B1. EVENTUALITIES AND THEIR STRUCTURE Since commonsense psychology deals with the things people think about and since people think about states and events in the world, we need to maximize the convenience of talking about such things. This is done by making states and events first-class individuals in the logic. That is, states and events are things in the world. They can be referred to by constants and variables in the logic. We "reify" states and events, from the Latin word "re(s)" for "thing"; we take them to be _things_. We will use the term "eventuality" to cover both states and events, after Bach (1981). Eventualities may be possible or actual. When they are actual, this will simply be one of their properties. To say that a state e actually obtains in the real world or that an event e actually occurs in the real world, we will write (Rexist e) That is, e really exists in the real world. If I want to fly, my wanting really exists, but my flying does not. On the other hand, it is often convenient _not_ to introduce eventualities, when they are not needed. Therefore, we will have two parallel sets of predicates -- primed and unprimed. The unprimed predicates will be the ordinary predicates we are used to in logical representations of language. For example, (give a b c) says that a gives b to c. When we assert this, we are saying that it actually takes place in the real world. The primed predicate is used to talk about the reified eventualities. The expression (give' e a b c) says that e is a giving event by a of b to c. This does not say that the event actually occurs, only that if it did, it would be a giving event. To say the same thing as "(give a b c)" says, we have to add that e really exists -- (Rexist e). Thus, the relation between the primed and unprimed predicates is given by the axiom schema (forall (x) (1) (iff (p x) (exists (e)(and (p' e x)(Rexist e))))) That is, p is true of x if and only if there is an eventuality e that is the eventuality of p being true of x and e really exists. (We use the term "axiom _schema_" because there will be a different axiom for each predicate p. The variable x here stands for all the arguments of p.) Note that the prime has no status in the logic. It is simply a notational convention for naming predicates. The predicate "Rexist" along with other modalities of existence is explicated further in Chapter B16. It will be useful to be able to say that something _is_ an eventuality. We will use the predicate "eventuality" for this. (eventuality e) The first question we need to settle about eventualities is how finely they need to be individuated. If John is running, he is also going. If e1 is an eventuality of John's running, there is an eventuality e2 of John's going. Are e1 and e2 the same or different? One possible model of eventualities is chunks of space-time. Under this interpretation John's running would be the chunk of space-time John occupies while he is running. The chunk of space-time he occupies while he is running can be the same as the chunk of space-time he occupies while he is going, so it would look like e1 and e2 should be the same. This model of eventualities is sometimes useful for fueling intuitions about eventualities, and in many cases it is a perfectly adequate way to think about eventualities. However, when we are modeling cognition, this is not adequate. Mary may believe that John is going but may not believe that he is running. If eventualities are going to be objects of belief -- as they are -- then there has to be a distinction between the running and the going. So we will take e1 and e2 to be distinct. Nevertheless, they are closely related. The event e2 occurs precisely because e1 is occurring. The running entails the going. To capture this relation, we introduce the predicate "gen", from the philosophical term "generates". We will say that there is a "gen" relation between e1 and e2. Moreover, we will say that whenever a running event e1 occurs, there is a going event e2 that it generates: (forall (e1 x) (2) (if (run' e1 x) (exists (e2) (and (go' e2 x)(gen e1 e2))))) Thus, we individuate eventualities very finely. If they can be described differently, they are different though perhaps closely related eventualities. There is a one-to-one correspondence between eventualities and predications in the logic. Many writers on semantics distinguish between states and events that obtain or occur in the world and the propositions that describe them. We do not. Because we individuate eventualities so finely, there is a one-one mapping between eventualities and predications, and we can use only eventualities and attribute to them properties that seem more appropriate for predications, e.g., they have predicates, arguments, and arities. Propositions are more coarse-grained than predications. For example, (p x) and (not (not (p x))) are different predications but the same proposition. Insofar as we need equivalences like this we will capture them in axioms about real existence and other modalities. One might object that cognitive predicates like "believe" are more properly applied to propositions or predications than to eventualities. But where this is so, the predicate applied forces the type of argument, and we can view the cognitive predicate as coercing the eventuality argument into a more propositional kind of entity. Thus, one can read "(believe John e)" as saying John _believes the proposition describing_ eventuality e. Because of this deterministic kind of coercion, we can ignore the distinction between propositions and eventualities, and deal only with the latter. It is often necessary to be able to refer to the participants in a state or event, or equivalently the arguments of a predication. For this we introduce a family of predicates. The first, "argn", says that some entity is the nth argument of the eventuality. For example, (forall (e x y z) (3) (if (give' e x y z)(argn x 1 e))) (forall (e x y z) (4) (if (give' e x y z)(argn y 2 e))) (forall (e x y z) (5) (if (give' e x y z)(argn z 3 e))) Note that we start numbering the arguments of primed predicates from 0 and of unprimed predicates from 1, so that a given entity will be the nth argument of both the primed and unprimed predicates. Thus, x is the 1st argument of both (give x y z) and of (give' e x y z). To complete the picture, we will say the eventualty is the 0th or "self" argument of itself. (forall (e x y z) (6) (if (give' e x y z)(argn e 0 e))) Axioms like this will in principle have to be written for every predicate. In practice, they would be handled by special mechanisms. The constraints on the arguments of "argn" are as follows: (forall (e x n) (7) (if (argn x n e)(and (nonnegInteger n)(eventuality e)))) That is, if x is the nth argument of e, then n is an integer and e is an eventuality. There are no constraints on x. Something is an eventuality if and only if it is the 0th argument of itself. (forall (e)(iff (eventuality e)(arg e 0 e))) (8) Sometimes, we only want to know that an entity is an argument, and don't care which argument it is. The predicate "arg" will express this relation. It is defined as follows: (forall (e x) (9) (iff (arg x e)(exists (n)(argn x n e)))) x is an "arg" of e if it is the nth argument for some n. Eventualities can be embedded in other eventualities. So to say that Mary believes John is tall, we might write (and (believe Mary e)(tall' e John)) or equivalently, (and (Rexist e1)(believe' e1 Mary e2)(tall' e2 John)) John is not directly an argument of the believing event, so "(arg John e1)" does not hold. But it is often convenient to talk about the looser relation that John bears to the believing event. For this we introduce the predicate "arg*". The statement "(arg* x e)" means that x is an argument of e, or an argument of an argument of e, or an argument of an argument of an argument of e, and so on. We can define it recursively as follows: (forall (x e1) (10) (iff (arg* x e1) (or (arg x e1) (exists (e2)(and (arg e2 e1)(arg* x e2)))))) For example, John is an arg* of the believing e1 because John is an arg* of the being tall e2 and e2 is an arg of e1. John is an arg* of e2 because John is an arg of e2. It will occasionally be useful to be able to talk about the predicate of an eventuality, or equivalently, of the unique predication that describes the eventuality. The predicate "pred" will express the relation between the predicate and the eventuality. We will use the unprimed predicate for this purpose, and we will simply assume the predicate names are constants in our logic referring to individuals in our domain of discourse. For example, (forall (e x y z) (11) (if (give' e x y z)(pred give e)) That is, if e is a giving event by x of y to z, then the entity we call "give" is the "pred" of e. The "pred" of an eventuality is a predicate. (forall (e p) (12) (if (pred p e)(and (predicate p)(eventuality e))) Predicates, and thus predications, have an arity, that is, a specific number of arguments, so we can also speak about the "arity" of an eventuality as well. Applied to an eventuality as something in the world rather than in a logic, the arity can be thought of as the number of participants in the eventuality that are designated as central. For example, for a giving event, we will designate the giver, the gift and the recipient as central, and say it has an arity of 3. The arity of predicates or eventualities can be described by axiom schemas like those for "argn" and "pred". For example, (forall (e x y z) (13) (if (give' e x y z)(arity 3 e)) Note that in the arity we do not count the 0th or self argument. The arity of an eventuality is a nonnegative integer. (forall (n e) (14) (if (arity n e)(and (nonnegInteger n)(eventuality e)))) All of axioms (3), (4), (5), (6), (11), and (13) are instantiations of axiom schemas. There will, in principle, be one set of these for each predicate in the language. All eventualities have this structure. (forall (e) (15) (iff (eventuality e) (exists (p n) (and (pred p e)(arity n e) (forall (i) (if (and (posInteger i) (leq i n)) (exists (x) (argn x i e)))))))) That is, if e is an eventuality, then it has a predicate p, an arity n and n arguments. Since the domain of discourse of the logic is the class of _possible_ individuals, this axiom does not say anything about whether or not the eventuality or its arguments exist in the real world. That has to be asserted separately with the predicate "Rexist". Moreover, it is possible to know some properties of an eventuality without knowing its whole structure as given in Axiom (15). We may know that something happened and that it was loud, without knowing that it was the event of a bookcase falling over. The idea of reifying events is usually attributed to the philosopher Donald Davidson (1967), although he was reluctant to reify states as well, and he did not individuate events as finely as we do. The linguist Emmon Bach (1981) recognized the need for a concept that covered both states and events and introduced the term "eventuality". A relatively brief exposition of eventualities as used here can be found in Hobbs (1985) and a more extensive exposition in Hobbs (1998). The latter contains a number of arguments for the need for eventualities, ways of looking at eventualities, and arguments for very fine individuation. Reification of states and events is a common device in artificial intelligence and linguistics. They are called "states of affairs" in the Head-driven Phrase Structure Grammar of Pollard and Sag (1994).They are called "situations" in the Cyc knowledge base, and at least resemble the situations of the Situation Semantics of Barwise and Perry (1983), though both of these differ substantially from the situations of the situation calculus of McCarthy and Hayes (1969) and Reiter (2001). The latter relate not to a characterization of a possible chunk of space-time, but rather describe the entire state of the world at a given instant. The next instant is another situation. This formalism is good for applications in which a single agent is the only thing that is effecting changes in the world, but it is extremely clumsy for representing natural language or expressing a rich theory of commonsense psychology. Predicates Introduced in This Chapter (eventuality e): e is an eventuality. (Rexist e): e really exists in the real world. (gen e1 e2): e1 generates or entails the existence of e2. (argn x n e): x is the nth argument of e. (arg x e): x is an argument of e. (arg* x e): x is an argument of e or an arg* of an argument of e. (pred p e): p is the predicate of e. (predicate p): p is a predicate. (arity n e): n is the arity or the number of arguments of e. In addition, we used but have not yet explicated the following predicates: (nonnegInteger n): n is a nonnegative integer. (posInteger n): n is a positive integer. (leq n1 n2): n1 is less than or equal to n2. These will be explicated in Chapter B9, but their meaning should be obvious in the meantime. References: Bach, Emmon, 1981. On Time, Tense, and Aspect: An Essay in English Metaphysics'', in P. Cole, ed., {\it Radical Pragmatics}, pp. 63-81, Academic Press, New York. Barwise, K. Jon, and John Perry, 1983. {\it Situations and Attitudes}. MIT Press, Cambridge, Massachusetts. Davidson, Donald, 1967. The Logical Form of Action Sentences'', in N. Rescher, ed., {\it The Logic of Decision and Action}, pp. 81-95, University of Pittsburgh Press, Pittsburgh, Pennsylvania. Hobbs, Jerry R. 1985. Ontological Promiscuity.'' {\it Proceedings, 23rd Annual Meeting of the Association for Computational Linguistics}, pp. 61-69. Chicago, Illinois, July 1985. Hobbs, Jerry R. 1998. The Logical Notation: Ontological Promiscuity.'' Chapter 2 of {\it Discourse and Inference}, available at http://www.isi.edu/~hobbs/disinf-tc.html McCarthy, John and Patrick Hayes, 1969. Some Philosophical Problems from the Standpoint of Artificial Intelligence'', in Donald Michie and Bernard Meltzer, eds.,{\it Machine Intelligence 4}, Edinburgh University Press, Edinburgh, Scotland. Pollard, Carl, and Ivan A. Sag, 1994. {\it Head-Driven Phrase Structure Grammar}, University of Chicago Press and CSLI Publications. Reiter, Ray, 2001. {\it Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems}, MIT Press, Cambridge, Massachusetts.