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.