Copredication describes certain forms of expression occurring in natural languages, where seemingly incompatible types of predicate are applied to an object. This typically happens when the object is identified by a polysemous count noun whose possible interpretations span two or more distinct ontological categories (such as 'book', which can be interpreted either as an informational artifact or a physical object). Copredication poses a significant challenge for theories of linguistic meaning because, although many copredication examples seem to be natural and easily understandable, they are very dificult to account for within established semantic theories and formal representations.
Ontological Mutability⋆
formal semantics and linguistics [
I shall also be considering whether previous work in the area of supervaluation semantics
and standpoint logic [
Consider the sentence “There is a book by Quine on my shelf.” Following a typical style of converting English into first-order logic, one might propose the following ‘naive’ classical representation: ∃[ Book() ∧ ByQuine() ∧ OnMyShelf()] (1)
However, interpretation of this formula is problematic. The predicateBook can either have
the sense of being an informational artifact or the sense of being a physical object. These are
surely distinct senses, since when we describe a book as being original or interesting we must
be speaking of its informational content, not the physical object; and conversely, when we say
a book is heavy or water-damaged, we must be talking about the physical object rather that
the information. In the example represented by formula1(), the authorship predicate,ByQuine,
applies to a book in the informational artifact sense: Quine originated the content of the book
but had no part in creating the physical object; whereas, the predicatOenShelf applies to a book
in the physical sense, a material object occupying a particular location. It is this application
of two predicates that seem to describe very diferent kinds of object, to what seems to be a
single referent that is calledco-predication. It typically occurs when there is some ambiguity in
a sortal concept word (i.e. a count noun such as ) which is used to form the object reference,
but can also occur with certain proper names (e.g. names of fictional characters — see [
The puzzle of co-predication is not so much that one can construct sentences in which copredication occurs. The real mystery is why sentences such as “There is a book by Quine on my self” of “Lunch was delicious but went on forever”1[] make perfect sense and do not even seem unusual.
A number of possible explanations of copredication have been proposed in the literature. In this section I outline some of the more prominent of these. One should note that only simple versions of the theories are presented here, in order to give an overview of possible approaches. The works referred to give considerably more elaborate accounts.
An initial response might be just to say that this is easily explained in terms of the ambiguity of ‘book’: it can be interpreted in the intellectual artifact sense within the context of the ‘by Quine’ predicate, and in the physical object sense within the context of the ‘on my shelf predicate.’ But, although this may make some sense as an informal explanation, it does not give any clear idea of how one might specify a coherent semantic interpretation of formul1a)(. The issue is not just thatBook has two senses but that the quantifying phrase “there is a book” seems to imply the existence of an object that satisfies both of the two predicates ‘by Quine’ and ‘on my shelf’. So the object would somehow have to be an instance of both of the two seemingly disjoint senses of ‘book’. This is a very diferent from what we would normally call ambiguity. Moreover, in ordinary cases of ambiguity, it is expected that they ambiguity gets resolved in the same way throughout a sentence. If this is not the case, the sentence usually seems bizarre or humorous — for example “There were three banks in the village: the two banks of the river and the Barclays bank on the main street.” For these reasons, no serious account of copredication suggests that it can easily be explained in terms of the familiar notion of ambiguity.
One approach to formulating a semantics that can accommodate such examples is to propose that that books are a type of object that has both a physical and an informational aspect, and that, when a predicate is applied to a book object, it must be interpreted with respect to the aspect appropriate to the type of predicate (authorship applying to the informational aspect and physical properties applying to the physical aspect of book).
The idea of dot types compounded from two apparently disjoint types was proposed and
elaborated by Asher [
Clearly, the introduction of dot types and objects carries significant ontological commitment and would require a fundamental modification of principles that are currently predominant in the design of ontologies.
Another way that one might account for copredications is by some mechanism ocofercion. As Asher explains, “a coercion is a function from one semantic value or one type to another that is employed when some problem arises in the construction of meaning”14[]. Programmers, will also be familiar with the closely related notion ocfoercion built into many programming languages. For example, a function that primarily operates on strings (e.g. a print function) may also accept a number argument, and if a number is given the function will automatically convert it to a string.
The potential of coercion to explain copredication is fairly obvious. If we consider formu1la, for this to be true we need to find a value that satisfies the three predicates Book, ByQuine and OnMyShelf. Suppose we take the ‘primary’ type ofBook to be book in the physical object sense. Then, if there are any books on my shelf, there will be some instances oBfook that are also instances ofOnMyShelf. But these physical books cannot by instances oBfyQuine which is a predicate applying to informational objects. However, we may propose that within the setting ByQuine() any argument that is not of typeInfo will, if possible be coerced into an object of type Info .
Of course, for this explanation to hold water, we need to have some plausible explanation as to how such a coercion might occur, and for KR purposes we would need some way to implement coercion with the apparatus of a formal language. Although such explanations are clearly not trivial, they certainly seem feasible. For instance, in the case of coercing a physical book object into an informational object we may make use of a relationManifests(, ) that holds whenever a physical book contains a physical inscription encoding the informational content . If we require that models for our formal language include an assignment specifying the Manifests predicate then this can be used to define coercion as a partial function from physical to informational books. Specifically, one might specify info() as a partial function determined by the Manifests relation in cases where there is a unique informational obje ct, for which Manifests(, ) is true. Moreover, the fact that not every physical book has information content with a unique author could explain why in some situations copredication does not permit a coherent interpretation.
In this section I consider another popular approach to modelling variable meanings and explain why, unless fundamentally modified, this kind of approach cannot explain copredication phenomena.
A well known approach to variability of meaning, and particularly to accounting for the
meanings of vague conceptual terms issupervaluation semantics [
Standpoint Semantics [
Unfortunately, when we try to apply supervaluation semantics (or standpoint semantics) to copredication examples, we soon become convinced that any account that follows the usual general principles of supervaluation semantics must fail. The problem is that supervaluation semantics considers the interpretation of a vague sentence in terms of all its possible classical interpretations. In other words in terms of all possible precise interpretationspr(ecisifications ). If Book has two alternative precise meanings then each precisification must interpretBook as either one or the other — i.e. either an informational artifact or as a physical object. But in any precisification that makes the first interpretation, the predicate OnMyShelf must be false and in any precisification making the second, the predicate ByQuine must be false. Hence, there will be no precisification where all three predicates can be true of any given object.
Recently, standpoint semantics has been elaborated in order to supportVaariable Reference Logic
(VRL) [
Unfortunately, although it might at first seem that we can use indeterminate objects to model
the referent objects in a copredication situation, this explanation does not work. To fully explain
the issue would require quite length exposition of the VRL semantics (see1[
The problem is that we still have to decide what kind of objects are instances oBfook according to diferent precisifications of that concept. And since book seems to be polysemous it is very natural to consider that it has an informational sense in some precisifications and a physical sense in others. But this means that, even though the semantics permits indeterminate objects, these objects must be either indeterminate informational objects or indeterminate physical objects, depending on the precise interpretation of book. So we would not have objects that are indeterminate between being an informational object or being an informational object.
The only way to avoid this problem seem to be to allow that there are objects that instances of some precise interpretation ofBook, but which are nevertheless indeterminate between being informational or physical objects. Formally, this is perfectly possible. However, the efect would be essentially the same as introducingInfo •Phys dot types into the representation. And once we do that, there is no need for any of the other apparatus of standpoint semantics or VRL.
From what we have seen so far it seems that both the approach usindgot types/objects and the approach making use ofcoercion might provide reasonable accounts of coercion. However, there is a further issue that still poses a considerable dificulty to establishing a plausible theory. This issue arises when copredication is combined with some numerical quantification of the number of objects involved in the situation. Consider the sentence: “There are two books by Quine on my bookshelf.” ∃∃ [ ≠ ∧
Book() ∧ Book( ) ∧ ByQuine() ∧ ByQuine( ) ∧ OnMyShelf() ∧ OnMyShelf( )]
The problem is that determining how many books by Quine are on my self depends upon whether I count in terms of informational artifacts or physical volumes. I might have two copies of the same book, or two book titles contained within the same volume (or some more complex combination of volumes and contents). Thus, a claim regarding the number of books can only have definite meaning once I choose what kind of book object I wish to count. But accepting this leads to a recapitulation of the original problem. Once I choose between informational and physical books, I am no longer dealing with entities that can support both the informational property of the books content originating from a particular author and also the physical property of being a physical object located on a particular shelf.
Neither the dot object or coercion approach provide any obvious way of addressing this
numerical quantification issue. In the case of dot objects, the prospects seem quite bad since dot
objects combine two aspects into a single object. So it seems hard to envisage that such objects
could be counted in diferent ways. Nevertheless, as outlined in the next section, Gotham 2[
Gotham [
Diferent senses have diferent individuation criteria giving rise to diferent counting principles. This diference can be located within the equality relation associated with a particular mode of individuation. According to Gotham the individuation criteria appropriate for counting objects in a given context are induced by the predicates in which they occur. Moreover, compound types are created compositionally when diferent types of predicate are applied to a quantified variable. Predicates that apply to an informational sense of book will give rise to counting in terms of informational objects and those applying to physical objects will induce counting in terms of physical volumes. And in the case of co-predication involving both informational and physical properties are ‘composed’ by taking the union of the equality relations for each mode. In the case of books, this means that, when one counts ‘informative’ ‘heavy’ books, one should count entities that are distinct both in terms of informational content and physical constitution.
Gotham’s theory provides a rather ingenious semantics that directly addresses the counting issues arising from copredication and also gives truth conditions for sentences involving compredication and counting that seem reasonable in relation to a variety of situations. However, there are also situation for which Gotham’s truth conditions seem rather strange. For example suppose a writer has written five novels, which for conciseness we simply call , , , and . Now suppose that on my shelf I have 2 copies of nov el and two copies of a collected work containing all five novels. How many books by are on my shelf?
According to my reading of Gotham1, the answer is 3. To see this we number the 4 physical volumes as 1–4. Then we consider the composite pairs consisting of physical volumes and informational novels. These are: 1A•, 2•A, 3•A, 3•B, 3•C, 3•D, 3•E, 4•A, 4•B, 4•C, 4•D, 4•E. We then have to find the maximal size of subset of these combinations such that no two members of that subset are associated either with the same volume or with the same content. There are actually several choices. One of these is:{1•A, 3•B, 4•C}. I find this very unintuitive. I would say that the number of books is ambiguous between 4 if we mean physical books and 5 if we mean informational artifacts. The fact that it gives specific and quite elaborate truth conditions for interpreting seemingly ambiguous sentences makes me uneasy with Gotham’s theory.
Having given an overview of some previous proposals and noting some of their limitations, I now present my own. The idea is that we can account for copredication phenomena by a combination of two interacting modes of semantic articulation: • There is variability of the meaning, and hence also the type and the individuation criteria, associated with the count noun that specifies the range of quantification. This variability can be captured by a precisification based semantics, similar to that used for Standpoint Logic. • A coercion mechanism is built into the interpretation of predication and comes into play whenever a predicate is found to operate on an argument that is not if the expected type. This mechanism will convert an argument object to a related object of the required type, on the basis that the given argument uniquely determines an associated object of the required type.
In order to see how these mechanisms would be implemented within a formal representation I make a few fairly modest modifications to the normal first-order logic syntax. (Probably these are not strictly necessary, but I think they make the representation clearer.) Firstly, we associate each quantifier with a count noun, in the fashion of asorted logic. And, keep numerical 1Whereas Gotham gives a very precise explanation of how we count objects with compound types, I found his presentation somewhat vague regarding which compound objects should be considered. From his examples, it seems that we consider each pair⟨, ⟩ , such that is a physical volume and is an informational work contained within volume , where each volume may contain one or several informational works. quantification simple, I will allow a numerical subscript to be added to the existential quantifier. Finally, I add coercion functions within the argument places of each predicate, indicating that the argument should be converted to the required type if possible. Hence, “There are exactly two books by Quine on my shelf” will be represented by: (∃2 Book ∶ )[ ByQuine(info()) ∧ OnMyShelf(phys()) ] (2)
So how should this be interpreted? We assume a standpoint semantics; but in this case it can be extremely simple, since the only term that exhibits semantic variability isBook, which we assume to have two precise interpretations corresponding to the information artifact and physical object senses of the word. Thus, we have precisifications info and phys. Since the Book predicate is attached to the existential quantifier, it is the interpretation ofBook that determines the counting criteria that will be applicable. Hence, the precisifcations info and phys are associated with quantifiying over diferent kinds of object with diferent counting criteria. However, whereas with the earlier versions of standpoint logic (and VRL) it was not possible for boBthyQuine and OnMySelf to both apply to any object in any precisification, we now have coercion functions as well. Hence, whichever interpretation ofBook is chosen, some value for can potentially satisfy both of the predicationsByQuine(info()) and OnMyShelf(phys()) . If we take Book in the physical sense then theinfo coercion function will come into play, whereas foBrook in the informational sense, thephys function will be needed.
What this means is that in the general standpoint∗‘’ comprising all precisifcationsm, i.e. { info, phys}, the formulae □ ∗(∃2 Book ∶ )[ ByQuine(info()) ∧ OnMyShelf(phys()) ] will be true provided that for either interpretation ofBook we can, using a suitable coercian function, ifnd to instances that satisfy both of the predicates. Morover, in cases where only one of the interpretations of book yields two satisifying instances, we will have the formul♦a∗(∃2 Book ∶ )[ ByQuine(info()) ∧ OnMyShelf(phys()) ] being true. And furthermore we can specify standpoints info = { info} and phys = { phys} corresponding to particular agents or contexts in which the interpretation ofBook is restricted to one of the two possible interpretations.
Copredication may seem like a relatively specific and peculiar oddity of language that only afects certain unusual linguistic constructions. However, after studying it at some length, I have come to the view that it may be more significant than it first appears. Once aware of its possibility, I became more sensitive to its occurrences, and found that shifts in ontological status of referent objects are much more common than I had appreciated. They afect a large number of very common words, including for example: meal, dance, film, celebration, library, nation, political party, to list but a few. Moreover, the ontological shifting that can occur may cover a very large number of distinct ontological categories.
It seems to me that it is actually the ontological shifting and diversity of interpretations that is the deep explanation of copredication and this presents challenges for knowledge representation in general, that are not at all specific to copredication examples. The real issue is that there are many very common natural language words to which we cannot straightforwardly assign any particular ‘ontological type’. If we try to make such a stipulation, any semantics based on this assignment will only explain a very limited subset of possible uses of the word. The concept ‘meal’ provides a good illustration of this ‘ontological mutability’. 8.1. Meal The word meal is vague in a huge number of ways. There are many considerations relating to ‘borderlne’ issues, such as the distinction between a few mouthfuls and a meal or whether there could be a meal where nobody actually ate anything (e.g. the dining hall was evacuated due to a fire alarm). But we are now interested only in polysemy relating to fundamental ontological type giving rise to diferent modes of individuation and counting. I suggest that the following six individuation modes can be distinguished.: (A) A type of eating event, usually diferentiated by time of day it is eaten, but also associated with a range of typical foods: • “Half board accommodation includes two meals. You can choose either breakfast and lunch or breakfast and dinner. Full board includes four meals: breakfast, lunch, afternoon tea and dinner.” (B) An eating event occurence which can be solitary but often involves multiple people participating in the same meal.
• “The conference programme included two meals: a reception bufet and a formal dinner” (C) An individual eating event: • “One pizza and one salad is enough food for three meals.” • “We ate together at the bistro. John loved his meal and we both liked the food, but I didn’t enjoy my meal as I kept thinking about the money I had lost.” Here, we could further diferentiate senses where we are thinking of the actual food from senses where we are thinking about other aspects of the experience. What is essential to this mode is that the ‘meal’ count will be the same as the number of people involved. (D) A physical food portion. That is, a quantity of food prepared for (and usually served to) an individual person during a particular time interval: • “The students ordered 8 meals between them — 5 pizzas and 3 omelettes. 2 of the students just ordered a beer.” • “The refectory prepares over 500 meals each day. Usually fewer than 20 go to waste.” (E) A physical food ensemble intended for a group of people: • Three meals had been laid out. One on each of the three tables. Each of the meals was for four people. (F) A particular food type or collection of food types often eaten together: • “Pancakes (with blueberries) is one of my favourite meals.”
Some senses of ‘meal’ with diferent modes of individuation.
• “The menu lists five set meals including two vegetarian meals” (G) Complex eating event types. One can potentially specify and enumerate a wide variety complex eating event types. Such meal types are listed in certain kinds of menu: • “The cruise ofers three special meals, which you can book as many times as you wish: oysters and champagne in your cabin, tea and cake for two on the poop deck, ifsh dinner at the captain’s table.
Meals individuated by mode (F) have a similar ontological status to those of mode (A) but are more specialised and are typically combined with food specifications akin to mode (F); and they support a counting mode that is distinct from either (A) or (F).
Although the current paper is largely exploratory and open ended it does bring to light some dificult issues that confront the task of representation and reasoning using a formal language that has similar expressive capabilities to natural language. I conclude with a couple of general observations: • Indeterminacy of natural language vocabulary afects involves a number of aspects and afects diferent elements of semantic structure in diferent ways. • Ontological mutability, in the form of shifting (or perhaps merging) of ontological category is revealed in the phenomenon of copredication, but is also pervasive in more subtle kinds of semantic variability. • Whereas supervaluationist semantics and standpoint theory are good for modelling ‘horizontal’ variability of linguistic terms (where a term’s meaning varies in its range of application but not in its fundamental ontological category); however, it is not good for modelling vertical variability of terms (whose meaning may shift in ontological category event within a sentence). • Coercion functions are required to understand many constructions found in natural language and may also be useful (perhaps essential) in the design of general purpose Knowledge Representation infrastructure, such as ontologies.
In further work I aim to further develop these ideas and incorporate them within formal languages with the aim of capturing the conceptual flexibility of natural languages in such a way as to be useful for KR-based AI applications.