v ", "if...then...",
written "A " P "or':
written "*".,and "if and on1 if", written "z") and
both quantrfiers ("for alY 'I, written "#+"I and
This will be the least
"there is", written M3").
altered portion of the logic.
The technical discussions in ,the sections
below will
ways
that ~~~~9Po~~~c"~11~~~1~~gc~~~~d~h~ndu~~~~
presuppose a complete escription of a first order
predicate logic. The informal descri tions should
overview for readers wR o lack this
conve
familYar!?y and motivate some of the less obvious
decisions embodied in the technical discussions.
A.
rnfarmaf Descr&.U?n
The guarded status of default generalizations
inherited through inferences, and from a
~%d~~
proposition, it must be possible to infer a
guarded version of any conclusion which could be
inferred from the un uarded version of the premise.
That is, if you coul% infer from "Roger flies" t;M;
will find
Roger, he
sees
"If my
fascinatindo4 then from "Presumably Roger flies"
you may in?& "Presumably, if my dog see;loR;ge;,,
yt
will find him fascinating," and so,on.
generally takes more thatfone premise.to warrant ye
the premises may
inference. How many
eneralizations? There are several ways to go on
9 his.
The grammar holds relatively few surprises.
In sim le terms, wherever in English one might
reasonait
1 prefix a clause with "There is reason to
believe T hat" or some similar construction, the
logic will ;llow its e u,;Fuptntof the clause to be
(for
operator p
governed
OUT
%
presumably"Y .
The only real decision involves whether to let
ropositions.
the operator govern formulas or only
While its use governing formulas !f. probabl
eliminable, it seems to me both innocuo6usand we1Y
motivated.
While "Birds
i3hl; f%%"l
becomes Ya(Bird(a) * p % E,e23
logic,
robabl
comes to *the same thin as "In
eneral !airds fP y;lo; Ya (Bird(a)) 3 Flies?a))l,,Iifn
9he long run,
R are said. Furthermore,
anything is a bird,
then it has wings
and
resumabl flies" must be considerably reformulated
g efore iY can be equivalent1 expressed without
having p bind a formula inszead of a sentence.
Hence p may directly govern either a formula or a
sentence.
B.
We could allow only one guarded premise per
inference. But suppose we have "In general, if A
then 8," and we have "There is reason to believe
that A." We would normally feel free to infer
"There is reason to believe that B." But this
uarded premises, and without
inference has two
further information, zhere is no wa to avoid usin
both guarded premises in a sing1 e inference an%
still get the conclusion.
contributing to it.
to give not only a guard, but also a sort of
certaint measure. Unfortunately, it's a very poor
measure Yor anything interesting. Here's why.
Let L be a standard first order predicate
calculus without equality and without function
SymbOlC3~with a Jaskow7p3i$tyle natural deduction
system
(Jaskowski,
'
containin explicit
introduction and elimination rule; for tl?
e standard
small set of
connectives and quantifiers
"bookkeeping“ rules. (The precise f&ma1 system is
unimportant.) The langua e of L is L, the logic
with defaults which we wi i!
1 build from L is LD, and
its language is CD.
The default operator is
written p.
Second, a single guarded roposition might be
used five or six times in tR e course of a long
Is the result any less certain for
derivation.
having that proposition enter man times ra;;th;
than once? B the time a chain of inYerences
taken place, T he number of ps on the front ma
disY in::
the number of
in reaching it.
&%%I~
p.
~%%v%%d
Throughout
the
discussion
below,
Roman
ca itals var over formulas and propositions of L,
wh rle tireekYetters Q and 0 var over formulas and
ropositions of LD, and a varf:
es of variables of
ED.
Def. The set of formulas of LD is the smallest set
satisfying the following clauses:
Third, there is no reason to sup ose that all
the original eneralizations are equaP ly reliable.
Five extreme9y strong generalizations probabl
x
provideIfbe~'gerc~yyr~ss;h,"" w,o=h~~l~':,";;~c~y=g
one.
probabilities -- we would be
reliability -- i.e.
dealing with statist cal generalizations and not
with defaults anyhow.
(a) All atomic formulas of L are formulas of
LD;
of
(b) For any formula 9
LD;
(c) For any formulas
are all formulas of LD:
9;
4
of LD,
O,UJ
-4;
p4
is a
of LD, the following
4 v *;
Oh*;
43
Furthermore, from "Generally, if A than B" and
"There is reason to believe that A," we don't
conclude "There is reason to believe that there is
reason to believe that B." We conclude that there
is reason to believe B. Stringing out ps does not
If
reflect an practice resent in English usage.
of
that pracY ice playe% a role in the kind
reasonin
we are
l~~gu~~~e~~uldltresleeecm
Seasonab9e to su pose??z?
I have tR erefore decided to "inherit" the
$&d
if any or all remises involved it, without
re ard to how many. 5 his is the force of the first
ru9e of inference below.
E 0.
(d) For any formula 4 of LD in which a
gpen , the following are also formulas of LD:
3a
formula
occurs
Ya 4;
4.
For any formula 4 of LD, (0 is a proposition of
LD if and only if 4 contains no open occurrences of
variables.
IV
-VI.
Even with the rule in that form, it would be
The
possible to generate long strings of
second rule says that if there are more tR in one
consecutive qualifications, they may be collapsed
into a single one.
.
.'SYS'&,M
As commented above. the deductive system is a
standard natural deduction system, with
the usual
five connectives
("not", written ,l*M
,
"and",
298
Devw
B.
We retain all the standard rules of inference
in L (generalizing them to allow for sentences of
LD and not just of L), and add the following two
rules.
rtpx: Su pose that UJ = {@l,...,*n) r LD, e' =
{WI' ,...,vn's t LD, 4 t LD and for all I, 1 i 1 5
n, vi' = oi or Qi' = pUri. Sup ose further that
erring 4 from Y.
some rule of inference warrants in!i
Then from rlr*you may infer po.
RpE:
From
pp4
you may infer
p4.
Given 4 s LD, UJ e LD, we sa that ry is
writI!
en 4 + ur)
provable from 4 (or 4 roves 1(1,
rovided that 01 can be lf
@rived from remises in 4
!ollowing the rules of inference in Phe usual and
obvious
least true for false V.
way.
This is the portion of the logic which deals
with issues of "meaning" and truth. The sense of
meaning involved is extremely primitive. For the
non-logical terms, it reflects only the function of
words as referring
to objects,
completely ne letting nat"oiZ?~Aov"rhey refer, but
even to wha? they refer in ordinary English.
Formal semantics are formal: they deal with what
can be said about inheritance of truth conditions
on the basis
of logical form alone, which throws
out the overwhelming ma ‘ority
of what we would
ordinarily mean when we ta 2k about meaning.
The semantics iven below re resents the most
complete departure 9rom standard Pogic in LD.
The
standard core of the system retains the familiar
semantic properties, but across the portion of LD
which is not in L,,the logic is three-valued. This
can be represented in man ways; I have chosen to
represent the truth vaY ues as non-em ty subsets
(instead of elements) of the set {t,f) oP familiar
truth values. Intuitively, (t) means "on1 true",
pd occurs where in ordinary ltgic one wou,,d
Y expect
similarly
(f) means
only false .
"Gewcomer" is {t,fa;dw",',"\can best be thought %
or both true and false.
*~~ke~w??$hi,y
LD p;ovides an intriguing face
for lovers of radical ;;;artures: it remains sound
traditional
law
of
while
violating
non-contradiction.
A.
&a&a&&g
that e(4) = A.
In particular, having defined the evaluation
function~ over ~ atomic Rroposttions. of LD in the
~K&Y;~F
(exce t for t e sub,titution of if) and
and ), we use the following induction
clauses:
.
lruth of !&es-
What does it mean for a default enoralization
to be true?
It does not mean that 9he statement
inside the p is true.
It does not even mean that
there is no reason to believe that it is false:
means something like
"There is reason pt"o
gePig:e that 4 is true but <here is also reason to
is
believe
that it is 'false." That sentence
consistent if unhelpful.
Part of what we want our default operator to
mean which cannot be modeled b
formal semantics:
neither c",u,w~i~y nort typicaYity is a
formal
state
certain
semantic
constraints on Row our p op~~at~~works.
An true proposition is evidence for itself.
(If we xnew that 4 is true, thenT;t w~~l~ccertainl
cannoT
willing to suppose it.
distinguish the truth value
of p4 %epending on
whether we know it or not.) Hence if 4 is true,
then p4 should be at least true. If 4 and ~1,are
both, only true, the;= clr~~ly .thefr conjunction
if either is only
??lzid b~he%?Yc~~ction
should
'
be only false.
and
~a~s;wiseSi~~la;on]unction sh;i;g b;oot;hrue
.
comments
other
connectives.
be
1.
For all 4 e LD, if 4 has the form pA for
L, then e(A) 2;e(4).
2.
For all 4 e LD, if 4 has the form ppv for some
0, s LD, then e(o) = e(pur).
3.
For a;: tj 0, s LD, we have
%a = {f) if e(4)=(t);
{tl if e(4)={f);
{t,f) if e(4)={t,f1.
et4
A
0)
=
if) if t 6?e(4)
or t d e(u);
ttl if f e!et41
and f d s(v);
(t,f> otherwise.
et4
v
rlr)
=
{f) if t e! e(4)
and t s!e(o);
{t) if f d e(4)
or f d et*);
{t,f) otherwise.
A e
e(4 =a 0) = <f) if f (re(4)
and t d e(u);
{tl if t d e(4)
or f 4! e(u);
(t,f) otherwise.
et4
299
f -4)
=
if et41 n et*) = A;
{tl if e(4) = e(v) = {t)
or e(4) = e(uu) = {f);
{t,f) otherwise.
{f)
will not constitute such a system:
tool for the system to use.
e('da 4) = if) if there is a u c U
such that e(4:u/a) = (f);
ttl ~~4four/(xa:l~
u{;, U,
I
;
{t,f) otherwise.
The logic presented here makes no pretense to
philoso hical depth.
As a logic, it is trivial;
as a pR ilosophical view ofos~i~l~lization. it is
perhaps the shallowest
.
But it can be
used to form non-triviaP systems modeling deep
views. From one point of view, that is what logic
is for.
e(Zla 4) = tf) ie;4fo;,~:l=u~fG>
U,
ttl if there is a G & U
such that e(4:u/a) = it);
{t,f) otherwise.
4.
instead it is a
VIII
If there is a 4 c ED such that e(e) = if) but
t E e(po), then for all UJ E: LD, f e a(~*).
m
. ‘2 2 '.*
Many thanks to Stuart Sha iro and to the SNePS
Research tiroupfor their many Relpful comments and
suggestions. This research was carried out in the
Department of Cornuter Science, State University of
New York at Buffas o, Amherst, New York.
(Notational remark: e(4:u/a) means the result of
evaluating 0, treating a as if it were a constant
and e(a) = u.)
The standard equivalences follow trivially
from these definitions. Now we define satisfaction
as follows: for all i = (U,e) an inter retation of
LD, and for all 4 c LD, i satisfies 4 Pwritten i (=
4) if and only if t s e(o).
REFERENCES
Cl1 Aronson, A. R., Jacobs,
B. E., and Minker, J.,
;'gnoQcoe3
on fuzzy deduction.'* JACH 27:4 (1980)
.
121 Davis,
reason
80.
As has been indicated from the outset, the
characteristic we are most concerned with for
im lementations
'c
logics in ~o~~h;~
lt'?Tthe % ollowing theor%
soundness.
follows trivially from i!
he soundness of standard
first order predicate logic:
133 Duda, R. O., Hart, P. E., Nillson, N. J., and
Sutherland, G. L., "Semantic network repre:entations in rule-based inference
Pattern - Directed Inference Systems,
systemsD IA"
Waterman and F. Ha es-Roth, editors, Academic
1?i78) pp. 203-223.
Press (New York,
Theorem The logic LL,is sound, i.e. for all UJ E;
LD,
4 c LD, if v k 4 then 0 b 4.
I41 Israel, D.J., "What's wron with non-monotonic
3" Proc. AAAI-80. Pa9 0 Alto, California,
;:;I%, 1980, pp. 99-101.
VII
A.
(51 Jaskowski, S
"On the rules of
formal logic:" Studia Logica 1 PY
of rem
[61 Martins, J. P., "Keasoning in Multi le Belief
Spaces," Technical Report 203, StaPe Universit
of New York at Buffalo, Amherst, New
Yorz , June 1983.
I73 McDermott, D. V. and Doyle, J., "Non-monotonic
logic I." Artif. Intell. 13:1-2 (1980) 41-72.
I81 McDermott, D., "Non-monotonic logic II." JACiY
29:l (19823 33-57.
I91 Nutter, J. T., "Defaults revisited, or "Tell
me if you're guessing." Proc. Cog. Sci. 4. Ann
Arbor, Michigan, August, 1982, pp. 67-69.
t103
OB+r
is imnortant not to overestimate what a
It
logic for defaiilt;f$yOying can do once we have
one.
As Israel
oints out, arriving at
conflicts
useful generalizations anii resolving
among conclusions of default reasonin both exceed
the sco e of logic. To those who won?er what help
the sysr em will be once it has deduced '34 A p--4",,
If the non-logical
the answer is, none at all.
ortion of our system contains heuristics
'
Phat conflrcting evidence indicates a neEzyf,",9
further investigation, and if your s stem further
ing, then the
contains some subsystem for investigaYcan
abilit to deduce statements of the kind above
be usex to trigger investigation; but this goes
beyond the logic alone.
It
does
not
such
a
osition in
Pi4).
As noteH;tlc;bove,the apf arently radoxicai
features
result
rom
truth-functional view of implication cat %%&ded
a deductive system based on relevance
g iyng
Such a
deductive system will
permit
.
subset of those
ins erences which are a
allowed under the tradition%"%iw presented here.
Martins ;;,983)b;;;eleveloped,avariant of relevance
which has been
logic
revision,
implemented in SNePS (Shapiro, 1979). An inference
s stem including default operators as described in
tK is paper is being im lemented on this belief
revision
Limitae.
ions of space prohibit
:;;'+;;,dis~%%?%~ here; for details, see Nutter
.
B.
"The mathematics of non-monotonic
Artif. Intell. 13:l-2 (1980) 73-
loaic is
300
Nutter,
J. T.!
"What else is.wrong with nonmonotonic logics? Represe;po;ional and informat ional shortcomings."
Cog. Sci. 5
Rochester, New York, May, 1983:
1111 Nutter, 9. T., "Default reasoning in A.I. systerns”
(1983b) draft.
Cl21 Reiter
Artif.
‘A&l “A
. 1
for default reasoning."
1980) 81-132.
1131 Shapiro, S., "The SNePS semantic network processing system" In Associative Networks,
N. V.
Findler, ed., Academic Press (New York, 1979)
pp. 791-'/96.