Ontology, Types and Semantics

Ontology, Types and Semantics Walid S. Saba1 1 Department of Computer Science and Computer Engineering, American University of Technology, Byblos, Lebanon P O Box 20 Abstract. In this paper we argue that many problems in the semantics of natural language are due to a large gap between semantics (which is an attempt at understanding what we say in language about the world) and the way the world is. This seemingly monumental effort can be grossly simplified if one assumes, as Hobbs (1985) correctly observed some time ago, a theory of the world that reflects the way we talk about it. We demonstrate here that assuming such a strongly-typed ontology of commonsense knowledge reduces certain problems to near triviality. 1 Introduction In ‘Logic and Ontology’ Cocchiarella (2001) convincingly argues for a view of “logic as a language" in contrast with the (now dominant) view of “logic as a calculus”. In the latter, logic is viewed as an “abstract calculus that has no content of its own, and which depends on set theory as a background framework by which such a calculus might be syntactically described and semantically interpreted.” In the view of “logic as a language”, however, logic has content, and “ontological content in particular.” Moreover, and according to Cocchiarella, a logic with ontological content necessitates the use of type theory (and predication), as opposed to set theory (and set membership), as the background framework. An obvious question that immediately comes to mind here is: what exactly is the nature of this strongly-typed ontological structure that will form the background framework for a new logic that has content? In our opinion, part of the answer lies in an insightful observation that Hobbs (1985) made some time ago, namely that difficulties encountered in the semantics of natural language are due, in part, to difficulties encountered when one attempts to specify the exact nature of the relationship between language and the world. While it has not received much attention, the crucial point that Hobbs makes is the observation that if one “assumes a theory of the world that is isomorphic to the way we talk about it” (emphasis added), then “semantics becomes very nearly trivial”. Given the multitude of difficult problems in the semantics of natural language (e.g., the semantics of the so-called intensional verbs, the semantics of nominal compounds, and the difficult problem of lexical ambiguity, to name just a few), a trivial semantics for natural language might seem a far fetched goal. However, as will be demonstrated in this paper, assuming a strongly-typed ontology that is isomorphic to the way we talk about the world, does indeed make semantics very ‘nearly’ trivial. S. Saba W. (2006). Ontology, Types and Semantics. In Proceedings of the 3rd International Workshop on Natural Language Understanding and Cognitive Science, pages 17-26 DOI: 10.5220/0002472300170026 Copyright c SciTePress 18 The picture we have in mind, depicted graphically as shown in figure 1 above, is a logic and a semantics that is grounded in an ontological structure that reflects our commonsense view of the world and the way we talk about it in our everyday language. The structure of this strongly-typed ontology is in turn to be discovered, rather than invented, using natural language itself, which is the best known theory of our (shared) commonsense knowledge. In this paper we first demonstrate that a number of problems in the semantics of natural language can be reduced to near triviality, assuming a strongly-typed ontology that reflects our commonsense knowledge exists. Subsequently, we briefly discuss the nature of this ontological structure and how it may be discovered. 2 Nominal Compounds The semantics of nominal compounds have received considerable attention by a number of authors, most notably (Kamp & Partee, 1995; Fodor & Lepore, 1996; Pustejovsky, 2001), and to our knowledge, the question of what is an appropriate semantics for nominal compounds has not yet been settled. According to Weiskopf (forthcoming), the problem of nominal compounds in the case of noun-noun combinations is largely due to the multitude of possible relations that are usually implicit between the two nouns. To illustrate, consider the following: abrick houseb = {x x is a house that is made of br ick } (1) adog houseb = {x x is a house that is made for a dog} (2) Thus, while a brick house is a house ‘made of’ brick, a dog house is a house that is ‘made for’ a dog. It would seem, then, that the relation implicitly implied between the two nouns differs with different noun-noun combinations. However, assuming the existence of a strongly-typed ontology might result in identifying a handful of patterns that can account for all noun-noun combinations. As shown in the fragment hierarchy shown above, it would seem that madeOf is the relation implicit between all [N 1 N 2 ] combinations whenever N 1 is a Substance and 19 N 2 is a Artifact (throughout, we use this Font for concept types in the ontology, and this font for predicate names. Thus, x::LivingThing means x is an object of type LivingThing and apple(x) means the predicate apple is true of x): N SubstanceN Artifact = λPλQ {x :: Artifact P( x) ∧ (3) ( ∃y :: Substance)(Q(y) ∧ madeOf ( x, y))} The following are some example instances of (3), denoting the meanings of brick house, wooden spoon, plastic knife and paper cup, respectively: brick house = {x :: Artifact house(x) ∧ E1} wooden spoon = {x :: Artifact spoon(x) ∧ E2 } plastic knife = {x :: Artifact k nif e( x) ∧ E 3 } paper cup = {x :: Artifact cup(x) ∧ E4 } where E1 ≡ ( ∃y :: Substance)( br ick (y) ∧ madeOf ( x, y )) E2 ≡ ( ∃y :: Substance)( w ood(y) ∧ madeOf ( x, y )) E3 ≡ ( ∃y :: Substance)( pl ast ic (y) ∧ madeOf ( x, y)) E4 ≡ ( ∃y :: Substance)(paper (y) ∧ madeOf ( x, y)) Note, however, that specific instances of Substance and specific instances of Artifact might require the specialization of the relation suggested in (3). For example, consider bread and knife, which are Substance and Artifact, respectively: bread :: FoodSubstance ≺ Substance (4) knife :: Tool ≺ ... ≺ Instrument ≺ Artifact (5) While knife combines with a raw Substance (Material), such as plastic, bronze, wood, paper, etc. with the relation madeOf, it combines with a FoodSubstance with the relation usedFor. Consider the following compounds of a FoodSubstance (which is a specific type of Substance) and some Instrument (a specific type of Artifact): butter knife = {x :: Instrument coffee mug = {x :: Instrument cereal box = {x :: Instrument k nif e(x) ∧ E1 } mug(x) ∧ E2 } bo x ( x ) ∧ E 3 } where E1 ≡ ( ∃y :: FoodSubstance)( but t er (y) ∧ usedF or ( x, y)) E2 ≡ ( ∃y :: FoodSubstance)(cof f ee(y) ∧ usedF or ( x, y)) E3 ≡ ( ∃y :: FoodSubstance)(cer eal (y) ∧ usedF or ( x, y)) 20 Although we cannot dwell on such details here, we should point out that since the purpose of an object of type Instrument (or more specifically, Tool) is to be used for something, the specific type of usage would in turn be inferred from the specific Instrument/Tool (e.g., cutting in the case of a knife, holding in the case of mug, etc.) 3 Intensionality and Compositionality Like nominal compounds of the form [Noun Noun ] the semantics of nominal compounds of the form [Adj Noun ] have also traditionally been problematic, and, more specifically, they have generally been considered as posing a serious challenge to the general program of compositional semantics in the Montague (1974) tradition. Recall that the simplest (extensional) semantic model for nominal constructions of the form [Adj Noun ] is that of conjunction (or intersection) of predicates (or sets). For example, assuming that red(x) and apple(x) represent the meanings of red and apple, respectively, the meaning of a nominal such as red apple is given as ared { } appleb = x r ed ( x ) ∧ appl e ( x ) (6) What (6) says is that something is a red apple if it is red and apple. This simplistic model, while seems adequate in this case (and indeed in many other instances of similar ontological nature), clearly fails in the following cases, all of which involve an adjective and a noun: former senator fake gun alleged thief (7) (8) (9) Clearly, the simple conjunctive model, while seems to be adequate for situations similar to those in (6), fails here, as it cannot be accepted that something is former senator if it is former and senator, and similarly for the case of (8) and (9). In general, therefore, more complex functions might be needed for other types of ontological categories. In particular, what we seem to have is something like the following: (10) ar ed appleb = {x r ed ( x ) ∧ appl e ( x )} (11) former senator = {x | WasButIsNotNowA _senat or (x)} a b a fake gun b = {x | LooksLikeButI sNotActuallyA _gun(x)} aalleged thief b = {x | IsNotButCouldPossiblyTurnOutToBeA _t hief (x)} (12) (13) It would seem, then, that different ontological categories require different compositional functions to compute the meaning of the whole from the meanings of the parts. In fact, the meaning (intension) of some compound might not be captured without resorting 21 to temporal and/or modal operators. This has generally been taken as an argument against compositionality, in that there does not seem to be an answer as to what the compositional semantic function F in aN 1 N 2 b = F (aN 1 b , aN 2 b) might be. We believe, however, that this is a fallacious argument in that the problem is not due to compositionality but in ‘discovering’ a number of semantic functions that could account for all nominal compounds of different ontological categories. For example, we argue that the following are reasonable definitions for the concepts fake, former and alleged: ( ∀x :: Physical)( f ak e( x) ≡df λP[( ∃y :: Physical)( ¬P ( x) ∧ P (y ) ∧ l ook sL ik e( x, y ))]) (15) ( ∀x :: Role)( f or mer ( x) ≡df λP[( ∃t )((t < now) ∧ P (x, t ) ∧ ¬P ( x, now))]) (16) (∀x :: Role)(al l eged( x) ≡df λP[¬P(x, now) ∧ (∃t )((t > now) ∧ ◊P(x, t ))]) (14) That is, ‘fake’ applies to some concept P as follows: a certain Physical object x is a fake P iff it is not a P, but it actually is similar (in certain respects) to some other Physical object, say y, which is actually a P. On the other hand, what (15) says is the following: a certain x is a former P iff x was a P at some point in time in the past and is not now a P. Finally, what (16) says is that something is an ‘alleged’ P iff it is not now known to be a P, but could possibly turn out to be a P in the future. It is interesting to note here that the intension of fake and that of former and alleged was in one case represented by recourse to possible worlds semantics – the case of (15) and (16), while in (14) the intension uses something like structured semantics, assuming that l ook sL ik e( x, y) which is true of some x and some y if x and y share a number of important features, is defined. What is interesting in this is that it suggests that possible-worlds semantics and structured semantics are not two distinct alternatives to representing intensionality, as has been suggested in the literature, but that in fact they should co-exist. Some additional points should also be raised here. First, the representation of the meaning of fake given in (14) suggests that fake expects a concept which is of type Artifact, and thus something like fake idea, or fake song, for example, should sound meaningless, from the standpoint of commonsense (One can of course say fake smile, but this is clearly another sense of fake. While fake gun refers to a gun (which is an Artifact) that is not real, fake smile refers to a dishonest smile, or a smile that is not genuine). Second, we should note that the representation of the meaning of former given in (15) suggests that former expects a concept which has a time dimension, i.e. is a temporal concept. Finally, we should note here that our ultimate goal of this type of analysis is to discover the various ontological categories that share the same behaviour. For example, conjunction, which as discussed above is one possible function that can be used to attain a compositional meaning, seems to be adequate for all nominal constructions of the form [A N] where A is a physical property (such as red, large, heavy, etc.) and N is a physical object (such as car, person, desk, etc.), as expressed in (17), which states that some adjectives are intersective, although it says nothing about the meaning of such adjectives: 22 aA N b = {x | A PhysicalProperty (x) ∧ N PhysicalThing (x)} (17) Similarly, an analysis of the meaning of former, given in (16), suggests that there are a number of ontological categories that seem to have the same behaviour, and could thus replace P in (15), as implied by the fragment hierarchy shown below. 4 On Intensional Verbs The argument we have been making thus far can be summarized as follows: assuming the existence of a strongly-typed ontology that reflects our commonsense view of the world and the way we talk about it, can help resolve a number of problems in the semantics of natural language. In this section we tackle the semantics of the so-called intensional verbs. Let us look at the following examples, which (Montague, 1969) discussed in addressing a puzzle pointed out to him by Quine: aJohn painted a elephant b = ( ∃x ) ( el ephant ( x ) ∧ paint ed ( j , x ) ) aJohn found a elephant b = ( ∃x ) ( el ephant ( x ) ∧ f ound ( j , x ) ) (18) (19) The puzzle Quine was referring to here was the following: both translations admit the inference (∃x)(elephant(x)) – that is, both sentences imply the existence of an elephant, although it is quite clear that such an inference should not be admitted in the case of (19). According to Montague, the obvious difference between (18) and (19) must be reflected in an ontological difference between find and paint in that the extensional type (e → (e → t )) both transitive verbs are typically assigned is too simplistic. While Montague’s solution to this problem was to suggest that some transitive verbs are intensional, we argue that the problem lies in the flat type structure assumed in Montague’s intensional logic. That is, we argue that a more complex type system is needed, one that would in fact yield different types for find and paint. One reasonable suggestion for find and paint, for example, could be as follows: find :: (eAnimal → (eThing → t )) (20) paint :: (eHuman → (eRepresentation → t )) (21) 23 Thus, instead of the flat type structure implied by (e → (e → t )) , the types of find and paint should reflect our commonsense belief that we can always speak of some Animal that found something (i.e., any Thing whatsoever), and of a Human that painted some illustration, or as we called it here a Representation. With this background, the proper translation of (18) and (19) and the corresponding inferences can now be given as follows: John painted a elephant = ( ∃x :: Representation)( el ephant ( x) ∧paint ed( j :: Human, x)) John found a elephant = (∃x :: Thing)( el ephant (x) (22) (23) So what do we have now? (23) clearly implies ( ∃x :: Thing)( el ephant (x)) . That is, (23) implies there is some object x of type Thing, of which the predicate elephant is true. Note, also, that el ephant (x) ⊃ (x :: Elephant ) , i.e., if el ephant (x) is true of some object x, then x must be object of type Elephant. Therefore, the expression ( ∃x :: Thing)( el ephant (x)) can be thought of as being is an abbreviation of ( ∃x :: Thing)( el ephant (x :: Elephant )) . That is, we have an expression that refers to an object x which is both, an Elephant and a Thing, which is fine, since Elephant ≺ ... ≺ Animal ≺ ... ≺ Thing as the fragment hierarchy shown below implies. In these cases we say that the two types unify. However ( ∃x :: Representation)( el ephant (x)) , which is what is implied by (22), is an abbreviation of the expression ( ∃x :: Representation)( el ephant (x :: Elephant )) (24) In (24) we are now referring to an object x which is of type Elephant and of type Representation. As shown in the fragment hierarchy of the figure shown above, these two types do not unify (neither is a subtype of the other). Instead of the subtype relation that exists between Elephant and Thing, another association must be inferred between Elephant and Representation. As shown in figure 4, an object of type Representation is a representation of something, and presumably, any Thing whatsoever (much like an object of type Story is a story about some Thing!) The conclusion of this discussion is that (22) and (23) given above result in the inferences given in (25) and (26), respectively: 24 ( ∃x :: Representation)( el ephant (x)) ( ∃x :: Thing)( el ephant ( x)) (25) (26) Eureka! Adding a rich type structure to the semantics seems to have solved Quine’s puzzle, as the correct inferences can now be made: if John found an elephant, then one could indeed infer that an actual elephant exists. However, the painting of an elephant only implies the existence of a representation (an illustration) of something we call an elephant! Stated yet in other words, (23) implies an elephant Thing exists, while (22) simply implies the existence of an elephant Representation. 5 Language, Logic and Ontology In this paper we have been making the following argument: semantics can be made ‘nearly’ trivial, as Hobbs (1985) correctly observed, if the semantics is grounded in strongly-typed ontological structure. The goal we have in mind is a formal system, much like arithmetic (or any other algebra) for concepts, as has been advocated by a number of authors, such as Cresswell (1973) and Barwise (1989), among others. What we are arguing for is a formal system that explains how concepts of various types combine, forming more complex concepts. To illustrate, consider the following: artificial :: NaturalKind → Artifact (27) flower :: Plant < ... < LivingThing ... < NaturalKind (28) What the above says is the following: artificial is a function that takes a NaturalKind and returns an Artifact (27); a flower is a Plant which is a LivingThing which in turn is a NaturalKind (28). Therefore, ‘artificial c’, for some NaturalKind c, should in the final analysis have the same properties that any other Artifact has. Thus, while a flower, which is of type Plant, grows, lives and dies like any other LivingThing, an artificial flower, and like any other Artifact, is not something that grows, lives and dies, etc., but is something that is manufactured, can be assembled, destroyed, etc. The concept algebra we have in mind should also systematically explain the interplay between what is considered commonsense at the linguistic level, type checking at the ontological level, and deduction at the logical level. For example, the concept artificial car, which is a meaningless concept from the standpoint of commonsense, is ill-typed since Car is an Artifact, and Artifact does not unify with NaturalKind – neither type is a sub-type of the other. The concept former father, on the other hand, which is also a meaningless concept from the standpoint of commonsense, escapes typechecking since father, which is a Role, is a type that former expects (recall the meaning of former given in (15) above). However, given the meaning of former, and an expression stating that 'once someone is a father he is always a father’, expressed as (∀x)(( ∃t 1 )( f at her ( x, t 1 ) ⊃ (∀t 2 )((t 2 > t ) ⊃ f at her ( x, t 2 )))) one can easily show that the concept former father, while it escapes type-checking, eventually results in logical contradiction at the logical level. A proper formulation of the nature of interplay 25 between language, ontology and logic, and correspondingly between commonsense, strong typing and deduction (as illustrated in figure 1 above), is what we believe is needed to ultimately have a ‘trivial semantics’ for natural language. Clearly, the most challenging task in this endeavour is nature of this ontological structure that reflects our commonsense view of the world and the way we talk about it (the middle layer in figure 1). But what is the nature of this ontological structure? First, since concepts we talk about in our ordinary language are public, then the structure of commonsense ontology must be shared, and subsequently this structure must be ‘discovered’ rather than ‘invented’. Second, natural language, which is the best known theory of commonsense, should itself be used as guide to discovering this structure. As we demonstrated above, an analysis of nominal compounds and so-called intensional verbs can give us important insights into the nature of this ontological structure that underlies natural language. In addition to the analysis of nominal compounds and intensional verbs, an analysis of verbs and adjectives that may or may not plausibly apply to nouns can also help us discover another piece of the puzzle. This process is very much similar to the process of type inferencing in higher-order, pure, polymorphic functional languages. First, we start this discussion by introducing a predicate app(p,c) which is taken to be true of a property p and a concept c iff ‘it makes sense to speak of the property p of c’. Consider now the following two sets P = {l ar ge, smar t } and C = {Table, Elephant} . A quick analysis of app(p,c) on the four adjective-noun combinations yields app(large,Table), app(large,Elephant), app(smart,Elephant), ¬app(smart,Table). That is, while it makes sense to say ‘large table’, ‘large elephant’, and ‘smart elephant’, it does not make sense to say ‘smart table’. This kind of analysis yields the following structure: Note that this structure was discovered and not invented. Note also that the decisions that lead to this structure, namely the application of the predicate app(p,c) on the four adjective-noun combinations could not be questioned. Moreover, the answer to these queries must be Boolean-valued – that is, while it could be a matter of degree as to how smart a certain elephant might be (which is a quantitative question), the qualitative question of whether or not it is meaningful to say `smart elephant’ is not a matter of degree. This has subsequently meant that the type-hierarchy we seek might be a strict binary tree with no multiple inheritance. For lack of space we cannot discuss this issue in more depth here. Instead, we refer the reader to (Saba, 2006) for more details on how such a ontology of commonsense knowledge might be discovered (rather than invented), using natural language itself as a guide in this process. 26 6 Concluding Remarks In this paper we have argued that many problems in the semantics of natural language are due to a large gap between semantics (which is an attempt at understanding what we say in language about the world) and the way the world is. This seemingly monumental effort can be grossly simplified if one assumes, as Hobbs (1985) observed some time ago, a theory of the world that reflects the way we talk about it. We have shown here that assuming such a theory (i.e., such a strongly-typed ontology of commonsense concepts) reduces certain problems to near triviality. Discovering such an ontological structure is clearly another matter. Clearly, however, since natural language is the best known theory of our (shared) commonsense knowledge, analyzing natural language and the way we talk about the world is the best avenue to discovering the nature of this ontological structure. Finally, and not withstanding some novel efforts to build such knowledge structures (such as Lenat & Ghua, 1990; Mahesh & Nirenburg, 1995; Sowa, 1995), we believe that the ontological structure that reflects our commonsense view of the world is shared, and thus attempts at inventing (rather than discovering) this structure have minimal chances of success. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Barwise, J. (1989), The Situation in Logic, CSLI Publications, Stanford. Cochiarella, N. B. (2001), Logic and Ontology, Axiomathes, 12, pp. 117-150. Cresswell, M. J. (1973), Logics and Languages, Methuen & Co., London. Fodor, J. & Lepore, E. (1996), The pet fish and the red herring: why concepts aren't prototypes, Cognition 58, pp. 243–276 Hobbs, J. (1985), Ontological Promiscuity, In Proc. of ACL-85, pp. 61--69, Chicago, Illinois, 1985. ACL. Kamp, H. & B. Partee. 1995. Prototype theory and Compositionality, Cognition, 57, pp. 129-191. Lenat, D. B. and Guha, R.V. (1990), Building Large Knowledge-Based Systems: Representation & Inference in the CYC Project. Addison-Wesley. Mahesh, K. and Nirenburg, S. (1995), A Situated Ontology for Practical NLP, In IJCAI-95 Workshop on Basic Ontological Issues in Knowledge Sharing, August 1995, Montreal, Canada. Montague, R. (1960), On the Nature of Certain Philosophical Entities, The Monist, 53, pp. 159-194 Montague, R. (1974), In Formal Philosophy: Selected Papers of Richard Montague, R. Thomasson (Ed.), Yale University Press. Pustejovsky, J. (2001), Type Construction and the Logic of Concepts, In P. Bouillon and F. Busa (Eds.), The Syntax of Word Meanings, Cambridge University Press. Saba, W. S. (2006), Language, Logic & Ontology – Uncovering the Structure of Commonsense Knowledge, accepted in International Journal of Human-Computer Studies – IJHCS (to appear). Sowa, J. F. (1995). Knowledge Representation: Logical, Philosophical, and Computational Foundations. Boston, MA: PWS Publishing. Weiskopf, D. A. (forthcoming), Compound Nominals, Context & Compositionality, Synthese, to appear.