THE REPRESENTATION OF INCONSISTENT INFORMATION IN A DYNAMIC MODEL-THEORETIC SEMANTICS 
Douglas B. Moran 
Department of Computer Science 
Oregon State University 
Corvallis, Oregon 97331 
ABSTRACT 
Model-theoretic semantics provides a 
computationally attractive means of representing 
the semantics of natural language. However, the 
models used in this formalism are static and are 
usually infinite. Dynamic models are incomplete 
models that include only the information needed for 
an application and to which information can be 
added. Dynamic models are basically approximations 
of larger conventional models, but differ is 
several interesting ways. 
The difference discussed here is the 
possibility of inconsistent information being 
included in the model. If a computation causes the 
model to expand, the result of that computation may 
be different than the result of performing that 
same computation with respect to the newly expanded 
model (i.e. the result is inconsistent with the 
information currently in the dynamic model). 
Mechanisms are introduced to eliminate these local 
(temporary) inconsistencies, but the most natural 
mechanism can introduce permanent inconsistencies 
in the information contained in the dynamic model. 
These inconsistencies are similar to those that 
people have in their knowledge and beliefs. The 
mechanism presented is shown to be related to both 
the intensional isomorphism and impossible worlds 
approaches to thi~ problem. 
I. INTRODUCTION 
In model-theoretic semantics, the semantics of 
a sentence is represented with a logical formula, 
and its meaning is the result of evaluating that 
formula with respect to a logical model. The 
model-theoretic semantics used here is that given 
in The proper treatment of quantification in 
ordinar~ English (PTQ) \[Montague 1973\], but the 
problems and results discussed here apply to 
similar systems and theories. 
From the viewpoint of natural language 
understanding, the conventional ~oO~l-theoretic 
semantics used in descriptive theories has two 
basic problems: (I) the information contained in a 
mod~ is complete and unchanging whereas the 
information possessed by a person listening to an 
utterance is incomplete and may be changed by the 
understanding of that utterance, and (2) the models 
are usually presumed to be infinite, whereas a 
person possesses only finite information. Dynamic 
model-theoretic semantics \[Friedman, Warren, and 
Moran 1978, 1979; Moran 1980\] addresses these 
problems by allowing the models to contain 
incomplete information and to have information 
added to the model. A dynamic model is a "good 
enough" approximation to an infinite model when it 
contains the finite subset of information that is 
needed to determine the meanings of the sentences 
actually presented to the system. 
Dynamic model-theoretic semantics allows the 
evaluation of a formula to cause the addition of 
information to the model. This interaction of the 
evaluation of a formula and the expansion of the 
model produces several linguistically interesting 
side-effects, and these have been labelled 
model-theoretic pra~matics \[Moran 19~0\]. 
One of these effects occurs when the 
information given by an element of the model is 
expanded between the time when that element is 
identified as the denotation of a sub-expression in 
the formula and the time when it is used in 
combination with other elements. If the expansion 
of the model is not properly managed, the result of 
the evaluation of such a formula can be wrong 
(i.e. inconsistent with the contents of the model). 
Two mechanisms for maintaining the correctness of 
the denotational relationship are presented. In 
the first, the management of the relationship is 
external to the model. This mechanism has the 
disadvantage that it involves high overhead - the 
denotational relationships must be repeatedly 
verified, and unnecessary expansions of the model 
may be performed. The second mechanism is similar 
to the first, but eliminates much of this overhead: 
it incorporates the management of the denotational 
relationship into the model by augmenting the 
model's structure. 
It is this second mechanism that is of primary 
interest. It was added to the system to eliminate 
a source of immediate errors, but it was found to 
introduce long-term "errors". These errors are 
interesting because they are the kinds of errors 
that people frequently make. The structure added 
to the model permits it to contain inconsistent 
pieces of information (the structure of a 
conventional model prevents this), and the 
mechanism provides a motivated means for 
controllin~ which inconsistencies may and may not 
be entered into the dynamic model. 
An important subclass of the inconsistencies 
provided by this mechanism are known as intensional 
16 
substitution failure and this mechanism can be 
viewed as a variant of both the "impossible" worlds 
\[e.g. Cresswell 1973: 39-41\] and the intenslonal 
isomorphism \[e.g. Lewis 1972\] approaches. Since 
intensionality alone does not provide an account 
for Intensional substitution failure, this 
mechanism provides an improved account of 
propositional attitudes. 
Finding the argument to which the ~-expression 
is applied before evaluating the ~-expression is 
not a viable solution for two reasons. First, some 
h-expressions are not applied to arguments, but 
they have the same problem with their denotations 
changing as the model expands. Second, having to 
find the argument to which a h-expression is 
applied eliminates one of the system's major 
advantages, compositionality. 
II. THE PROBLEM 
Dynamic models contain incomplete information, 
and the sets, relations, and functions in these 
models can be incompletely specified (their domains 
are usually incomplete). In PTQ, some phrases 
translate to ~-expressions; other ~-expressions are 
used to combine and reorder subexpressions. The 
possible denotations of these ~-expressions are the 
higher-order elements of the model (sets, 
relations, and functions). For example, the proper 
name "John" translates to the logical expression 
(omitting intensionality for the time being): 
(I) \[~ P P(j)\] 
where P ranges over properties of individuals and 
has as its denotation the set of properties that 
John has. The sentence "John talks" translates to: 
(2) \[~ P P(j)\](talk) 
This formula evaluates to true or false depending 
on whether or not the property that is the 
denotation of "talk" is in the set of properties 
that John has. 
The dynamic model that is used to evaluate (2) 
may not contain the element that is the denotation 
of "talk". If so, a problem ensues. If the 
formula is evaluated left-to-right, the set of 
properties denoted by the ~ -expression is 
identified, followed by the evaluation of "talk". 
This forces the model to expand to contain the 
property of talking. The addition of this new 
property expands the domain of the set of 
properties denoted by "John", thus forcing the 
expansion of the characteristic function of that 
set to specify whether or not talking is to be 
included. However, because the relationship 
between the Z-expression for "John" and the set of 
properties denoted is maintained only during the 
evaluation of the ~-expression (there is no link 
from the denotation back to the expression that it 
denotes), there are no restrictions on how the set 
is to be expanded. Thus, it is possible to define 
the property of talking to have John talking and to 
expand the set previously identified as being 
denoted by "John" to not include talking, or vice 
versa. If such an expansion were made, the 
inconsistency would exist only in the evaluation of 
that particular formula, and not in the model. 
Subsequent evaluations of the sentence would 
recompute the denotation of "John" and get the 
correct set of properties. 
This is not a problem with the direction of 
evaluation - the argument to which the ~-expression 
is applied may occur to the left of that 
~-expression, for example: 
(3) \[R R R(talk)\](AP P(j)) 
(note: (3) is equivalent to (2) above). 
III. THE FIRST MECHANISM - EXTERNAL MANAGEMENT 
The mechanism that evaluates a formula with 
respect to a model has been augmented with a table 
that contains each ~-expression and the ima6e of 
its denotation in the current stage of the dynamic 
model. When the domain of the ~-expression 
expands, the correct denotational relationship is 
maintained by expanding the image in the table 
using the ~-expression, and then finding the 
corresponding element in the model. If the element 
in the model that was the denotation of the 
h-expression was not expanded in the same way as 
the image in this table, a new element 
corresponding to the expanded image is added to the 
model. This table allows two ~-expressions that 
initially have the same denotation to have 
different denotations after the model expands. 
Since the expansion of elements in the model is 
undirected, an element that was initially the 
denotation of a ~-expression may expand into an 
unused element. The accumulation of unused 
elements and the repeated comparisions of images in 
the table to elements in the model frequently 
imposes a high overhead. 
IV. THE SECOND MECHANISM - AUGMENTING THE MODEL 
The second mechanism for maintaining the 
correctness of the denotations of ~-expressions 
basically involves incorporating the table from the 
first mechanism into the model. In effect, the 
R-expressions become meanin6ful names for the 
elements that they denote. These meaningful names 
are then used to restrict the expansion of the 
named elements; once an element has been identified 
as the denotation of a ~-expresslon, it remains its 
denotation.* 
In the first mechanism, when the domain of two 
~-expressions does not contain any of the elements 
that distinguish them, they will have the same 
denotation, and when such a distinguishing element 
is added to the model, the denotations of the two 
h-expressions will become different. With 
meaningful names, this is not possible because the 
denotational relationship between a h-expression 
* Meaningful names are also useful for other 
purposes, such as generating sentences from the 
information in the model and for providing 
procedural - rather than declarative - 
representations for the information in the model 
\[Moran 1980\]. 
17 
end its denotation in the model is permanent. 
Since the system cannot anticipate how the model 
will be expanded, if it is possible to add to the 
domain of two h-expresslons an element that would 
distinguish their denotations, those expressions 
must be treated as having distinct denotations. 
Thus, all and only the logically-equivalent 
expressions should be identified as having the same 
denotation. If two equivalent expressions were not 
so identified, their denotations would be different 
elements in the model and this would allow them to 
be treated differently. For example, if "John and 
Mary" was not identified to be the same as "Mary 
and John", it would be possible to have the model 
contain the inconsistent information that "John and 
Mary talk" is true and that "Mary and John talk" is 
false. If two non-equivalent ~-expressions were 
identified as being equivalent, they would have the 
same element as their denotation. When an element 
that would distinguish the denotations of these two 
expressions was added to the model, the expansion 
of the element that was serving as both their 
denotations would be incorrect for one of them and 
thus introduce an inconsistency. 
This need to correctly identl~y equivalent 
expressions presents a problem because even within 
the subset of expressions that are the translations 
of English phrases in the PTQ fragment, equivalence 
is undecldable \[Warren 1979\]. It is this 
undecidability that is the basis of the 
introduction of inconsistencies into the model. To 
be useful in a natural language understanding 
system, this mechanism needs to have timely 
determinations of whether or not two expressions 
are equivalent, and thus it will use techniques 
(including heuristics) that will produce false 
answers for some pairs of expressions. It is the 
collection of techniques that is used that 
determines which inconsistencies will and will not 
be admitted into the model.* 
V. PROPOSITIONAL ATTITUDES AND 
INTENSIONAL SUBSTITUTIONAL FAILURE 
Intensional substitution failure occurs when 
one has different beliefs about intensionally- 
equivalent propositions. For example, all theorems 
are intenslonally-equlvalent (each is true in all 
possible worlds), but it is possible to believe one 
proposition that is a theorem and not believe 
another. The techniques used by the second 
mechanism to identify logically-equivalent formulas 
can be viewed as similar to Carnap's Intensional 
isomorphism approach in that it is based on finding 
equivalences between the constituents and the 
structures of the expressions being compared. This 
mechanism can also be viewed as using an 
* While the fragment of English used in PTQ is 
large enough to demonstrate the introduction of 
inconsistent information, it is viewed as not being 
large enough to permit interesting claims about 
what are useful techniques for testing 
equivalences. Consequently, this part of the 
mechanism has not been implemented. 
"impossible" worlds approach: if two 
intensionally-equivalent formulas are not 
identified as being equivalent, the mechanism 
"thinks" that it is possible to expand their domain 
to include a distinguishing element. Since the 
formulas are equivalent in all possible worlds, the 
expected distinguishing element must be an 
"impossible" world. 
The presence of intensional substitution 
failure is one of the important tests of a theory 
of propositional attitudes. This mechanism is a 
correlate of that of Thomason \[1980\], with the 
addition of meaningful names to intensional objects 
serving the same purpose as Thomason's additional 
layer of types. 
VI. REFERENCES 
Cresswell, M. J. (1973) Logic and Languages, 
Methuen and Company, London. 
Friedman, J., D. Moran, and D. Warren (1978) "An 
interpretation system for Montague grammar", 
American Journal for Computational Linguistics, 
microfiche 74, 23-96. 
Friedman, J., D. Moran, and D. Warren (1979) 
"Dynamic interpretations", Computer Studies in 
Formal Linguistics report N-16, Dept. of Computer 
and Communication Sciences, The University of 
Michigan; earlier version presented to the October 
1978 Sloan Foundation Workshop on Formal Semantics 
at Stanford University. 
Lewis, D. (1972) "General semantics", in 
D. Davidson and G. Harman (eds.) (1972) Semantics 
of Natural Language, D. Reidel, Dordrecht, 169-218; 
reprinted in B. H. Partee (ed.) (1976) Monta6ue 
Grammar, Academic Press, New York, 1-50. 
Montague, R. (1973) "The proper treatment of 
quantification in ordinary English" (PTQ), in 
J. Hintikka, J. Moravesik, and P. Suppes (eds.) 
(1973) Approaches to Natural Language, D. Reidel, 
Dordrecht, 221-242; reprinted in R. Montague (1974) 
Formal Philosophy: Selected Papers of Richard 
Monta~ue, edited and with an introduction by 
Richmond Thomason, Yale University Press, New 
Haven, 247-270. 
Moran, D. (1980) Model-Theoretic Pra~matics: 
Dynamic Models and an Application to Presupposition 
and Implicature, Ph.D. dissertation, Computer 
Studies in Formal Linguistics, Dept. of" Computer 
and Communication Sciences, The University of 
Michigan. 
Thomason, R. H. (1980) "A model theo~ for 
propositional attitudes", Linguistics and 
Philosophy, 4, I 47-70. 
Narren, D. (1979) Syntax and Semantics in Parsin%: 
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