Probabilistic Foundations for Operator Logic B. H. Slater The British Journal for the Philosophy of Science, Vol. 44, No. 3. (Sep., 1993), pp. 517-530. Stable URL: http://links.jstor.org/sici?sici=0007-0882%28199309%2944%3A3%3C517%3APFFOL%3E2.0.CO%3B2-F The British Journal for the Philosophy of Science is currently published by Oxford University Press.
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Brit. 1. Phil. Sci. 44 (1993), 5 I 7-530 Printed in Great Britain
Probabilistic Foundations for
Operator Logic
B. H. SLATER
1 2 3 4 5 6
Introduction Privileged Access Some Probabilistic Definitions The Case of Knowledge lrnrnediate Consequences The Parallel w i t h Obligation
I INTRODUCTION
I give in this paper a doxastic and epistemic logic based on probability theory. A deontic logic of this kind has been provided before, by George Schlesinger (Schlesinger [I985 (a)], [ l 9 8 5 (b)]),and I will have a word to say later in further explication of that. Also, it is well known that Frank Ramsey based his account of probability on belief (Ramsey [19 78]),so he formulated the reverse connection to one established here. But I take my start from a basic question in modern philosophy, which enables me, in the first place, to deal with belief and knowledge in the same way as obligation, and specifically permits the doxastic logic to be easily extended to an epistemic one. That basic question in philosophy, which underlies much of epistemology, was traditionally answered by empiricists in a well known way (Ayer [1936], [1956]). Our knowledge of the world, they held, was formed from a series of inferences made on the basis of immediate experiences. These immediate experiences, providing 'sense data', were the unshakable foundations for our knowledge of the external world. The story will be familiar to most readers, even if they have been persuaded that the theory of knowledge is more adequately dealt with within, say, Idealism. In particular, from within Idealism it may seem to be easier to construct a theory of moral knowledge, and social concepts generally. Now it will be a central part of the present account that we do not have access to objective sense data-a difficulty which sometimes persuades people that Idealism must be right. Also there will be room within the theory for choice and varying culturally determined concepts-further features traditionally associated with the imaginative base
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of Idealism. But the theory of belief, knowledge and obligation which will emerge is instead stamped more with twentieth century movements like Existentialism than with either of the two traditional rival philosophies. For it places action at the base of epistemology in place of any kind of experience, whether sensory or imaginative. What is firm and unshakable, in the first place, if anything is, is an action, not a sensation, and, even less, an image. What we have privileged access to is not something given but something made. 2 P R I V I L E G E D ACCESS
It will be useful to have some definitions at the outset, of some key concepts in epistemology: Knowledge, Belief, and Wonder. Writing 'Kap' for the operator expression 'a knows that p', we can say that, according to empiricism, a defining feature of an immediate experience is that it is a part of consciousness, and so is one about which the owner of the experience must have complete knowledge. Alston (Alston [197l]) formulated this 'Omniscience' by making p 3 Kap, necessary, when 'p' describes the immediate experience in question. Because of the properties of knowledge, however, this expression for basic experiences may be tightened so as to make it necessary that
-
p Kap. As I said, it is part of my brief to attack empiricism on this matter, and one can do so very firmly with this point. For not only, as a result, is the knowledge operator redundant in this context, the full equivalence leaves the supposed experience entirely without any content. For the experience is revealed to be no more than just the knowing of itself. Now it is sometimes said, regarding taking omniscience as defining certain experiences, that it presumes the experiences in question cannot be too subtle. Thus someone seeing magenta, it has been said, need not recognize it as such, and the empiricist defence to this charge is commonly "Well even if I don't know it is magenta, at least I know it is that colour!' To this maybe there is an inclination to respond 'He is left with that personal ostension even if the colour is crude', or 'How can he even be sure it is a colour? Personal ostension cannot define anything' (cf. Hacker [1972], p. 233f., Kenny [1973], p. 182f.).But the above equivalence gives us a stronger point to make than these. For the content of all 'immediate experiences' is shown to be quite empty by that equivalence, so it is a logical point which settles the matter, and absolutely. Maybe an omniscient being contingently knows everything, but if it is a definitional part of some event that it be known, that can only be 'navel gazing'.
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But while there must be vacuity with basic experiences, there can be no vacuity with basic actions. If you believe your hand is rising, and it is subject to your will, then your hand is rising. But even if your hand is rising you need not believe this, since, for instance, you may be asleep. So no corresponding equation between the belief and its object is obtainable with the cognition involved in conation. If we write 'Bap' for the operator expression 'a believes that p', and take 'p' to describe a movement entirely subject to a's will, we can follow Alston again and define 'Infallibility' by making it necessary that Bap 2 Kap, which entails the necessity of B~PIP for the specific 'p's in question. But while that anchors the mind in behaviour, at a n appropriately basic level, there is no reverse requirement, p 2 Bap, reducing the possession of the belief completely to the physical movement. Thus the movement can occur without the mental element-it is then simply not an action. But we must be careful to rule out the inactive imagination, allowed within Idealism, when we thus link basic actions and beliefs. For beliefs invariably are linked with actions, since one cannot be said to have a belief unless one acts accordingly. And the same is not true of imaginings. Moreover, it is a special kind of belief that is involved with basic actions, which makes the link, in that case, even more immediate. Actions are generally thought of as the result of a complex of beliefs and desires. Schematically, when wanting W, and believing action A will lead to obtaining W, we do action A. The case of basic actions, however, cannot be construed on quite this model, since there is no prior action then needed to obtain the desired end, and so no causal belief about how to satisfy any want. Being intentional, however, means that another kind of belief is central to basic actions, namely the belief just that we are doing them. It is that belief alone which, as above, logically guarantees the action. Maybe we often do such actions 'just because we want to', but the relation between any such want and the resultant action is not logical but causal. The relation between the action and the belief that we are doing it is not a causal relation. This kind of belief is expressed in the action itself, by the action being done knowingly, i.e. 'willingly'. So possession of the belief is not separate from or antecedent to the associated behaviour. This point needs to be appreciated to understand the difference between Belief and Thought. Certainly one's hand need not move if one merely thinks one's hand is moving, but that is expressly because thoughts, unlike beliefs, need not be apparent in overt bodily behaviour. Thoughts, after all, may just be
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expressed in words, and so may be insincere, or impotent, i.e. not lead to the appropriate deeds. In connection with such insincere and impotent thoughts, there is therefore one further important concept in the current series. The notion of 'wonder' characterizes a third alternative, where there is neither belief nor knowledge, but just thought. Thus, for example, in the above cases of basic actions, there is a one-way gap in between the belief and the movement. So while with basic actions if a believes that p, then p, and if a believes that l p then l p still p or l p may occur without either Bap or Balp. But we say, if someone neither believes that p nor believes that l p , that he wonders whether p. So the exceptional case where a's hand goes up although he does not believe it is a case of this wonderment. We enter 'wonderland' characteristically when we are asleep, though if, while awake, one's hand is 'asleep' one can have the same kind of detached experience. One of the failures of Idealism is that it did not separate out this area of the mind from more practical areas. More recent twentieth century traditions, however, have sieved out such non-functional parts of the mind, while basing their positive account of knowledge and obligation on action. This points to the centrality of belief in the above sense. Ramsey, it will be remembered, required beliefs to be evident in practical action, if only to the extent that they were related to the individual's preparedness to bet. But we go further than that here, since preparedness to bet only certainly relates to subjective assessments of belief, and so does not take into account beliefs the agent may be unconscious of, or indeed mistaken about. Many other philosophers have recently essayed such more 'activist' views, including Wittgenstein, for instance: Giving grounds, however, justifying the evidence, comes to a n end;-but the end is not certain propositions' striking us immediately as true, i.e. it is not a kind of seeing on our part; it is our acting which lies at the bottom of the language game (Shiner [1977/8], p. 105).
3
SOME PROBABILISTIC DEFINITIONS
Now 'wondering whether p' may clearly be defined probabilistically as that state of awareness where the probabilities of p happening and not happening are the same. When the probability is one-half, there is no belief that p nor belief that l p . The states of positive belief, therefore are simply when the probabilities are not one-half. So we can start to formulate exactly some of the above ideas by separating out three cases, as follows,
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521
which specifically formulate belief, wonder and disbelief, respectively. Knowledge is then clearly given by
though much more will have to be said about this key concept later. We can then define 'thought' by the obverse relation
This covers knowing p, believing p, wondering whether p, and most ways of disbelieving p, but excludes knowing otherwise, i.e. excludes
So there is knowledge when and only when there is no thought of the reverse. Contrariwise, there is most often a state of 'uncertainty' when one both thinks that p and thinks that l p , which state is more general than, but includes the case of wonder. We may also include in the latter case matters about which one simply has no opinion, i.e. whose probability is one half by the principle of indifference. It remains to remind readers of the axioms of probability theory, before we tie in these probabilistic expressions with expressions for different individuals' beliefs. In fact we rieed only say (Rescher [1968], p. 183) that, given a domain D of statements closed under the statement generating operations of propositional and predicate logic, then, for all p in D
If p is necessary then pr(p)= 1,
+
If l(p.q) is necessary then pr(p v q) = pr (p) pr (q). These axioms are sufficient to define a probability measure, although, importantly, they do not finalise pr(p v q) when p and q are compatible. That probability is formally indeterminate, being simply not greater than the sum of pr(p) and pr(q),but not less than each separately. This looseness will feature when we come to assess how knowable probabilities are, in section 6, but we shall see that the more determinate aspects of the probability calculus are themselves already sufficient to tell us a great deal about knowledge and belief, in Section 5 .
4
T H E CASE O F K N O W L E D G E
But clearly, to get expressions specifically for individuals' beliefs we must, first of all, relativise the above probabilities in some way. But using the looseness in the definition of probability to allow self assessments of beliefs in terms of subjective probabilities would run into trouble with the definition of know-
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ledge. A person's own avowal of certainty would then establish it. More plausible is the quasi-empiricist move, which would conditionalize the probabilities on the immediate knowledge available to each individual. So long as we do not take the subject's immediate knowledge to be given to him as a n experience, or derived from his imagination, but instead see it as created by him, through his actions, we will have escaped the major philosophical difficulty with this approach which we identified at the start. For individual a we may then write.
to express the fact that he has a belief that p. Here LY describes a's immediate knowledge, and so the conditional probability is defined, i.e. not pr(cc)= 0. As a result, we get a n objective estimate of the probability of p, from a's perspective, i.e. conditional on a's immediate knowledge, although what is used as the probability measure may vary, as before. Being objective others must be able to judge it on the overt behaviour of a, and so, again, his own self-estimate is not necessarily a guide to it. What someone believes he believes he does not necessarily believe, since pr(Bap/r) > does not entail Bap. But the major technical problem with the above formulation of belief is this: how can it be extended to give a n account of knowledge? How can
3
express a's knowledge that p, i.e. entail that p? For might it not be the case, while that expression held, that, say, for individual b
Then we would seemingly be committed to saying p was both true and false, or at least 'true for a', 'false for b' etc. The rationalist solution to this general problem of personal access to objective truth was to limit all that can be known to the necessary truths of mathematics and logic, i.e. things independent of a and b. But that produces a 'modal collapse', with no discrimination between 'Necessarily, p' and 'p'. The empiricist solution, as we have seen, was to limit the conditions in the probability judgements to immediate experiences, now shown to be vacuous. The subjective idealists, giving a spurious legitimacy to imagination and the resultant wonder, believed knowledge did not have a guaranteed intersubjective consistent form, and so saw no solution to the problem, with many of them even ending up accepting, as true, contradictions, But other idealists, notably Kant, saw another way out of the impasse, and it is easy to see that line of thought is all we need to follow here. For the logic of the matter only requires, if pr(p/a) = 1 entails p, that it rule out pr(lp/B) = 1, for all p, i.e. for all statements of the type which
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can occur in place of 'a' in this entailment. But since, in particular, we must have
what that means is that LY must be true, while we must not have, for instance, i.e. that P is incompatible with r. Yet that compatibility is automatically achieved, since also, P must be true, i.e.
But quite generally, if pr(P/r)>O then, given pr(a)> 0, one cannot have pr(p/cx)= 1 and pr(lp/b) = 1, though, of course one can still have pr(p/a)>+ and pr(lp/P) > To operate with a concept of knowledge on the above lines, therefore, people must just be presumed to have beliefs which are based on facts. Incompatible beliefs are then not ruled out (justifying the subjective idealists to some extent), but all proper knowledge will be consistent. This might look like a form of causal theory of knowledge, but the present transcendental deduction from the possibility of consistent knowledge has a conclusion which is not a causal hypothesis. Its regulative conclusion merely requires we treat basic actions as providing the ultimate unshakable facts upon which we found any firm beliefs.
t.
5
IMMEDIATE CONSEQUENCES
Now, as was mentioned before, there are already some clear advantages of the present account of knowledge and belief, deriving from the more determinate aspects of the probability calculus, which we may check through before proceeding further. (i) Perhaps the prime virtue of the present account is that it allows the 'logicality' of the mind to be quite limited. Other theories have required 'deductive omniscience', i.e. the power to generate all inferences. But it is not a theorem, on the present account of belief, for instance, that
so whatever follows from p.q is not a conclusion which needs to be drawn when p and q are believed separately. In technical terms, Adjunction is not automatic. (ii) What is a theorem, however, is
and this traditionally has been a source of worry (Cresswell [1985], Ch. I), since everyone as a consequence believes all necessary truths. But fear of this
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consequence is much reduced once it is realized Adjunction is not given. Distinctions amongst learners of mathematics and logic can then still be made-though in a more practical content. For it is now respected as quite reasonable that people don't always 'put two and two together'. Hence someone who believes he has two apples, and also believes he has a further two apples need not believe he both has two apples and a further two apples-and only from the latter belief does the belief immediately follow that he has four apples. (iii) Thus what is also a theorem is
+
allowing Ba(x = 2 2) to entail Ba(x = 4), and also, for instance, Ba(p.q) to entail Bap. But this means we have a characterization of the 'obvious' inferences commonly taken to delimit the extent to which the mind can be guaranteed to operate. The inferences it alone draws are the irnrnediate ones. The mind thus does not draw all consequences of all beliefs, but indeed without some consequences being inferred what could be the content of any single belief? That content, we can now see, is given simply by producing, and unpacking the implications of that single belief. Even so there theoretically could be a n infinity of such implications, which runs counter to some views of the mind-as a physical instrument running occurrent brain processes. Clearly the product of any such instrument must be finite. But the 'producing' and 'unpacking' of implications are on our part, they do not describe any temporal brain process on the part of the agent. For, as before, having a belief is a matter of relating to the world in a certain way. But from this it logically, and not causally follows that the subject has another belief, i.e. relates to the world another way. So what we unpack or produce are merely third party re-descriptions of part of the agent's behaviour pattern. (iv) Likewise we can often tell the agent does not have a certain belief. For on the present definition we have Bap 2 TBalp, and we also have
Hence we resolve the further question whether beliefs in the impossible are possible. They are impossible, in both ways that this can be taken. On the present definition of thought, however, contradictory choughts are not ruled out. For the quite consistent pr(p/a)= p r ( l p / a ) =+, means that a both thinks that p and thinks that l p . But still the agent cannot think that p.lp, since P~((P?~P =)0./ ~ ) The point gives us a solution to Kripke's puzzle about belief (Kripke [ I 9 791). For if Kripke's Pierre says 'Londres est jolie' and 'London is not pretty' then, at
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least, his mind is clear to the extent that it is guaranteed to lack a thought which would lead to absurdity-that London both is and isn't pretty. And that perhaps eases Kripke's reluctance to saying that, despite Pierre's impeccable reasoning abilities, he still has contradictory thoughts. He certainly does have contradictory thoughts, even though he cannot adjoin them, as thoughts. What is more in question though is whether he has contradictory beliefs. For it is now clear that the disquotational principles Kripke uses to locate Pierre's beliefs shows Pierre is insincere in at least one of the things he says. Sincerity is not a matter of stressed expression, after all, it is a matter of whether the assertion leads to the appropriate behaviour. So the inconsistency of Pierre's thoughts expressly shows that only thought can be involved in at least one side of the case. The question is, what will Pierre do when he gets to London-love it or hate it? Whichever he does he goes against his other word. So his thinking, at least part of the time, must be idle; and perhaps all the time it is idle, if he is not put in a situation where he has to act. (v) In this connection we may remember that Ramsey's theory of probability, which we are largely reversing, has sometimes been criticized for presuming beliefs are 'rational'. But beliefs are by definition 'rational', at least to the extent of being consistent, though they may fail to be entirely 'logical' in other ways, because of the absence of Adjunction. Thus while it is never possible to have Bap. Balp, it is quite possible to have Bap. Baq. Bal(p. q). Thus, as has often been remarked, one might well believe, individually, each of a series of propositions, but still have a proper suspicion that at least one of them is false. But notice that we avoid here standard 'Dutch Book' difficulties with subjective assessments of probability (cf., for instance, Van Fraassen [1984]),because we are using objective judgements of belief, which may be at odds with the agent's self-assessments (cf. Mellor [1980]). A quasi-Freudian 'unconscious' thus naturally enters our present picture of the mind, with the agent's beliefs about his beliefs not necessarily being reliable. (vi) Of more importance for later issues is the related matter that there is no difficulty now with the 'rationality' in the standard definition of knowledge. Indeed, the present analysis shows the way out of Gettier's problem with the standard definition (Gettier [1963]). For, from (iii) we see, for example, that if a believes there is a man in the room, he believes there is a person in the room. But maybe there is only a woman in the room, although a believes there is a man in the room, and cannot be blamed for believing this, i.e. he has taken all reasonable care. Do we want to say that a knows there is a person in the room, since he justifiably believes this, and it is true? But even when one has taken all
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the care one could reasonably be expected to take one needn't (strictly, absolutely) know, simply because the case may be 'unreasonable', and one may still have been deceived, as the skeptic will continually remind us. Hence, as Gettier showed, perfect knowledge is not a matter of justified true beliefwhen 'justified' is taken in the quasi-moral sense just indicated, i.e. that in which it is linked with such legal concepts as having taken all reasonable care. However, there is a logical, and not quasi-moral justification in the present concept of knowledge which leads to a re-establishment of 'justified true belief' as a proper definition of knowledge. For it is easily seen, for example, that (Kap. Kaq) 3 I
