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.