H o w to Represent Opaque Sentences in F i r s t O r d e r Logic
Bijan Arbab
I B M Corporation
Los Angeles Scientific Center
2525 Colorado Ave. 3rd floor
Santa Monica, California
90406
Abstract
T h i s p a p e r presents a m e t h o d f o r a p p l y i n g
s t a n d a r d inferencing mechanisms to a broader
class o f sentences t h a n t h a t w h i c h was p o s s i ble before.
T h e l o g i c of proposition surrogates
allows representation of a n d reasoning w i t h a
class of sentences ( t h e so c a l l e d o p a q u e sentences) t h a t pose s p e c i a l d i f f i c u l t i e s f o r s t a n d a r d logics.
W i t h i n t h i s class are sentences
w i t h one o r m o r e o c c u r r e n c e s o f such w o r d s a s
know, believe, aware, search, hunt, etc.
It is shown that standard formal (programm i n g ) l a n g u a g e s , e.g. f i r s t o r d e r l o g i c , c a n b e
extended w i t h p r o p o s i t i o n surrogates to deal
w i t h f a c t s t h a t have t r a d i t i o n a l l y been expressed i n m o d a l o r v a r i o u s o t h e r p r o p o s e d l o g ics. I t has been a r g u e d t h a t such f a c t s c a n n o t
a d e q u a t e l y b e expressed i n s t a n d a r d logics; t h e
f i n d i n g s a n d r e s u l t s r e c o r d e d h e r e , h o w e v e r , are
to the contrary. P r o p o s i t i o n surrogates can be
added, in a conservative manner, to s t a n d a r d
a u t o m a t i c reasoning systems.
P r o p o s i t i o n s u r r o g a t e s a n d t h e i r h i s t o r i c a l dev e l o p m e n t are p r e s e n t e d . A n i n f e r e n c e e n g i n e
based o n t h e logic o f p r o p o s i t i o n s u r r o g a t e s i s
t h e n o u t l i n e d a n d a p p l i e d t o some p r o b l e m s i n
t h i s area.
1
Introduction
In Be griff sschrift, a f o r m u l a l a n g u a g e f o r p u r e t h o u g h t
m o d e l e d u p o n t h a t o f a r i t h m e t i c , G o t t l o b Frege ( 1 8 7 9 )
takes i d e n t i t y t o b e a r e l a t i o n b e t w e e n n a m e s o r signs o f
objects:
substitution in functions
A n u m b e r of t h e o r e m p r o v e r s 1 t h a t are based on such
logics i m p l e m e n t t h e a x i o m s o f e q u a l i t y e i t h e r d i r e c t l y o r
b y v a r i o u s m e t h o d s a n d rules o f i n f e r e n c e . I t i s because
of the axioms of equality t h a t certain conclusions can
f o l l o w f r o m p r e m i s e s . F o r e x a m p l e i f i t i s t r u e t h a t eight
is greater t h a n five
(1)
a n d t h a t t h e a t o m i c n u m b e r o f o x y g e n i s eight
(2)
i t c a n t h e n b e c o n c l u d e d , o n t h e bases o f 1 , 2 a n d t h e
axioms of equality, t h a t :
(3)
w h i c h t r u t h f u l l y expresses t h e p r o p o s i t i o n t h a t t h e
atomic n u m b e r of oxygen is greater t h a n five. In the
p a p e r t i t l e d On Sense and Denotation, Frege ( 1 8 9 2 ) expresses d i s s a t i s f a c t i o n w i t h h i s e a r l i e r choice o f t h e i d e n t i t y r e l a t i o n . He explaines in d e t a i l why a name cannot
a l w a y s b e r e p l a c e d b y a n o t h e r o f t h e same t r u t h - v a l u e o r
content, in the view of the invariance of the t r u t h of the
w h o l e sentence. I t i s o f course a s s u m e d t h a t d e c l a r a t i v e
sentences d e n o t e a t r u t h value ( e i t h e r t r u e o r false) a n d
express a p r o p o s i t i o n ( t h e o b j e c t i v e c o n t e n t w h i c h is capable of being the c o m m o n property of m a n y ) , j u s t as
names have a d e n o t a t i o n (the p a r t i c u l a r object n a m e d )
a n d a sense ( t h e m a n n e r a n d c o n t e x t o f p r e s e n t a t i o n ) .
A j d u k i e w i c z ( 1 9 6 7 ) i l l u s t r a t e s t h e same p o i n t w i t h t h e
following example:
If it is true t h a t N e w t o n knew t h a t eight is greater
than five
(4)
m e a n t h a t the sign A and the sign B have the
same conceptual content, so that we can everywhere put B for A and conversely.
T h i s i s t h e basic i d e a b e h i n d t h e a x i o m s o f e q u a l i t y w h i c h
are a s s u m e d b y v a r i o u s systems o f l o g i c . F o l l o w i n g i s a n
explicit list of the axioms of equality: reflexivity x = x,
s y m m e t r y x = y —> y — x, t r a n s i t i v i t y x = y^y = z—>
x = z, s u b s t i t u t i o n in p r e d i c a t e s
t h e n i t c a n b e c o n c l u d e d , o n t h e basis o f 2 , 4 a n d t h e
axioms of equality that:
(5)
w h i c h i s c e r t a i n l y n o t t r u e since i t expresses t h e p r o p o s i tion that N e w t o n knew that the atomic number of oxygen i s g r e a t e r t h a n f i v e ( a f a c t w h i c h was b e y o n d his k e n ) .
1
For example, see the work of Wos and Robinson (1969)
and more recently D i g r i c o l i and Harrison (1986).
458
Automated Deduction
How is i t , then, that a sound system of logic admits false
conclusions based on true premises and standard rules
of inference?
According to the terminology of Church (1983), this
problem is called the parados? of the name relation. A
number of radically different solutions have been proposed to solve the paradox of the name relation. Following are various contrasting views whence the source of
the problem lies and how it should be solved.
2
Philosophical Views
2.1
Sense a n d D e n o t a t i o n
Frege's (1892) solution to this paradox revolves around
the idea that names, sentences, or signs have associated
w i t h them a sense (the proposition expressed) which is
no less relevant than the denotation. He also identifies
three different contexts, ordinary, direct, and indirect,
in which names can be used. In an ordinary context,
names have their customary denotation and sense. The
direct context is what is now known as the use-mention
distinction: words name (denote) other words 3 . In an
indirect context, names denote their customary sense,
not their customary denotation, and have an indirect
sense which is different than their customary sense.
The paradox is resolved since formula 4 is about the
customary sense of the number eight, not its customary
denotation, and formula 2 is about the customary denotation of the number eight, not its customary sense.
Therefore, formula 2 does not warrant the substitution
of f{o) for 8 in formula 4. Frege did not present a formulation, similar to that provided in Begriffsschrift, for
the logic of sense and denotation.
Church presents three different alternatives under
which a formulation of the logic of sense and denotation
can be carried out. The three alternatives-Alt (2), Alt(1),
Alt(0}~ correspond to different sets of assumptions under which two sentences can be considered to have the
same sense or express the same proposition. That two
sentences S and 51 have the same sense if and only if
S ~ S1 is logically valid is called Alt(S). A stronger
criterion of identity between senses, Alt(l), is that S is
convertible to 51 according to the rules of lambda calculus. The strongest criterion of identity between senses,
Alt(O), is that 5 and 51 differ at most by one or more
alphabetic changes of bound variable, or one or more
interchanges of synonymous notations. Two names are
synonymous if they have the same denotation as well as
the same sense.
A sound system of axioms characterizing two of these
alternatives, Alt(2) and Alt(l), has been specified by
Church (1973, 1987). M c C a r t h y (1979) also presented
a first order theory of individual concepts and propositions based on Frege's solution 4 . Formulation of the logic
2
Carnap (1956) used the word antinomy, but the word
paradox is preferable since no apparent contradiction occurs
in the absence of any further assumptions.
3
In writing, quotation marks or italics are used for direct
contexts.
4
It differs, however, from Frege's solution in that the latter
calls for an infinite hiearchy of senses where as the former
of sense and denotation remains open under Alt(O), the
strongest alternative under which two sentences can be
considered to express the same proposition. The solution presented in this paper is under Alt(O), however, it
differs f r o m the logic of sense and denotation.
2.2
Contextual Descriptions
Russell's (1905) 5 solution to the paradox eliminates
names altogether from the language, and introduces contextual descriptions. The relevant distinction is that contextual descriptions have no meaning of their own; however, every sentence in which they occur has a meaning.
It was commonly believed that the theory of contextual
description can be used to resolve the paradox of the
name relation as well as other paradoxes. By providing
counter examples Church (1983) demonstrates that, if
intensionality is to be avoided, then the theory of contextual descriptions cannot be adopted as a solution to
the paradox of the name relation. Contextual descriptions, however, remain useful for solving a variety of
other problems.
2.S
Nonclassical Logics
The notion of possible worlds has recently received a lot
of attention f r o m philosophers because it can be used to
provide an analysis of necessity and possibility. More recently it has also been applied to propositional attitudes
such as believing and knowing. A number of different
modal logics based on the possible world models have
been proposed.
There are disagreements, however, among philosophers regarding the nature of these possible worlds.
Some say that possible worlds combine the actual world
w i t h other worlds that contain only things similar to
those in the actual world. Others say that a possible
world is described by a set of propositions, such that
each proposition or its negation is a member of the set.
Some of the modal logics based on possible world semantics unnecessarily commit the agents to be what Hintikka (1975) called logically omniscient6. The strongest
objection to nonclassical logics is the lack of efficient i n ferencing mechanisms. Construction of efficient inference engines for modal logics must also address the computational complexities of logics that are based on the
possible world models.
2.4
Proposition Surrogates
This paper presents a modification, Arbab (1988), of the
solution first proposed by Ajdukiewicz (1960) and later
formalized by Church (1983). The solution follows the
allows only a single level. An infinite array of senses is called
for since various levels of indirection (Pat knows that Newton
knew that ...) can easily be formed.
5
In 1903, Russell had outlined a different solution to the
paradox of the name relation. Russell (1905), however, flatly
states that the Russell (1903) solution is very similar to
Frege (1892), and both are shown to be unsatisfactory. The
particular line of reasoning presented by Russell (1905) remains unclear to this author!
6
An exception to this is Church's (1951) formulation of
the logic of and sense and denotation under Alt(2) which is
also based on the possible world models.
Arbab
459
philosophy t h a t there is n o t h i n g inherently w r o n g w i t h
the u n d e r l y i n g logic (either the rules of inference or the
axioms of e q u a l i t y ) ; therefore, it is unnecessary to con­
s t r u c t new logics, e.g., m o d a l l o g i c s , o r t o a b a n d o n ( o r
weaken) t h e a x i o m s o f e q u a l i t y . T h e source o f t h e p r o b ­
l e m lies i n h o w n a t u r a l l a n g u a g e sentences are t o b e f o r ­
mulated in the f o r m a l language. In short, if well-formed
f o r m u l a s c o r r e s p o n d i n g t o E n g l i s h sentences are w r i t t e n
c o r r e c t l y , t h e n p a r a d o x i c a l c o n c l u s i o n s w i l l n o t arise. F o r
e x a m p l e , 4 is n o t t h e c o r r e c t r e p r e s e n t a t i o n of t h e f a c t
t h a t N e w t o n k n e w t h a t e i g h t i s g r e a t e r t h a n f i v e , since
i t leads t o p a r a d o x i c a l c o n c l u s i o n s . W h a t a r e , t h e n , t h e
well-formed formulas corresponding to n a t u r a l language
sentences?
T h e a n s w e r i s based o n t h e i d e a t h a t sentences d e n o t e
t r u t h - v a l u e s a n d express p r o p o s i t i o n s . T h e p r o p o s i t i o n
expressed by a sentence is, of course, i n d e p e n d e n t
of
the p a r t i c u l a r n a t u r a l language in which it happens to
b e w r i t t e n a n d c a n b e expressed b y sentences i n differ­
ent l a n g u a g e s . For e x a m p l e , t h e t w o sentences eight is
greater than five, in E n g l i s h , a n d Acht ist groBer als funf,
i n G e r m a n , b o t h express t h e p r o p o s i t i o n t h a t e i g h t i s
greater t h a n five. One m e t h o d of encoding propositions
w i t h i n a f o r m a l language is presented in this paper. T h e
e n c o d i n g s are c a l l e d proposition surrogates, since in t h e
f o r m a l l a n g u a g e t h e y p l a y t h e r o l e o f t h e p r o p o s i t i o n ex­
pressed b y a sentence. T h e a l g o r i t h m f o r c o n s t r u c t i n g
p r o p o s i t i o n surrogates can t h e n be added to any f o r m a l
l a n g u a g e . I n t h i s p a p e r a n i n f e r e n c e e n g i n e based o n t h e
proposition surrogate solution is presented and applied
t o some e x a m p l e s .
3
where /, o, gt, a n d 5 correspond, respectively, to the
a t o m i c n u m b e r o f e l e m e n t s , t h e c o n s t a n t oxygen, t h e r e ­
l a t i o n greater than, a n d t h e c o n s t a n t f i v e . T h e o r d e r e d n t u p l e o c c u r r i n g i n t h e second p o s i t i o n o f t h e knew p r e d ­
icate is a p r o p o s i t i o n surrogate. I t s first m e m b e r corre­
s p o n d s t o t h e f o r m o f t h e f o r m u l a , a n d t h e rest c o n t a i n
the p r i m i t i v e constants, function, and predicate symbols
t h a t o c c u r i n t h e f o r m u l a . F o r every about clause i n
,
there is an appropriate constant, f u n c t i o n , or predicate
name in the proposition surrogate.
T h e f o r m u l a w h i c h c o r r e s p o n d s t o a p r o p o s i t i o n sur­
r o g a t e can b e o b t a i n e d b y a p p l y i n g t h e f i r s t m e m b e r o f
t h e p r o p o s i t i o n s u r r o g a t e w h i c h i s a l w a y s a l a m b d a ex­
p r e s s i o n c o n t a i n i n g t h e p a r t i c u l a r form o f t h e f o r m u l a
t o t h e rest o f t h e m e m b e r s w h i c h are a l w a y s t h e p a r ­
t i c u l a r constants o c c u r r i n g i n t h e f o r m u l a . T h e f o r m u l a
w h i c h c o r r e s p o n d s t o t h e above p r o p o s i t i o n s u r r o g a t e
is gt(f(o), 5), w h i c h expresses t h e p r o p o s i t i o n t h a t t h e
atomic n u m b e r of oxygen is greater t h a n five.
T h e second m e a n i n g , Ω, of 6 can i n f o r m a l l y be s t a t e d
as f o l l o w s :
Newton knew
about the atomic n u m b e r of oxygen,
a b o u t the r e l a t i o n greater t h a n ,
about the n u m b e r five,
that the atomic number of oxygen is
greater t h a n five.
Sentence 6 c a n be f o r m u l a t e d on t h e basis of Ω as f o l l o w s :
(8)
Solution
A j d u k i e w i c z ( 1 9 6 0 , 1967) argues t h a t i f sentences are
i n a n i n d i r e c t c o n t e x t , t h e n t h e y are a m b i g u o u s . I t i s
p r i m a r i l y t h i s a m b i g u i t y t h a t leads t o p a r a d o x i c a l c o n ­
clusions. T h e r e are a t least t w o d i f f e r e n t m e a n i n g s t h a t
can be a t t r i b u t e d to s u c h sentences. L e t us c a l l these _
a n d Ω . T h e p a r a d o x arises because w e u n d e r s t a n d t h e
sentence o n t h e basis o f
b u t f o r m u l a t e i t o n t h e basis
o f Ω . T h e s o l u t i o n , t h e n , i s t o f o r m u l a t e t h e sentence
a c c o r d i n g t o t h e u n d e r s t o o d m e a n i n g . For e x a m p l e , t h e
first m e a n i n g ,
, of t h e sentence
Newton
knew
the atomic number of oxygen is greater than
that
five
(6)
c a n i n f o r m a l l y be s t a t e d as f o l l o w s :
T h e d i s t i n c t i o n between the two meanings can now be
m a d e clear b y e x a m i n i n g t h e d i f f e r e n t n u m b e r o f about
clauses. A c c o r d i n g t o
, Newton knew about the atomic
n u m b e r o f e l e m e n t s [ / ] , o x y g e n [o], a n d t h a t t h e value
of the atomic n u m b e r of oxygen [/(o)] is greater t h a n
five. If sentence 6 is u n d e r s t o o d a c c o r d i n g to
, then its
t r u t h value i s f a l s e h o o d , since k n o w l e d g e o f t h e a t o m i c
n u m b e r o f elements [ / ] can n o t b e a t t r i b u t e d t o N e w t o n . Sentence 6 u n d e r s t o o d a c c o r d i n g to Ω, h o w e v e r , is
t r u e . T h e second m e a n i n g , Ω, of 6 does n o t a t t r i b u t e
explicit knowledge of the a t o m i c n u m b e r of elements [/]
o r o x y g e n [o] t o N e w t o n : t h e y d o n o t i n d i v i d u a l l y a p ­
pear a m o n g the p r i m i t i v e constants of the p r o p o s i t i o n
surrogate.
T h e difficulty w i t h this solution to the p a r a d o x of the
n a m e r e l a t i o n lies i n t h e w a y p r i m i t i v e c o n s t a n t s o f t h e
f o r m a l i z e d l a n g u a g e are h a n d l e d : a n e q u a l i t y r e l a t i o n
between p r i m i t i v e constants of the formalized language
can b e u s e d t o r e i n t r o d u c e t h e p a r a d o x . C o n s i d e r , f o r
e x a m p l e , t h e t w o sentences:
Newton knew
a b o u t the a t o m i c n u m b e r of elements,
about oxygen,
about the relation greater t h a n ,
about the number five,
that the atomic n u m b e r of oxygen is
greater t h a n five.
J o h n b e l i e v e d t h a t D r J e k y l l was a g e n t l e m a n
Sentence 6 c a n be f o r m u l a t e d on t h e basis of $ as f o l l o w s :
Dr Jekyll is Mr Hyde
(9)
(10)
Sentence 9 is f o r m a l i z e d u n d e r A l t ( O ) as:
(7)
The existence of propositions are not effected by the fact
that they can or can not be expressed in a particular n a t u r a l
or formal language.
460
Automated Deduction
(11)
w h e r e John, gent, a n d dj are p r i m i t i v e c o n s t a n t s of t h e
language corresponding to J o h n , gentleman, and the Dr
J c k y l l respectively.
u s u a l w a y as:
Sentence 10, i s f o r m a l i z e d i n t h e
(12)
In f o r m u l a 1 1 , it is possible to s u b s t i t u t e rah f o r dj on
t h e basis o f t h e a x i o m o f e q u a l i t y a n d f o r m u l a 12, t h u s
a r r i v i n g at the conclusion:
(13)
which corresponds to the paradoxical conclusion:
J o h n b e l i e v e d t h a t M r H y d e was a g e n t l e m a n
(14)
An extensional solution to the p r o b l e m of p r i m i t i v e
constants is o u t l i n e d below. T h e idea of pointers to cons t a n t s (address o f a p a r t i c u l a r cell w i t h i n t h e m e m o r y o f
a c o m p u t e r ) a n d the associated operators ( o b t a i n i n g the
address a n d d e - r e f e r e n c i n g ) i s w e l l - k n o w n i n t h e f i e l d o f
c o m p u t e r science. T h e a n a l o g y 8 b e t w e e n p o i n t e r s a n d
w h a t Frege (1892) called t h e sense of a n a m e can be
used t o c o n s t r u c t a s o l u t i o n t o t h e p r o b l e m o f p r i m i t i v e
constants of proposition surrogates.
After a while,
that
wisest announces
that his spot
• T h a t each wise m a n k n o w s t h a t t h e r e is at least one
w h i t e spot is expressed by
N o t e t h a t | P J is s h o r t h a n d for t h e p r o p o s i t i o n
surrogate of P under A l t ( 0 ) obtained according to
t h e a l g o r i t h m p r e s e n t e d i n t h e last s e c t i o n .
The
c o r r e s p o n d i n g P r o l o g clause is:
C h u r c h ' s 1983 a l g o r i t h m f o r o b t a i n i n g t h e p r o p o s i t i o n
s u r r o g a t e u n d e r A l t ( 0 ) i s m o d i f i e d s o t h a t every o c c u r rence of a p r i m i t i v e c o n s t a n t , say c, in the p r o p o s i t i o n
s u r r o g a t e is r e p l a c e d by @c ( t h e p a r t i c u l a r p o i n t e r to c ) ;
a n d every o c c u r r e n c e o f a b o u n d v a r i a b l e i n t h e b o d y o f
the l a m b d a t e r m corresponding to the p r i m i t i v e constant
c, is r e p l a c e d by a p p l i c a t i o n of t h e + ( d e - r e f e r e n c i n g )
o p e r a t o r t o t h a t v a r i a b l e . For e x a m p l e , 9 w i l l n o w b e
f o r m a l i z e d u n d e r A l t ( 0 ) as:
a n d a l t h o u g h dj — rnh, it does n o t f o l l o w
Thus, the paradoxical conclusion:
the
is white. How does he know?9
T h e s o l u t i o n t o t h i s p u z z l e requires a wise m a n t o r e a s o n
a b o u t w h a t o t h e r wise m e n k n o w a n d d o n o t k n o w , f r o m
observations and the king's announcements. T h e puzzle
solved here is a c t u a l l y a s i m p l i f i e d v e r s i o n of t h e o r i g i n a l
p u z z l e . T h e s i m p l i f y i n g a s s u m p t i o n s arc t h a t t h e r e are
o n l y t w o wise m e n , a n d t h a t a f t e r some t i m e t h e f i r s t
wise m a n announces t h a t h e c a n n o t t e l l t h e c o l o r o f h i s
s p o t , w h e r e u p o n t h e second wise m a n says his o w n s p o t
is w h i t e . T h e f o l l o w i n g is a p a r t i a l list of a f o r m u l a t i o n of
the p u z z l e i n f i r s t - o r d e r logic, a n d P r o l o g m o d i f i e d w i t h
p r o p o s i t i o n surrogates ( t h e c o m p l e t e list c a n b e f o u n d i n
A r b a b (1988)).
N o t e t h a t ps(S, P) is t r u e i f f P is t h e p r o p o s i t i o n
s u r r o g a t e of S. A l s o , if P is u n b o u n d t h e n t h e P r o l o g i n t e r p r e t e r w i l l c o m p u t e the p r o p o s i t i o n s u r r o gate of S a n d b i n d it to P.
T h a t each wise m a n k n o w s t h a t t h e o t h e r wise m a n
k n o w s t h a t there is at least one w h i t e s p o t , is expressed b y
(15)
—
T h e c o r r e s p o n d i n g P r o l o g clause is:
4
Example
L e t us assume, t h e n , t h e a v a i l a b i l i t y of a t w o place p r e d i c a t e ps(SyP) such t h a t P is t h e p r o p o s i t i o n s u r r o g a t e
c o r r e s p o n d i n g t o S u n d e r A l t ( 0 ) , for t h e exact d e t a i l s
a n d t h e P r o l o g code see A r b a b ( 1 9 8 8 ) . T h i s e x a m p l e i s
called t h e The Wise Man Puzzle a n d has b e e n used to
test t h e r e p r e s e n t a t i o n a l a b i l i t y o f f o r m a l i s m s for k n o w l edge r e p r e s e n t a t i o n .
A certain King wishes to determine which of his
three wise men is the wisest. He arranges them
in a circle so that they can see and hear each
other and tells them that he will put a white or
black spot on each of their foreheads, hut that
at least one spot will be white. In fact, all three
spots are white. He then offers his favor to the
one who will first tell him the color of his spot.
Notice the doubly-indirect c o n t e x t - thus the two
calls to t h e ps p r e d i c a t e .
• T h a t A does n o t k n o w t h a t he has a w h i t e s p o t ,
since he m a d e an a n n o u n c e m e n t to t h i s effect, is
expressed by
(19)
T h e c o r r e s p o n d i n g P r o l o g clause is:
non(know(a,
• T h a t D k n o w s t h a t A does n o t k n o w t h a t he has
a w h i t e s p o t , since B h e a r d A's a n n o u n c e m e n t , is
expressed b y
T h e c o r r e s p o n d i n g P r o l o g clause is:
8
This is not to suggest that Frege's (1892) sense of a name
is simply a pointer to t h a t name, only that pointers can play
the logical role of the sense of a name.
w(a))).
9
T h i s puzzle has been a t t r i b u t e d , by Konolige (1986), to
an unpublished note by M c C a r t h y .
Arbab
461
6
Acknowledgement
T h i s w o r k was c a r r i e d o u t a t U C L A u n d e r t h e s u p e r v i s i o n o f P r o f e s s o r S t o t t P a r k e r o f t h e C o m p u t e r Science
D e p a r t m e n t , a n d Professor A l o n z o C h u r c h o f t h e P h i losophy and M a t h e m a t i c s Departments. T h e author is
g r a t e f u l f o r t h e i r c o n t r i b u t i o n s o f ideas a n d c o r r e c t i o n s
m a d e over t h e years. Needless t o say, a l l r e m a i n i n g err o r s are a t t r i b u t a b l e solely t o t h e a u t h o r .
References
5
Conclusion
I t was s h o w n t h a t t h e c o m m o n i n t e r p r e t a t i o n o f t h e i d e n t i t y r e l a t i o n , i.e., a r e l a t i o n i n v o l v i n g o n l y d e n o t a t i o n o f
n a m e s , leads t o t h e p a r a d o x o f t h e n a m e r e l a t i o n . T h i s
is the source of inconsistencies w h e n a classical set of
a x i o m s a n d rules o f i n f e r e n c e are a s s u m e d b y t h e f o r m a l
l a n g u a g e . T h e p h i l o s o p h i c a l p o i n t o f v i e w d e f e n d e d here
is t h a t the e l i m i n a t i o n of the paradox of the name relat i o n r e q u i r e s n e i t h e r a m o d i f i c a t i o n o f t h e classical set o f
axioms, n o r of the rules of inference.
A n i n f e r e n c e e n g i n e based o n t h e l o g i c o f p r o p o s i t i o n
s u r r o g a t e s c a n p l a y a n i m p o r t a n t role i n t h e f i e l d o f m a c h i n e i n t e l l i g e n c e , f o r i f a m a c h i n e i s ever t o i n t e r a c t i n t e l l i g e n t l y w i t h o t h e r agents, m a c h i n e s o r h u m a n s , t h e n
i t m u s t b e a b l e t o r e p r e s e n t a n d reason w i t h f a c t s a b o u t
t h e agents' s t a t e o f m i n d . A n a g e n t ' s s t a t e o f m i n d , o f
course, i n c l u d e s , b u t i s n o t l i m i t e d t o , f a c t s a b o u t i t s
k n o w l e d g e , beliefs, awareness, a n d e x p e c t a t i o n s .
T h e l o g i c o f p r o p o s i t i o n s u r r o g a t e s c a n b e used t o r e p resent a n d r e a s o n w i t h such f a c t s . A n i n f e r e n c e engine
based o n t h e logic o f p r o p o s i t i o n s u r r o g a t e s enables a
m a c h i n e n o t o n l y t o r e p r e s e n t , b u t also t o discover, l o g i c a l consequences o f s u c h f a c t s .
462
Automated Deduction
[1]
K a z i m i e r z A j d u k i c w i c z . I n t e n s i o n a l expressions. I n
J e r z y G i e d y m i n , e d i t o r , Kaztmierz Ajdukicwicz The
scientific
world-perspective
and
other essays
19311963, pages 3 2 0 - 3 4 7 , D . R e i d e l P u b l i s h i n g C o m p a n y , 1967.
[2]
Kazimierz Ajdukicwicz.
A m e t h o d of e l i m i n a t i n g
i n t e n s i o n a l sentences a n d s e n t e n t i a l f o r m u l a e .
In
Atti
del
XII
Congresso
Internazional
di
Filosofia,
pages 1 7 - 2 4 , 1960.
[3]
Bijan Arbab.
A Formal Language for Representation of and Reasoning about Indirect Context.
PhD
thesis, U n i v e r s i t y o f C a l i f o r n i a a t Los A n g e l e s , 1988.
[4]
Rudolf Carnap.
Meaning and Necessity. T h e U n i v e r s i t y o f C h i c a g o Press, second e d i t i o n , 1956.
[5]
Alonzo C h u r c h . Intensionality a n d the paradox of
the name relation.
In Themes from Kaplan, S t a n f o r d Press, 1988. T h e c o n t e n t o f t h i s p a p e r was p r e sented a s a n i n v i t e d l e c t u r e a t a j o i n t s y m p o s i u m o f
the A.P.S. and the Association for Symbolic Logic
i n B e r k e l e y , C a l i f o r n i a , M a r c h 1983.
[6]
V. J. Digricoli and M. C. Harrison. Equality-based
binary resolution.
Journal of the ACM, 3 3 ( 2 ) : 2 5 3 289, 1986.
[7] G o t t l o b Prcge. B e g r i f f s s c h r i f t , a f o r m a l l a n g u a g e ,
modeled u p o n that of arithmatic, for pure thought.
In J e a n v a n H e i j e n o o r t , e d i t o r , From Frege to Godel,
pages 1 - 8 2 , H a r v a r d U n i v e r s i t y Press, 1977. o r i g i n a l l y p u b l i s h e d i n 1879.
[8] G o t t l o b Frege.
O n sense a n d m e a n i n g .
In
A . P . M a r t i n i c h , editor,
The Philosophy of Language,
pages 2 1 2 - 2 2 0 , O x f o r d U n i v e r s i t y Press, 1985. o r i g i n a l l y p u b l i s h e d i n 1892.
[9] J . H i n t i k k a . I m p o s s i b l e p o s s i b l e w o r l d s v i n d i c a t e d .
Journal of Philosophical Logic, 4 : 4 7 5 - 4 8 4 , 1975.
[10]
K u r t K o n o l i g e . A Deduction Model of Belief.
gan K a u f m a n n P u b l i s h e r s , 1986.
Mor-
[11] J o h n M c C a r t h y . F i r s t o r d e r t h e o r i e s o f i n d i v i d u a l
concepts and propositions. In B. M e l t z e r a n d D.
M i c h i e , e d i t o r s , Machine Intelligence 9, pages 1 2 0 147, E l l i s H o r w o o d , 1979.
[12]
B e r t r a n d Russell.
On d e n o t i n g .
Mind, 1 4 : 4 7 9 - 4 9 3 ,
1905.
[13] L a r r y A . W o s a n d G . A . R o b i n s o n . P a r a m o d u l a t i o n a n d t h e o r e m p r o v i n g i n 1st o r d e r theories w i t h
e q u a l i t y . I n B . M e l t z e r a n d D . M i c h i e , e d i t o r s , Machine Intelligence 4, pages 1 3 5 - 1 5 0 , E d i n b u r g h U n i v e r s i t y Press, E d i n b u r g h , S c o t l a n d , 1969.