B16. MODALITY
1. The Predicates "Rexist" and "atTime"
Eventualities exist in a Platonic universe of possible individuals --
entities, states and events. If they happen to actually occur in the
real world, that is one of their properties, and we express it with
the predicate "Rexist". Real existence can be thought of as one
possible mode of existence. There are others. The eventuality could
be part of someone's beliefs but not occur in the real world. It
could exist in someone's imagination. It could exist in some
fictional universe. It could be merely possible rather than real. It
could be likely. It could be unlikely, or not real, or impossible.
An especially important modality is "happening at a particular time"
or "atTime".
We have gone to some effort in this book to simplify the logical
formulas as much as possible. Using unprimed predicates when possible
and primed predicates only when necessary has been one way of doing
this. Another issue we debated was whether to include time arguments
in every predication, and we decided against it in order to keep the
notation relatively simple. But that raises the question: Just what
is the relation among unprimed predicates, "Rexist", "atTime" and
temporal arguments.
The relation between unprimed predicates and "Rexist" is
straightforward, as expressed in Axiom Schema B1.1, repeated here.
(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. The
expression "(p x)" is really an abbreviation for
(and (p' e x)(Rexist e))
The relation between "atTime" and temporal arguments is similarly
straightforward. Suppose p' is a predicate taking an eventuality
argument e and some sequence of other arguments x -- "(p' e x)" -- and
p_t is a corresponding predicate with no eventuality argument but with
an extra time argument t, saying that p is true of x at time t. Then
the equivalence of the two representation styles would be captured by
the following axiom schema.
(forall (x t) (2)
(iff (exists (e) (and (p' e x)(atTime e t)))
(p_t x t)))
The relation between "Rexist" and "atTime" is trickier to describe.
The predicate "Rexist" is fine for describing a world that does not
change. The expression "(Rexist e)" simply means that the
eventuality holds or is happening in that world. But in a world that
changes, it is not obvious what "Rexist" means. Let's work into it
with an example. Suppose we want to write an axiom that says that if
someone smiles, they are happy.
(forall (x) (if (smile x)(happy x))) (3)
The smiling and the happiness have to be true at the same time. The
implication does not hold if x smiles in April and is happy in July.
We can relate "Rexist" to time if we say that it means that the
eventuality exists at some stipulated instant we call "Now". We will
say that an instant is Now by means of the expression "(Now t)". Now
exists and is unique.
(exists (t) (Now t)) (4)
(forall (t2 t2) (if (and (Now t1)(Now t2))(equal t1 t2))) (5)
Then for something to really exist is for it to happen or hold Now.
(forall (e t) (6)
(if (Now t)
(iff (Rexist e)(atTime e t))))
Then the illustrative axiom (3) says that if x smiles now, x is happy
now. The two times have to be the same.
But since any time can be Now, the expression of the axioms in terms
of "Rexist" or, equivalently, unprimed predicates, is merely a way of
anchoring all the predications in a single moment of time. Axiom (3)
holds for any possible Now, and hence for any possible instant. Thus,
we can go from
(forall (x) (if (smile x)(happy x)))
to
(forall (e1 x)
(if (and (smile' e1 x)(Rexist e1))
(exists (e2)(and (happy' e2 x)(Rexist e2)))))
by Axiom Schema (1); and then to
(forall (e1 x t)
(if (and (Now t)(smile' e1 x)(atTime e1 t))
(exists (e2)(and (happy' e2 x)(atTime e2 t)))))
by Axiom (6); and then to
(forall (x t)
(if (smile_t x t)(happy_t x t)))
by Axiom Schema (2) and because any time t can be Now.
2. Positive Modalities
Of the various modalities, the most important are what can be called
"positive modalities". Generalizations can be captured at this level,
and we will do so by introducing the predicate "PosMod" as a cover
term for all positive modalities. "PosMod" takes two arguments.
Although we will always use "PosMod" applied to eventualities, there
is no reason it cannot be applied to other entities as well. Its
second argument will be a predicate that labels the modality.
(forall (e p) (7)
(if (PosMod e p)
(predicate p)))
For example, "Rexist" is a positive modality.
(forall (e) (if (Rexist e)(PosMod e Rexist))) (8)
The predicate "Rexist" as the second argument of "PosMod" labels the
modality as real existence.
The principal property of positive modalities is that Modus Ponens can
be applied within the modality. If an eventualty exists in some
modality, and that eventuality implies another eventuality, the second
eventuality also exists in that modality.
(forall (e1 e2 m) (9)
(if (and (PosMod e1 m)(imply e1 e2))
(PosMod e2 m)))
Thus, if e1 really exists and e1 implies e2, then e2 also really
exists. Possibility and likelihood are also positive modalities. If
e1 is possible and e1 implies e2, then e2 is also possible. If e1 is
likely and e1 implies e2, then e2 is also likely.
Negation is clearly not a positive modality. Flying implies moving,
but the fact that I am not flying does not mean that I am not moving.
An agent's belief is also not a positive modality, in this strict
sense. The agent may not believe the implication. We could introduce
the notion of a positive epistemic modality, of which knowledge and
belief are instances,
(forall (a e) (if (believe a e) (PosEpMod a e believe))) (10)
Then the implication in the antecedent of the Modus Ponens rule would
have to not merely hold but would have to be believed. There are
problems with this, however. We will return to this issue in Chapter
C1.
3. Possibility and Necessity
Possibility is with respect to a set of constraints. For example, is
it possible to put an X in the central square in tic-tac-toe if your
opponent has already put an O there? No, if you accept the rules of
the game. But if the only constraints you accept are the laws of
physics, then, yes, you can. It is possible.
Thus, the predicate "possible" has two arguments, an eventuality and a
set of constraints.
(forall (e c) (11)
(if (possible e c) (and (eventuality e)(eventualities c))))
For something to be possible with respect to a set of constraints is
for those constraints not to rule it out. An eventuality is possible
if and only if the constraints do not imply a negation of the
eventuality.
(forall (e c) (12)
(iff (possible e c)
(forall (e1)
(if (not' e1 e)
(not (if (Rexist c)(Rexist e1)))))))
That is, e is possible with respect to a set of constraints c just in
case whenever e1 is some negation of e, then it is not the case that
c's real existence implies e1's real existence.
When we hear a statement that something is possible, part of the job
of interpreting it is deciding from context what the set of
constraints is, and in discourse statements of possibility are
frequently accompanied by an indication of the constraints.
Possibility is a positive modality.
(forall (e c) (if (possible e c)(PosMod e possible))) (13)
Necessity is similarly with respect to a set of constraints. An
eventuality is necessary if the set of constraints implies it. If my
king is in check in chess, is it necessary for me to move it out of
the way or interpose another piece? Yes, if I accept the rules of
chess. But if I accept only the laws of physics, no. I don't have to
play the game at all.
The predicate "necessary" has two arguments, an eventuality and a set
of constraints.
(forall (e c) (14)
(if (necessary e c) (and (eventuality e)(eventualities c))))
An eventuality e is necessary with respect to a set of constraints c
if and only if c implies e.
(forall (e c) (15)
(iff (necessary e c)(imply c e)))
Necessity is a positive modality because of the transitivity of
"imply".
(forall (e c) (if (necessary e c)(PosMod e necessary))) (16)
It is a theorem that if an eventuality is possible, its negation is
not necessary.
(forall (e c) (17)
(if (and (possible e c)(not' e1 e))
(not (necessary e1 c))))
Impossibility is the negation of possibility.
(forall (e c) (18)
(iff (impossible e c)
(and (eventuality e)(eventualities c)
(not (possible e c)))))
Impossibility is obviously not a positive modality.
4. Likelihood
Possibility is one common judgment we make about eventualities in
situations of uncertainty. Likelihood is another. We use the term
"likelihood" to stand for the commonsense notion for which probability
is the cleaned-up, mathematical version. We will take likelihood to
be a more qualitative notion, of which mathematical probability will
be a specialization, but which also covers the vague judgments we make
in everyday life, as when we say it is likely to rain without having
any real mathematical basis for the judgment.
We could define likelihoods as elements in the closed interval [0, 1],
and this is certainly one possible model of the axioms we propose.
But to allow only this model would be artificial. Instead, we will
say there is a scale of likelihoods, maybe only a partial ordering,
with a qualitative structure on it. The predicate "likelihood" will
express a relation between these likelihoods and eventualities. In
addition, like possibility, likelihood is with respect to an implicit
set of constraints that in a sense defines the sample space. Making
the constraints an argument allows us to relate likelihood to
possibility and to entailment.
(forall (d e c) (19)
(if (likelihood d e c)(and (eventuality e)(eventualities c))))
Normally, e will be an eventuality type, but we will not require
this.
The d values will be elements on a scale. For now we will assume we
have a "lowerLikelihood" relation. Then the scale will be the scale
with likelihoods as its elements and the "lowerLikelihood" relation
as its partial ordering. We will call this the "likelihoodScale".
(forall (s) (20)
(iff (likelihoodScale s)
(exists (s1 e d1 d2)
(and (forall (d)
(iff (member d s1)
(exists (e c) (likelihood d e c))))
(lowerLikelihood' e d1 d2)
(scaleDefinedBy s s1 e)))))
We have introduced three predicates, "likelihood", "likelihoodScale",
and "lowerLikelihood", but these are really only ways of referring to
different aspects of likelihoods. We have not yet begun to nail down
the content of the notion. We will take some initial steps now.
If something has a nonzero likelihood, then it is possible. That is,
we cannot prove from the constraints that it does not occur. However,
we also cannot prove from the constraints that it does occur. Either
possibility is consistent with the constraints. Given an eventuality
e and a set of constraints c, we can ask what other set s of
eventualities would have to obtain in order to entail that e also
obtains. Let's call this set "alsoRequired". The expression
"(alsoRequired s e c)" says that s is a set of eventualities that will
entail the real existence of e, over and above c.
(forall (s e c) (21)
(if (eventuality e)
(iff (alsoRequired s e c)
(and (eventualities s)(eventualities c)
(exists (e1)
(and (and' e1 c s)(imply e1 e))))))
Now suppose whenever we add a set of eventualities to get e1 to exist
it also gets e2 to exist. Then we can say that e2 is at least as
likely as e1. That is, the more we have to assume will happen, the
less likely it is. We can capture this observation in terms of subset
consistency if we construct a relation between the likelihood of an
eventuality and the set of eventualities that are also required to
make it obtain. Let us extend the "alsoRequired" relation to take
likelihoods as well as eventualities as their second arguments.
(forall (s d c) (22)
(if (likelihood d e c)
(iff (alsoRequired s d c)(alsoRequired s e c))))
We have now constructed a relation between likelihoods and sets. But
the subset relation corresponds with a higher likelihood, not a lower
likelihood, so we will define a reverse likelihood scale and say that
that is subset-consistent with this relation.
(forall (s) (23)
(iff (revLikelihoodScale s)
(exists (s1)(and (likelihoodScale s1)(reverse s s1)))))
Finally, we can say that a reverse likelihood scale is
subset-consistent with respect to the alsoRequired sets.
(forall (s e0 s1 d c e) (24)
(if (and (revLikelihoodScale s)(alsoRequired' e0 s1 d c)
(likelihood d e c)(inScale d s))
(subsetConsistent s e)))
This development could be recast in terms of possible worlds, had we
developed an approach to possible worlds. If the set of possible
worlds in which e2 occurs is a subset of the set of possible worlds in
which e1 occurs, e1 is more likely. We have presented the
propositional equivalent of this.
The second set of constraints on determining likelihoods comes from
combining likelihoods for component eventualities to determine
likelihoods of compositie eventualities. In particular, we can say
something about how likelihoods operate under "and", "or" and "not".
We cannot be as precise about this as we can in probability theory,
but we can say that the likelihood of the "and" e of two eventualities
e1 and e2 is less than or equal to the likelihood of each of them.
(forall (s e e1 e2 d d1 d2 c) (25)
(if (and (likelihoodScale s)(inScale d s)
(inScale d1 s)(inScale d2 s)
(likelihood d1 e1 c)(likelihood d2 e2 c)
(and' e e1 e2)(likelihood d e c))
(and (leqs d d1 s)(leqs d d2 s))))
The "or" e of two eventualities e1 and e2 is greater than or equal to
each of them.
(forall (s e e1 e2 d d1 d2 c) (26)
(if (and (likelihoodScale s)(inScale d s)
(inScale d1 s)(inScale d2 s)
(likelihood d1 e1 c)(likelihood d2 e2 c)
(or' e e1 e2)(likelihood d e c))
(and (leqs d1 d s)(leqs d2 d s))))
If the likelihood of an eventuality is the top of the likelihood
scale given constraints c, then it is entailed by c.
(forall (d e c s) (27)
(if (and (likelihood d e c)(likelihoodScale s)(top d s))
(necessary e c)))
If the likelihood of an eventuality is the bottom of the likelihood
scale, then it is impossible given the constraints.
(forall (d e c s) (28)
(if (and (likelihood d e c)(likelihoodScale s)(bottom d s))
(impossible e c)))
If something has a likelihood above the bottom of the likelihood scale
with respect to a set of constraints, it is possible with respect to
those constraints.
(forall (d e c s) (29)
(if (and (likelihood d1 e c)(likelihoodScale s)(bottom d s)
(lts d d1 s))
(possible e c)))
A word about the interaction of likelihood and time: We make
likelihood judgments about past, present and future eventualities.
It is likely that John Wilkes Booth acted alone.
It is likely that Kim secretly loves Pat.
It is likely that the Democrats will win the next presidential
election.
But usually it is the likelihood of future events that is at issue.
To say that it is likely to rain tomorrow we do not say that it is
likely that the rain really exists -- that would be now -- but that
the "tomorrow-ness" of the rain really exists. It is the "atTime" or
other temporal proposition to which we assign a likelihood.
A likelihood scale consists of elements that are all likelihoods and
hence have a common property; the scale can then be a potential
ground.
(forall (s) (if (likelihoodScale s)(ground s))) (30)
If the likelihood of an eventuality e is d, we can say that e is at d
on the likelihood scale.
(forall (d e c s) (31)
(if (and (likelihood d e c)(likelihoodScale s)(inScale d s))
(at e d s)))
The likelihood scale is conceptually vertical. We talk about
likelihoods being higher or lower.
(forall (s) (if (likelihoodScale s)(vertical s))) (32)
Thus, we can talk about the likelihood of something increasing.
One eventuality is more likely than another with respect to a set of
constraints if its likelihood is higher.
(forall (e1 e2 c) (33)
(iff (moreLikely e1 e2 c)
(exists (s d1 d2)
(and (likelihoodScale s)(inScale d1 s)(inScale d2 s)
(likelihood d1 e1 c)(likelihood d2 e2 c)
(lts d2 d1 s)))))
An eventuality is "likely" with respect to constraints c if its
likelihood d is on the high end of the scale of likelihoods.
(forall (e c) (34)
(iff (likely e c)
(exists (s d s1)
(and (likelihood d e c)(likelihoodScale s)
(Hi s1 s)(inScale d s1)))))
If an eventuality is likely, its negation is not likely.
(forall (e e1 c) (35)
(if (and (likely e c)(not' e1 e))
(not (likely e1 c))))
The predicate "likely" represents a positive modality.
(forall (e c) (if (likely e c)(PosMod e likely))) (36)
That is, if e1 is likely and e1 implies e2, then e2 is also likely.
Predicates Introduced in this Chapter
(Now t): t is an instant stipulated to be Now.
(PosMod e p): The positive modality p holds of eventuality
e.
(possible e c): e is possible with respect to a set of
constraints c.
(necessary e c): e is necessary with respect to a set of
constraints c.
(impossible e c): e is impossible with respect to a set of
constraints c.
(likelihood d e c): d is the likelihood of e really existing,
given constraints c.
(likelihoodScale s): s is a scale of likelihoods.
(lowerLikihood d1 d2): d1 is a lower likelihood than d2.
(alsoRequired s e c): s is a set of eventualities which when added
to constraints c entails e or an
eventuality with likelihood e.
(revLikelihoodScale s): s is the reverse of a scale of likelihoods.
(moreLikely e1 e2 c): e1 is more likely than e2 given constraints c.
(likely e c): e is likely, given constraints c.
The predicate "PosEpMod" is explicated in Chapter C1.
(PosEpMod e a p): The positive epistemic modality p holds of
eventuality e for agent a.
The predicates "smile" and "happy" were used only for illustrative
purposes.