Mental Spaces from a Functional Perspective

COGNITIVE SCIENCE 11, 1-21 (1987)

Mental Spaces from a Functional Perspective

JOHN DINSMORE

Southern Illinois University at Carbondale

In his book Mentol Spaces, Fouconnier develops o powerful theory of human

knowledge representation and linguistic processing thot handles o variety of

problems in linguistics ond the philosophy of language in a simple, uniform, and

intuitively plausible way. However, he hos little to soy about the structure or

general role of mental spaces in cognition. The present paper proposes that men-

tal spaces ore a means of organizing knowledge in support of a general inference

method, simulofive reasoning, found in various guises both in logic and in Artifi-

cial Intelligence. The structuring required to fulfill this role allows us to make a

wide voriety of predictions which seem to be borne out by evidence from natural

language. In attributing o specific function to mental spaces, this paper suggests

that the theory of mentol spaces defines a potentiolly significant paradigm for

knowledge representation in Artificial Intelligence.

1. INTRODUCTION

In Fauconnier’s, at long last, published book Mental Spaces (Fauconnier, 1985), he describes a theory of human knowledge representation and linguistic processing that provides a simple and uniform account of a wide variety of problems that have long perplexed both linguists and philosophers of language. Among these problems are those of referential opacity and transparency, specificity of reference, definite reference in discourse, the interaction of quantifiers with modalities, the projection problem for presuppositions, the semantic processing of counterfactuals, and the use of comparatives in modal contexts.

Fauconnier’s theory is particularly important in its identification of the role of cognitive factors, especially principles for organizing knowledge and procedural strategies for semantic interpretation, in what is often loosely termed the “logic” of natural language. It is by now widely accepted that such cognitive factors are important in associating meanings to discourses. What is not so well recognized, the work of Johnson-Laird (1983) being a notable exception, is the importance of these factors in the semantic interpretation of lower-level structures like quantifiers and modalities. Accord- ingly, we find that accounts of these problems tend to rely heavily on ideas from formal logic like “semantic scope”, and abstract from cognition to treat language as a purely formal system. Fauconnier’s work, like that of

Correspondence and requests for reprints should be sent to the author at the Department of

Computer Science, Southern Illinois University at Carbondale, Carbondale, IL 62901.

2 DINSMORE

Johnson-Laird (1983), is a radical departure from that tradition. What is remarkable is how much more powerful and simple Fauconnier’s account of these problems is.

The cognitive model Fauconnier advocates is quite simple. Mental spaces are domains used for consolidating certain kinds of information. Such domains can represent, for instance, the world defined by a picture or a work of fiction, a particular person’s view of the world, a situation located in time and/or space, some individual’s hopes, a hypothetical or imagined situation, etc. Within spaces objects may be represented as existing and relations may be represented as holding between those objects, regardless of the status of those objects and relations in the real world.

Spaces are evoked and accumulate information during the processing of discourses, primarily in response to the occurrence of linguistic structures called space builders such as “Fred believes -, ” “In 1929 9-, ” “Ifs, then -,” and so forth. For instance, the processing of the sentence “Max believes that Susan hates Harry” involves either creating or locating a belief space for Max and representing the information that Susan hates Harry in that space. The processing of the sentence “In that movie, Clint Eastwood is a villain” involves adding the information that (the character played by) Clint Eastwood is a villain in that space.

For Fauconnier, spaces are relatively self-contained with relatively little external structure. However, objects in different spaces might be related by a connector, a particular psychologically salient relation, for instance that existing between Clint Eastwood and the character he plays in the movie space of the last paragraph. One significance of connectors for discourse semantics, formulated as the identity principle, is that if two objects a and b are related by a connector then a description of a can be used in referring to b. This is what allows us to refer to the character played by Clint Eastwood as “Clint Eastwood.” Aside from this, a principle of space optimization is used to inherit information from one space to another, and a special mentor relationship is posited to capture the strong dependence of hope spaces on belief spaces. These structures are justified only in intuitive terms.

The present paper does not presuppose familiarity with the details of Fau- cannier’s work but is nevertheless intended to complement and to strengthen it. The focus is on developing the external structure of spaces in terms of a specific conception of the function of mental spaces in representing and reasoning about knowledge. It will attribute a semantics tospaces, that is, it will tell us what it means for a given proposition to be represented as true in a specific space. And it will tell us how mental spaces support efficient reasoning. These functional considerations will allow us to make sense of much of the structure Fauconnier attributes intuitively to mental spaces and will lead to some proposals about human cognition. It will thereby significantly extend the range of linguistic phenomena the theory of mental spaces can

MENTAL SPACES 3

explain. Finally, it will integrate the idea of mental spaces into the concerns of knowledge representation in Artificial Intelligence.

2; THE FUNCTION OF MENTAL SPACES

It’s a happy occurrence when a cognitively plausible system of memory organization or knowledge representation finds functional motivation in its support of efficient reasoning. An instance of this is the schema, which was originally proposed as a mechanism of human cognition, but which has been found to have useful counterparts in the frames and scripts (e.g., Bobrow & Winograd, 1977; &hank & Abelson, 1977) in Artificial Intelligence systems. The thesis of this paper is that mental spaces are functionally motivated in their support of a general reasoning technique that will be called simulative reasoning. Simulative reasoning requires a partitioning of knowledge into distinct spaces and additionally assumes that the contents of each space effectively simulate or model a possible reality, or a part of a possible reality, and therefore represents a meaningful domain over which normal reasoning processes work. The results of localized simulative reasoning are meaningful because they lead to global consequences which can be made ultimately to bear on the real world.

Parenthetically, I should state that I make no general commitment here to the nature of “normal reasoning processes” in humans, for instance whether humans reason primarily by chaining through if-then rules or by building analogical models. What a normal rule of inference is understood to be is not particularly important, as long as we assume that they at least approximate criteria for logical consistency. Nevertheless, for expository purposes the discussion of simulative reasoning will use terms, such as “proposition,” and reference rules, such as modus ponens, borrowed from traditional logic,

2.1 Simulative Reasoning The efficiency of the method of simulative reasoning is realized in numerous AI systems that in one way or another partition knowledge into separate knowledge bases and employ normal rules of inference locally in individual knowledge bases, generally called spaces or contexts. For instance, a be/ief space is frequently used to consolidate the set of propositions some person believes to be true. As Creary (1979) points out, once a separate knowledge base has been set up for a particular person and any propositions explicitly known to be believed by that person are added to that knowledge base, the system can simulate the thinking of the agent by using its own reasoning facilities to derive further beliefs of the agent. The results are likely to re- flect the person’s belief system very closely. Cohen and Perrault (1979) show the usefulness of such knowledge bases in a system which infers the in-

4 DINSMORE

tentions of participants in natural language discourse in order to interpret their utterances and plan its own. Martins (1983) describes a further application in a system which coordinates planning among several people, for instance which in the planning of a meeting in which different participants have different beliefs relevant to an appropriate time to hold the meeting.

Spaces are also often used to support reasoning about hypothetical or potential situations. The CONNIVER (McDermott & Sussman, 1972) context mechanism is convenient in planning systems for modeling the results of alternative actions, any of which would result in a situation almost like the previous situation except for changes introduced by the action itself. It thereby allows alternative hypothetical actions to be reasoned about inde- pendently. A number of other knowledge representation systems support very general mechanisms for creating separate data bases for representing belief systems, fictional stories, times, and so forth. Among these are Hen- drix’ Partitioned Semantic Networks (Hendrix, 1979), KL-ONE (Brachman & Schmolze, 1985) and NETL (Fahlman, 1979).

Aside from the use of spaces and simulative reasoning in AI knowledge representation systems in Artificial Intelligence, these concepts are evident in the traditional deductive methods of symbolic logic. A conditionalproof is the deduction of an expression of the form “p-q” by first assumingp, in effect creating a space base in which p is represented as true along with the original givens, and showing that q can be deduced. A proof by reductio ad absurdurn of an expression p proceeds by first assuming -p, again in effect creating a space like the original, but this time with -p added, and showing that a contradiction results. Case analysis allows one to use a disjunction p V q in the deduction of any expression r by alternatively assuming p and assuming q and showing that r can be derived in either case. Finally, reasoning with universal quantifiers typically involves an application of universal instantiation to replace a quantified variable with a constant in some expression, thereby effectively creating a knowledge base with a new constant about which something is assumed to be given, and using the resulting expression to deduce an expression containing that constant. An application of universal generalization then replaces the constant in the deduced expression by a universally quantified variable, in effect translating the deduced expression into an expression in the original knowledge base. In each of these methods a localized deduction with a specially contrived space leads to the recognition of some global consequences which would be difficult to derive by other means.

Simulative reasoning is a highly efficient inference technique insofar as it treats difficult inferences over a potentially large set of complex propositions as relatively easy inferences over a small set of simple propositions. If a system that uses spaces knows a set of propositions to be true concerning George’s beliefs, “George believes that Fred’s phone number is 123-4567,”

MENTAL SPACES 5

“George believes that Mary has the same phone number as Fred,” and so forth, it will represent the objects of George’s beliefs in a belief space for George. This belief space will contain a highly constrained set of relatively simple propositions. Normal inferences can then be performed over this set. To continue the current example, the system can use the propositions we are assuming to be true in George’s belief space to infer locally that Mary’s phone number is 123-4567. This last inference with respect to George’s belief space corresponds exactly to the inference of the more complex global consequence that George believes that Mary’s phone number is 123-4567, true with respect to the real world. Dinsmore (1987) considers further implications of this example.

More explicitly, the power of simulative reasoning results from two factors. First, the information relevant to local reasoning is strongly delimited. Rather than considering the entire knowledge base in drawing inferences, only the propositions represented in a specific space are considered. This helps control combinatorial problems, well known in Artificial Intelligence.

Second, inferences themselves are made simpler, and accordingly require fewer deductive steps. Rather than inferring “George believes that Mary’s phone number is 123-4567” directly from “George believes that Fred’s phone number is 123-4567” and “George believes that Mary’s phone number is Fred’s phone number,” a system that makes effective use of spaces and simulative reasoning instead infers “Mary’s phone number is 123-4567” locally from “Fred’s phone number is 123-4567” and “Mary’s phone number is Fred’s phone number.” Notice that whereas the latter inference involves no more than a simple application of Substitution of Equals; the former inference over complex propositions will involve a large number of deductive steps, which will require the use of at least two meaning postulates about belief, roughly:

(believe(x,p) & believe(x,q))- believe(x, p KL q), (believe(x,p & (p-q))-believe(x,q).

In summary, the use of spaces in support of simulative reasoning is well- motivated functionally. But it will be appreciated that in all of these cases the success of the reasoning process depends crucially on two factors: The first is the ability to set up spaces and distribute knowledge over spaces appropriately in support of simulative reasoning. The second is the ability to recognize the global consequences that result from simulative reasoning. The rest of this section concerns these problems.

2.2 Partitioning Knowledge A result of simulative reasoning is a proposition inferred for a particular, relatively isolated space. This result is useful because it is based on propositions that in some sense belong in that space, and because it leads to con-

6 DINSMORE

elusions that have some applicability outside of that space. Both of these preconditions are satisfied if we assume that the meaning of a given space is a specification of what must be true in reality in order for a proposition to be true in that space. The truth of p in any given space thus corresponds to the truth of some proposition in the real world. For instance, p is true in George’s belief space because “George believes p” is true in the real world. The truth of the proposition that Moriarty has invented an air gun in Sherlock Holmes’ belief space corresponds to the truth of the proposition in the real world that in the Sherlock Holmes stories, Sherlock believes that Moriarty invented an air gun. This proposition is true in the real world because in reality A. Connan Doyle described Holmes claiming that Moriarty invented an air gun in writing one of the Sherlock Holmes stories. The meaning of a particular space will be called its context.

In every case the context of a space can be regarded as a propositional function, i.e., a function from propositions to propositions. For instance, the context of George’s belief space is a function that takes a proposition p and maps it onto the proposition that George believes p. In setting up a space we consolidate a set of propositions, each of which is appropriately mapped by the context of the space onto something presumed to be true in the real world. To recognize the global consequence of a new proposition derived in the space by simulative reasoning we use the same context to map the proposition onto something thereby inferred to be true in the real world.

Propositional functions correspond roughly to Fauconnier’s space builders. For readability, I will continue to use Fauconnier’s convention in referring to such functions by syntactic patterns in English or some form of logic containing “ -” as a place-holder for a propositional expression. For instance, the function that maps the proposition p onto the proposition that George believes p is represented “George believes that -. ” The function that maps the proposition p onto the proposition q-p is represented “q- -, ” and so forth.

Notice that since propositional functions are defined in semantic terms, the meanings of English sentences that don’t embed propositional expres- sions (sentences) can also be expressed’ partially in terms of propositional functions. For example, “George wants a hamburger” -does not embed a sentence. But this sentence has roughly the same meaning as “George wants it to be the case that he has a hamburger.” This last sentence makes the propositional function involved in the meaning of either sentence easier to recognize. The recognition of propositional functions is in terms of meaning, not syntactic form.

The real world, as Fauconnier suggests, is just another space. Accordingly, it is possible to relativize the idea of context somewhat. The truth of a proposition in the real world may correspond directly to the truth of a simpler proposition in a more localized space sl. It will be convenient to say that, recursively, the truth of a proposition in sl may correspond to the truth of a

MENTAL SPACES 7

simpler proposition in a still more localized space ~2. Further spaces can be similarly nested in ~2, and so forth.

For instance, the truth in the real world of the modal proposition that George believes Mabel has been dating Fred corresponds to the truth in George’s belief space, that Mabel has been dating Fred. The truth in George’s belief space of the proposition that if penguins ate blueberries they would be able to fly, corresponds to the truth of the proposition that penguins can fly in ~2, an “if penguins ate blueberries -” space which exists in George’s belief space. This last truth would also correspond to the truth in the real world of the doubly complex modal proposition that George believes that if penguins ate blueberries, they would be able to fly. Observe that in this scheme there is bound to be more than one belief space for George. George has a belief space in the real world which represents George’s actual beliefs. Likewise, George has a belief space in Fred’s belief space which represents the set of beliefs Fred attributes to George, correctly or incorrectly. A proposition p is true in this last space if Fred believes that George believes p.

In each case, the truth of a proposition p in s2 is systematically related to the truth of a propositionf(pl in sl, wherefis a propositional function. We will say that the propositional propositionfis the context of s2 relative to sl. This function expresses the criterion for truth in s2 in terms of truth in sl. It also suggests that any propositionp derived in ~2 can be used immediately in sl by using the contextfof s2 relative to sl to map it onto a proposition true in sl.

The process of distributing knowledge over spaces according to context will be called knowledge partitioning. For instance, the knowledge that George believes p is represented by the assertion of p in a “George believes that -” space. Knowledge partitioning seems to belong to the processes by which mental models are constructed in Johnson-Laird’s (1983) theory. The reciprocal process of using contexts to access knowledge, thereby real- izing global consequences of simulative reasoning, will be called context climbing. For instance, if 4 has been derived in George’s belief space relative to the real world, and is therefore represented there as true, then by context climbing the proposition that George believes that 4 is available as a true proposition in the real world.

In summary, contexts are propositional functions that relate the truth of some proposition in one space to the truth of some proposition in another space (ultimately in the real world). They are necessary to represent the meaning of a space, and to make the results of simulative reasoning available to other spaces (ultimately to the real world). Contexts thereby specify what it means for a proposition to be true in a specific space.

2.3 Restrictions on Contexts It is of crucial importance to recognize that not every propositional function can be a legal context. The reason is that context climbing would lead to er-

8 DINSMORE

roneous global consequences in many cases. Suppose that a propositional function f is the context of s2 relative to sl, but that f is such that even if f(pI is true and p immediately and logically entails 4, it is not necessarily the case that f(q) is true. “It is not the case that -” is such a function. By simulative reasoning, normal rules of inference apply locally to s2 and, by context climbing, their results are meaningfully interpreted with respect to the parent space. If f(p) is true in sl, p is true in ~2. If p immediately and logically entails 4, then 4 is true in ~2, from which it follows by context climbing that f(q) is true in sl. But this last result is erroneous. Therefore, our assumption that there is a space defined by a context f must be wrong. The space thus defined would not make any sense as an appropriate domain for simulative reasoning.

The problem with propositional functions like “It is not the case that -” is that they are not entailment preserving in the following sense:

Definition of Entailment Preserving: For any propositional function f, f is entailment preserving if f is not the identity function and for any propositions p and Q, if f(p & (p-q)) is true, then f(q) is true.

Without constraining contexts to entailment preserving, or at least nearly entailment preserving propositional functions, simulative reasoning would be pointless, since its global consequences could not be reliably derived. It is not necessary to limit contexts to those that are strictly entailment preselv- ing. I assume that human reasoning is essentially non-monotonic (Bobrow, 1?80), that inferences are commonly made by default when counterevidence is lacking. As long as a propositional function is nearly entailment preserving, in the sense that its use as a context will result in the derivation of correct global consequences most of the time, then its use as a context is heuristically justified, at least as a source of default inferences. Therefore, we impose the following restriction on contexts:

Restriction on Legal Contexts: Any context is entailment preserving, or nearly entailment preserving.

For instance, ‘Seorge believes that -” is a legal context, because it is nearly entailment preserving, since when George believes that p entails 4 and that p is true, it is at least plausible that George believes q. Similarly, “it is possible that -” is a legal context because if p entails 4 then the proposition that p is possible entails the proposition that Q is possible; if 4 were impossible so wouldp be. Other legal contexts are “p- -,” ‘;p VP,” “if it were the case that p, then -, ” “in the novel Fletch -,” “every paleontologist is such that -,” “In 19th century America, -, ” and so forth.

An additional example of a propositional function which is not a legal context is “ -- r.” If p entails q and p- r is true, it is not necessarily the

MENTAL SPACES 9

cause that q-r is true. For instance, “(Warren ate a bug)- (We’d better call the doctor)” does not entail “(Warren ate something)-(We’d better call the doctor).” It does not make sense to define a space in which all propositions are true which (in the real world, say) imply that we’d better call the doctor. Such a space wouldn’t form an appropriate domain for simulative reasoning.

In summary, the usefulness of simulative reasoning can only be realized if contexts are limited to entailment preserving propositional functions. Another way to arrive at the same conclusion is to recognize that knowledge partitioning and simulative reasoning are a means of making what would be a set of meaning postulates in alternative knowledge representation schemes implicit in the way knowledge is partitioned. These are meaning postulates that would associate the property of entailment preservation with concepts like “believe,” “possible,” and so on.

2.4 Distributive Contexts The Restriction on Legal Contexts guarantees the correct derivation of global consequences from the immediate results of simulative reasoning by context climbing for any space that is defined by a legal context. It does not yet appropriately constrain knowledge partitioning. It turns out that certain contexts can and should be assumed to bind unique spaces to a given space, while others must be assumed to define more than one space.

Let’s consider some examples of the first type. Letfbe the context “Elmer believes - .” Intuitively, if Elmer believes p and Elmer believes q in space s, then p and q should be true in the same space bound to s by f. That is why we have no trouble referring to the belief space of Elmer. The justification for our intuition is that p and q fall appropriately into a common domain of inference; for any proposition r that we can infer from p and q we have the plausible inference that Elmer believes r. For instance, the truth of “Elmer believes Bugs is a rabbit, ” and of “Elmer believes all rabbits like carrots,” should allow the inference “Elmer believes Bugs likes carrots.” This inference follows by context climbing from a simple local inference only if “Bugs is a rabbit” and “all rabbits like carrots” are true in the same space. We are justified in assuming these to be true in the same space because “Elmer believes p” and “Elmer believes q” together entail “Elmer believes (p & q).” So it is appropriate to assume that the context “Elmer believes -” defines a unique space.

Another example of a context that defines a unique space is “p--.” If “p-q” is true in space s and “p-r” is true in s, then q and r should be true in a common space defined by “p-- ” to s. Again, the justification is that “p- (q & f)” follows.

Let’s consider, on the other hand, some contexts that must define non- unique spaces. Suppose that “it is possible that p” and “it is possible that q” are both true in s. Then there might be a space bound to s by “it is possi-

10 DINSMORE

ble that -” in which p is true, and there might be a space bound to s by “it is possible that -” in which 4 is true. If we assume that the space is the same in both cases, then “p & q” will be true in that space, and “it is possible that (p & 4)” will be true in s by context climbing. However, in many cases the derivation of such a global consequence would be erroneous. Let p be “Platypuses bear living young,” and Q be “Platypuses lay eggs.” In the interests of logical consistency we must assume thatfmust be able to define more than one space relative to s.

The property that distinguishes contexts that define unique spaces from contexts that do not, is the ability to distribute over conjunction. The former are therefore called distributive contexts, formally defined as follows

Definition of Distributive: For any propositional function f, f is distributive if for any propositions p and q, (f(p) & f(q)) = f(p & q).

A requirement that distributive contexts define unique spaces tends to optimize the consolidation of information into domains appropriate for simulative reasoning, while the same requirement for non-distributive contexts would frequently result in logical inconsistency. So we assume:

Distributive Constraint: For any distributive context f and for any space s, there is no more than one space with the context f relative to s.

Additional examples of distributive contexts which, by the Distributive Constraint, can define unique spaces are “every elephant is such that -,” “in 1982 , -,” and “all paleontologists believe that __ .” Additional examples of non-distributive context’s which define non-unique spaces are “some elephant is such that -, ” “there was a time when -9 ” and “some paleontologist believes that -.”

In summary, contexts that are not distributive must be capable of de- fining more than one space relative to some given space, while contexts that are distributive should be used to define unique spaces. Using a context to define a unique space is in fact a means of making what would be a set of meaning postulates in an alternative knowledge representation scheme, implicit in the knowledge partitioning. These are meaning postulates associated with concepts like “believe” and “in 1982 -,” and so on, that capture the property of distributiveness.

2.5 Settings Fauconnier introduces the principle of Space Optimization to account for the apparent inheritance of information in one space from some other space under certain conditions. In this section I show that the inheritance of information of one space by another is motivated by logical considerations which become apparent once the idea of contexts of spaces has been made explicit. Specifically, information about one space, sl, can immediately be assumed

MENTAL SPACES 11

to apply directly in another space, ~2. because of the semantic properties of their respective contexts. In such a case I will say that sl is in the setting of ~2.

In light of the functional motivation of this phenomenon the structure of the inheritance relationship will be constrained. We will see that this structure is somewhat different than that expressed by Fauconnier in his Space Optimizaton principle. For instance, conditions will become apparent under which two types of inheritance must be distinguished: absolute in which all information carries over from one space to another without exception, and defaulf in which information carries over with certain exceptions.

Consider a “p- -” space s2 relative to sl. If a proposition q is true in sl, then p-q is necessarily true in sl. So, (by the Distributive Constraint) q is true in ~2. Therefore, by definition s2 is in the setting of sl; a “p- -” space s2 defined relative to sl is a space exactly like sI, but in which p is assumed to be true.

Consider an “it is possible that -” space s2 relative to sl. Such a space is much like a “p- -” space with respect to inheritance; it also should inherit from the father space. Suppose, for instance, that “p-q” is true in sl and that p is true in ~2. Since by context climbing “it is possible that p” is true in sl, “it is possible that q” should be deducible in sl. If s2 inherits absolutely from sl then “p-q” is true in ~2, where q can then be deduced locally. By context climbing “it is possible that q” will be inferred in the real world. The assumption that an “it is possible that -” space inherits absolutely from its parent always yields the correct inferences.

As a final example, consider a space sl relative to which a “p--” space s2 and a “George believes that -” space s3 are defined. Suppose another “George believes that -” space s4 is defined relative to ~2. Then s4 will inherit from ~3. This is because for any proposition q, “George believes that q” entails “p-(George believes that q).”

In summation, an absolute inheritance relation exists between two spaces sl and s2 if the set of propositions true in sl is logically a subset of the set of propositions true in s2.

The content of one space can depend crucially on the content of another as a function of the semantics of the respective contexts and yet not exhibit absolute inheritance. This is the case for counterfactual or “if S were true, then -” spaces, as opposed to “S- -” or simple “if S is true, then -” spaces.

Suppose that the proposition that if I had looked in my wallet I would have found a dollar is true in the real world. This would presumably be true ifin the real world there was a dollar in my wallet. If this modal proposition is true in the real world, then there is an “if I had looked in my wallet, -” space in which it is true that I found a dollar. I submit that that space is very much like the real world, except that the proposition that I looked in my wallet is true in that space, while it is false in the real world. In particular,

12 DINSMORE

the proposition that there was a dollar in my wallet is true in the “if I had looked in my wallet , -” space, as well as the proposition that if one looks in something, one generally finds whatever is there. These propositions are inherited from the real world. What makes the proposition that I found a dollar true in this space, is the fact that it may be locally inferred from the propositions inherited from the real world and the proposition that I looked in my wallet.

The kind of inheritance involved in this case cannot be absolute, since the proposition that I did not look in my wallet (as well as a number of other propositions closely related to this one, such as the proposition that I did not open my wallet) are not inherited. However, most propositions are inherited. Intuitively it would be true in the real world that if I had looked in my wallet the earth would (still) orbit the sun, the U.S. Senate would still be debating the budget, and so forth. Such cases require a weaker form of inheritance, default inheritance.

It is easy to see how the ability to create a model of a world which inherits from the real world or from another model of the world in this way would have a strong function for a cognitive agent which is able to reduce the difference in the contents of such spaces through deliberate action: it allows simulative reasoning to play a role in planning sequences of actions. This is in fact the primary application of context-layered data base system such as CONNIVER which implement an inheritance mechanism that seems to work in a manner consistent with default inheritance of the kind discussed here.

Default inheritance appears also to be important to the content of fictional spaces. Consider the space of the Nero Wolf detective stories. In this space a number of propositions are false that are true in the real world, such as the proposition that Nero Wolf doesn’t exist. However, most of what is true in the real world carries over to the Nero Wolf space. In fact the Nero Wolf stories would be incomprehensible if this were not the case, since such information must underlie virtually all the reasoning about the situations, events, and so forth, about the Nero Wolf space.

2.6 Summary The theory of knowledge partitioning and simulative reasoning developed in this section builds on the theory of mental spaces by attributing simulative reasoning as the primary function of mental spaces. From the necessity of adequately supporting this function it recognizes that certain clear mechanisms and restrictions should or must be present. Among these are contexts, context climbing, the Restriction on Legal Contexts, the Distributive Con- straint, and absolute and default settings. The structure of spaces is thus more carefully constrained than in the theory of mental spaces.

The conclusion I wish to draw has direct implications for Cognitive Science: humans do in fact make use of these mechanisms and restrictions.

MENTAL SPACES 13

The next section demonstrates the viability of these conclusions. These mechanisms and restrictions have direct implications for Artificial Intelli- gence as well, which are described more formally in two separate papers (Dinsmore, 1986; Dinsmore, 1987).

3. SOME CONSEQUENCES OF THE MODEL

In this section the predictions of the functional model of mental spaces proposed in the last section are demonstrated for three types of linguistic phenomena. It will be shown that the additional structure of the theory extends the predictive power of the theory of mental spaces and in some cases cor- rects some false claims.

3.1 Discourse Focus The concept of context introduced in the previous section contributes significantly to the formulation of an adequate semantics for discourse. It is apparent that people are able to put together a coherent discourse by de- scribing the objects and relations of any space as if those objects and relations belonged to the real world. The same kinds of inferencing involved in processing discourse about the real world would seem to be involved in processing discourses about other spaces, since the structures of the discourses seem to be similar. I will call the space with respect to which a discourse is understood thefocus space for that discourse. The most obvious kind of example is provided in fictional discourses. However, another interesting example is found in a discourse that is initiated like this:

(1) Arthur believes it is the duty of everyone to fight what he thinks is an invasion of space frogs. Before this situation gets out of hand, every homeowner should defrog his own yard, taking care to __

After the initial shift in focus space, the discourse could continue indefi- nitely with the implicit understanding that we are talking about Arthur’s belief space rather than about the real world. This space has become active and inference processes relevant to discourse understanding apply locally to this domain. However, every proposition expressed in this discourse (in- cluding immediate entailments of propositions) can be mapped implicitly onto a proposition asserted to be true in the real world by using the context “Arthur believes that -.” This mapping is a crucial component of the actual knowledge expressed in this discourse. However, it is not actually necessary that each proposition be explicitly mapped by the hearer as part of the semantic processing of each individual sentence. An advantage of organizing knowledge in terms of spaces is that if the information explicitly represented in the discourse is asserted directly in the focus space, the real knowledge expressed could be retrieved subsequently by context climbing.

14 DINSMORE

This would be necessary, for example, in answering questions about the real world of the type “Does Arthur believe that p?”

The following discourses are analogous to (1).

(2) In chess, each player tries to capture the opponent’s king. If he moves a piece into a position that attacks the king, then he must say “check.” If his own king is attacked by a piece of the opponent’s, then he must either make a move to avoid the attack, or resign. . .

(3) Last month I attended the 23rd Annual Conference on Shoe Tech- nology. Someone gave a wonderful paper on toe awareness. An exhibi- tionon...

In (2) each proposition expressed is implicitly mapped onto a real-world proposition using the context “each chess player is such that -. ” Here the active space corresponds to what in some knowledge representation systems would be a frame or schema for the prototypical chess player. In (3) each proposition is implicitly mapped using the context “at the 23rd Annual Conference on Shoe Technology, -, ” which constrains the truth of each proposition temporally and locally. The focus space represents that particular situation.

It is interesting to notice that, in contrast to discourses like (l), (2) and (3), we don’t find discourses like (4) in which every proposition expressed is implicitly mapped into a proposition about the parent space by using the function “it is not the case that -.”

(4) It is not the case that there is an invasion of space frogs. It is the duty of every homeowner to defrog his own yard. . .

This cannot mean that in the parent space it is not the duty of every homeowner to defrog his own yard, and so on. This observation is an immediate consequence of the Restriction on Legal Contexts, which tells us that “It is not the case that -” is not a legal context and therefore cannot be used to define the focus space of a discourse.

Notice, incidentally, that “it is not the case that -” can be used to bring a space into focus for subsequent discourse, as in (5).

(5) It is not the case that there is an invasion of space frogs. It would be the duty of everyone to fight the invasion. Every homeowner would have to defrog his own yard. . .

In (5), as in (l), (2) and (3), every subsequent proposition is mapped implicitly onto a proposition in the real world. But (5) uses the legal context “if there were an invasion of space frogs then __ ,” not the propositional function “it is not the case that -.” The Restriction on Legal Contexts seems to condition the shift to an alternative focus space. Although “It is not the case that -” might be called a space builder in Fauconnier’s sense,

MENTAL SPACES 15

since it causes a space to be set up or accessed, its role in knowledge partitioning is considerably different from that of a space builder like “Arthur believes that -. ”

3.2 Conditionals and Counterfactuals If we represent default inheritance in a system that uses spaces we can capture the logic of contexts of the form “if it were the case that p. then -.” It has been observed (Lewis, 1973) that the counterfactual behaves differ- ently than standard material implication. If “p-q” and “q-r”, then it follows that “p-r.” But if it is true that “ifp were the case then q” and “if q were the case then r”, it does not follow that “if p were the case then r.” For example, (6) and (7) together do not entail (8).

(6) If J. Edgar Hoover had been born a Russian, then he would have been a Communist.

(7) If J. Edgar Hoover had been a Communist, he would have been a traitor. (8) If J. Edgar Hoover had been born a Russian, then he would have been a

traitor.

The difference between material implication and the counterfactual seems to be captured precisely in the difference between absolute and default settings between spaces. Thus, if “p-q” is true in sl then q is true in ~2, the ‘p-- ” space in sl. If “q-r” is true in sl as well, then “q-r” is necessarily true in ~2, since s2 inherits from sl absolutely. Since q and “q-r” are true in ~2, r is true in s2 by modus ponens. Since r is true in ~2, “p-r” is true in sl by context climbing. Hence, the observed property of material implication. On the other hand, the analogous inference fails for the counterfactual. If “if p were the case then q” is true in sl, then q is true in ~2, the “if p were the case then -” space in sl. If “if q were the case then r” is true in sl, then it is not necessarily true in ~2. For the current example, we observe that the truth of (7) in the real world is at least partially a consequence of the fact that J. Edgar Hoover was born in a strongly anti-communist country. But in s2 he would be born in a Communist country, so (7) would be incompatible with the antecedent proposition used to define that space; it would be one of the propositions that does not carry over from sl and ~2. Therefore, we cannot locally deduce that J. Edgar Hoover was a traitor in s2 or that (8) is true in sl.

The difference between material conditionals and counterfactuals seems also to be reflected in the difference between hoping and wishing. We have seen that propositional functions like “Fred hopes that -” and “Fred wishes that -” are not legal contexts since they are not entailment preserving. (For example, “Fred’s dumb cat has fleas” entails that Fred’s cat is dumb, but “Fred hopes that his dumb cat has fleas” does not entail that

16 DINSMORE

Fred hopes that his cat is dumb.) Nevertheless, spaces may be involved in the representation of these functions.

Intuitively, in formulating a wish or hope, Fred makes a comparison between what he believes to be the state of the world and what the world would be like if some proposition, in the current example that his dumb cat has fleas, is true (in the case of hope) or were true (in the case of wish). He then concludes that such a space is preferable to the real world. More formally, we notice that “Fred hopes p” is synonymous to “Fred believes that if p, then G” where G is something like “he (or people in general) is fortunate (or more fortunate than otherwise)“. If this proposition is true, then “if p, G” is true in Fred’s belief space, and G is true in the “if p then -” space in Fred’s belief space. “Fred wishes that -” is similar to “Fred hopes that -“, except that the counterfactual “if p were true, then -” space in Fred’s belief space is involved.

Thus, the information of the proposition “Fred hopes that p” may be distributed over a total of three spaces, even though “Fred hopes that -” is not itself a legal context. The “hope space” in which p is true is actually an “if p then -” space in Fred’s belief space. So, spaces and simulative reasoning can play a significant role even in representing and thinking about modalities that do not directly correspond to legal contexts.

An interesting observation about the subjunctive (“would” in English) is that it seems invariably to be used in clauses which are taken to say something about a space that is (a) defined by a conditional, i.e., “if S”, and (b) inherits from its parent by default. This is regardless of where that clause occurs syntactically, for example whether it is a complete sentence, or em- bedded in a larger sentence. (6) above is one example. Another is “It would be the duty of everyone to fight the invasion” occurring in the discourse with the “If there were an invasion of space frogs then -” focus space described earlier. Notice that “wish” also requires subjunctive, as in “George wishes that he were the winner” while “hope” does not. This is confirming evidence for the analysis of wish and hope spaces respectively as “if p were true, then -“, and “if p is true, then -” spaces in belief spaces.

The analysis of wish and hope outlined above immediately and explicitly captures the dependence of so-called wish and hope spaces on belief spaces that Fauconnier informally characterizes in terms of the otherwise unmoti- vated mentor relation which Fauconnier posits to account for the observation that wish and hope spaces seem to be related by Space Optimization to belief spaces instead of to the father spaces. In the present analysis a belief space is the father of a hope or wish space, so the mentor relation and the treatment of hope and wish as a special case is not needed in the extended model.

From the prediction of the present analysis that the information in a “hope” or “wish” sentence is distributed over three spaces rather than two,

MENTAL SPACES 17

some additional consequences follow. For instance, Fauconnier points out two readings for the following sentence:

(9) John Paul hopes that a former quarterback will adopt needy children.

On one reading, there is a specific quarterback in the real world that John Paul hopes will adopt needy children (whether or not John Paul knows that this person is a quarter back). On a second reading, John Paul imagines a situation in which a quarterback, it doesn’t matter which one, adopts needy children. These two readings are predicted in terms of Fauconnier’s Idenfify Princip/e, which attributes the description alternatively to the real world (or the speaker), and to the hope space. The present analysis predicts that three spaces are involved in the interpretation of (9). The other space is John Paul’s belief space. Therefore we predict a third reading by the Identity Principle, one in which the description is attributed to this space. Indeed this reading exists. It is the one in which John Paul has a particular person in mind who he believes is a former quarterback (whether or not he actually is a quarterback in the real world, or for that matter whether or not he actually exists in the real world), and he hopes that that particular person will adopt needy children.

3.3 The Projection Problem for Presupposition Presuppositions are parts of the content of sentences that are normally taken as given or irrefutable in the utterances of sentences. Thus, “The King of France is bald” is said to presuppose that there is a King of France and to assert that that individual is bald. It would normally be used in a context in which it is already understood, or given, that there is a King of France, that is, in which the presupposition is satisfied. For about 15 years, linguists have been intrigued by what has been called the projection problem forpre- supposition, roughly that of predicting the presuppositions of complex sentences as a function of their structures and of the presuppositional constructions that they embed. For instance

(10) Max has just stopped smoking.

presupposes that Max has smoked. But whereas

(11) It is possible that Max has just stopped smoking.

retains that presupposition,

(12) Fred believes that Max has just stopped smoking.

presupposes only that Fred believes that Max has smoked, and

(13) If Max has smoked, then Max has just stopped smoking.

loses the presupposition altogether.

18 DINSMORE

An almost complete account of the projection problem for presupposition falls out under the assumption that the simple presuppositions associated with presuppositional constructions are uniformly satisfied locally in the relevant spaces. The Restriction on Legal Contexts and the Distributive Constraint entail a set of explicit predictions of the presuppositions of complex sentences not otherwise available. In fact, they allow predictions that include almost the whole set of observations that Karttunen and Peters (1979) make with respect to the projection problem for presupposition in descriptive terms.

For instance, we can predict that (12) above presupposes that Fred believes that Max has smoked. According to the theory of mental spaces the presupposition that Max has smoked will be satisfied locally in the “Fred believes that -” space in which “Max has just stopped smoking” is asserted to be true. If “Max has smoked” is given in this space, then by context climbing “Fred believes that Max has smoked” is necessarily given in the parent space.

On the other hand, we can see why the following

(14) It is not the case that Max has just stopped smoking. (15) If Max has just stopped smoking, then he must be pretty jumpy.

do not, in analogy with (12), presuppose that it is not the case that Max has smoked or that if Max has smoked then he must be pretty jumpy. We have seen that things like “it is not the case that -, ” and “If -1 then he must be pretty jumpy” are not legal contexts. Therefore, there is no more local space in which to satisfy the presupposition. The presupposition is instead satisfied directly in the focus space.

Next, we have seen that a necessary condition for the satisfaction of the presupposition in (12), that Max has smoked, is that “Fred believes that Max has smoked” be given in the focus space. Analogously, a necessary condition for the satisfaction of the presupposition of (11) is that “It is possible that Max has smoked” be given in the focus space.

However, (11) and (12) are crucially different. Since “Fred believes that -” is a distributive context it is also a sufficient condition for the satisfaction of the presupposition that “Fred believes that Max has smoked” be given, since “Max has smoked” is, by the Distributive constraint, thereby given in the one and only “Fred believes that -” space. This accounts for the observation that a sentence like (12) will presuppose something like “Fred believes that Max has smoked.”

On the other hand, it is not a sufficient condition for the satisfaction of the presupposition of (11) that “it is possible that Max has smoked” be given. If this is given, then “Max has smoked” must be given in some “It is possible that -” space, but not necessarily in the appropriate one, since “it is possible that -” is non-distributive and there is therefore no unique “it is possible that -” space.

MENTAL SPACES 19

What condition, then, is sufficient for the satisfaction of the presupposition of (1 l)? We have seen that in the setting of a possibility space is its father space. That is, if “Max has smoked” is given in the focus space, then it is likewise given in any “It is possible that -” space defined relative to the focus space. Therefore, if “Max has smoked” is given in the focus space, the satisfaction of the presupposition of (11) is guaranteed. This accounts for the observation that a sentence like (11) will presuppose “Max has smoked” rather than the weaker “It is possible that Max has smoked.”

A particularly interesting set of observations for the projection problem for presupposition involves “hopes” and “wishes.”

(16) Fred hopes that Max has just stopped smoking.

(16) is observed to presuppose not that Max has smoked, nor that Fred hopes that Max has smoked, but that Fred believes that Max has smoked. We have seen that the meaning of (16) is appropriately represented as an assertion of the form “If Max has just stopped smoking, then G” in a “Fred believes that -” space. This assertion itself involves an “If Max has just stopped smoking, then -” space within the belief space. Accord- ingly, we predict that the presupposition will be satisfied in the belief space for the same reasons the presupposition of (13) was satisfied in the focus space. It follows that (16) presupposes that Fred believes that Max has smoked for the same reason that (12) has this presupposition.

This account of the projection problem for presupposition has necessarily been somewhat sketchy because of space limitations. A more thorough demonstration can be gleaned from Dinsmore (1981).

4. CONCLUSIONS

In this paper I have addressed the questions of what mental spaces are, what their <function is, and how they are structured. I have contended that mental spaces allow humans to exploit the very general inference method of simulative reasoning. Furthermore, I have shown how the structure of mental spaces derives their structure from this function.

The significance of this paper, I feel, is not only that it provides the theo- retical underpinnings for the study of mental spaces as a phenomenon significant in human cognition, but also that it integrates the study of mental spaces into the field of knowledge representation and reasoning.

The study of mental spaces as a cohesive and pervasive organizational device is a powerful new idea. I conclude by summarizing some of its immediate and potential consequences, first for the study of human cognition, and second for the study of functionally motivated knowledge representation systems.

We have seen that our understanding of mental spaces provides a simple account of a wide variety of problems that have haunted linguists and philos-

20 DINSMORE

ophers of language for years. Among these are the problems of referential opacity and transparency, specificity of reference, definite reference in discourse, the interaction of quantifiers and modalities, the projection problem for presupposition, the semantic processing of counterfactuals and other modalities, and the use of comparatives in modal contexts. How far this list can be extended is an open question for future research. For instance, it will probably be fruitful to explore how the insights we have gained about mental spaces bear on the use of the various moods, tenses, and aspects (a current topic of my research) in natural language, or what the use of focus spaces and our ability to switch focus spaces in discourse have to say about the structure of discourse. Further areas where the study of mental spaces might bear fruit are the ontogeny and structure of fiction, pretending, lying, and game-playing. Fauconnier mentions the relevance of Goffman’s work in this regard.

We have also seen how mental spaces support efficient reasoning about complex information. The apparent ubiquity of spaces in human information processing as well as its specific uses in AI systems and in logic suggest the power of simulative reasoning as a general inference method. How far its usefulness can be extended is an open question. For instance, it will be interesting to see how valuable some of the ideas developing out of research on mental spaces are in solving the problems of representing and reasoning about time, or in supporting analogical or metaphorical reasoning.

Significantly, work on mental spaces from a functional perspective has suggested a small set of primitive structural concepts and constraints that seem to be required by any system making effective uses of spaces. Among these are contexts associated with spaces, the Restriction of Legal Contexts, the Distributive Constraint, and absolute and default settings. A computa- tional knowledge representation system called Space Probe (Dinsmore, 1986) which is based on knowledge partitioning and simulative reasoning has been implemented for the purpose of exploring further the implications of this new area of research.

REFERENCES

Bobrow, D. (Ed.). (1980). Special Issue on non-monotonic logic. Artificial Intelligence 13: 1,2.

Bobrow, D., & Winograd, T. (1977). An overview of KRL, a knowledge representation language. Cognitive Science I, 3-46.

Brachman, R., & Schmolze, J. (1985). An overview of the KL-ONE knowledge representation system. Cognifive Science 9. 171-216.

Cohen, P., & Perrault, C.R. (1979). Elements of a plan-based theory of speech acts. Cognitive Science 3, 177-212.

Creary, L. (1979). Propositional attitudes: Fregean reasoning and simulative reasoning. In Proceedings of the S&h Infernational Joint Conference on Artificial Intelligence (pp. 176-181).

MENTAL SPACES 21

Dinsmore. J. (1981). The inherifance ofpresupposifion. Amsterdam: Benjamins. Dinsmore, J. (1986). Foundations of knowledge partitioning. Unpublished manuscript.

Dinsmore, J. (in press). Spoceprobe: A sysrem for representing modul know/edge. In Pro- ceedings of the lnrernational Symposium on Methodologies for Inrelligenr Systems. New York: ACM SIGART.

Fahlman, S.E. (1979). NETL: A system for representing and using real-world knowledge. Cambridge, MA: MIT Press.

Fauconnier, G. (1985). Menial spaces: Aspects of meaning consrrucrion in norural language. Cambridge, MA: Bradford/MIT Press.

Hendrix, G. (1979). Encoding knowledge in partitioned semantic networks. In N. Findler

(Ed.), Associative networks (pp. 51-92). New York: Academic. Johnson-Laird, P.N. (1983). Men&l models. Cambridge, MA: Harvard University Press.

Karttunen, L., & Peters, S. (1979). Conventional implicatures. In C.K. Oh, & D. Dineen (Eds.), Presupposition (pp. I-56). New York: Academic.

Lewis, D. (1973). Counrerfucruals. Cambridge, MA: Harvard University Press.

Martins, J. (1983). Reasoning in multiple belief spaces. (Tech. Rep. No. 203). Department of Computer Science, SUNY, Buffalo.

McDermott, D., & Sussman, G.J. (1972). The CONNIVER reference monuol (Tech. Rep.

Memo 259), MIT AI Lab. Schank, R., & Abelson, R. (1977). Scripts, plans, goals and understunding. Hillsdale, NJ:

Erlbaum.

Mental Spaces from a Functional Perspective

Documents

Transcript of Mental Spaces from a Functional Perspective