“Knowing oneself” and other problems in epistemic logic

by

J A A K K O H I N T I K K A

[University of Helsinki)

1. It may not appear very graceful of me to cavil at Professor Hector-Neri Castaiieda’s perceptive and generous review of my book Knowledge and Belief‘ in the Journal of Symbolic Logic, vol. 29 (1964), pp. 132-134. I am nevertheless prompted to do so by Professor Castaiieda’s emphasis on the significance of the book as an impetus for further work. Such work might be hampered, it seems to me, if I did not make explicit certain points which I had not succeeded in making clear in the book itself, to judge from Castaiieda’s review. This gives me also a chance to make some constructive suggestions concerning the problems Castaiieda raises.

2. Castaiieda says repeatedly that I have formalized “very strong senses of ’know’ and ‘believe’ of which probably there are no human instances”. The basis of this statement is illustrated by the fact that sentences of the form

(11 “a knows that p 2 a knows that q”

are on my assumptions valid (or, as I prefer to say, self-sustain- ing) as soon as p logically implies q. This fact is taken by Casta- iieda to entail the consequence that, in the sense in which I have used the notion of knowledge in the book, whoever knows some- thing always knows all the implications of what he knows. If

’ Knowledge and Belief. An lntroduction to the Logic of the Two Notions. Contemporary Philosophy Series, ed. by Max Black, Cornell University Press, Ithaca, New York, 1962. 1 - Theoria, 1 : 1966

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this consequence really followed, then it would indeed be true to say that I have presupposed an unnaturally strong sense of knowledge.

However, the consequence follows only if the notion of self- sustenance (validity) is interpreted as truth on every possible occasion (in every possible situation). This interpretation is not the only natural one, and it is in so many words ruled out in my book (see pp. 34-38). Instead, it is proposed that self-sustenance be interpreted as truth in every “epistemically perfect world”, that is to say, in every possible world whose inhabitants all follow up the consequences of what they know far enough to see each particular consequence of what they actively know. If this interpretation of the metalogical notion of self-sustenance is adopted, together with the parallel interpretations of other basic metalogical concepts, then there is no objection to saying that the sense of knowing we are dealing with is essentially our or- dinary sense of knowing. (Strictly speaking, it is the strong sense of knowing in which knowledge is contrasted to true belief.) In brief, instead of sticking to the normal senses of our metalogical notions and imposing a peculiar interpretation on the object- Ianguage notions of knowledge and belief, we might as well keep as close to our commonplace notions of knowledge and belief as possible and modify the metalogical notions instead-notions which have the character of artificial helps to theoretical under- standing anyway.

Although Professor Castaiieda’s criticism is thus somewhat inaccurate, it is nevertheless perfectly fair. There is a price to be paid for the kind of modification suggested above (and in my book). This price is that results formulated in terms of self- sustenance (validity), defensibility (consistency) and other meta- logical notions are not automatically applicable to what people may be said to know or believe on the different occasions that arise in this actual world of ours. They are applicable only in so far as our world approximates, or can be made to approximate, an “epistemically perfect world”. Whether this is the case has to be decided separately in each case. There is not much one can say in general except that metalogical results that can only be

KNOWING ONESELF 3

established by long chains of reasoning usually cannot be as- sumed to be applicable.

In so far as Castafieda’s remarks are intended to call attention to this drawback, they are in order. I agree that it would be eminently worth while to try to develop a formalized logic of the concepts of knowledge and belief which would be freer of this disadvantage than the systems put forward in Knowledge and Belief. In such a system, ( 1 ) will be valid (self-sustaining) only if q logically follows from p by certain especially simple modes of argument. In fact, it seems to me that it would be natural to restrict the validity of (1) to the cases in which the conditional “ p 3 q” is what I have called a surface tautology.z A formalization of the resulting logic of knowledge remains to be carried out.

This suggestion is made plausible by the close connection which there seems to obtain between the notion of surface tautology and the notion of (explicit) meaning. In general, we cannot be sure that a man who knows that p will admit that he knows that q even when “p 3 q” is a surface tautology. However, if it is true to say that the tautological consequences of p (in the sense of surface tautology) only bring out the (explicit) meaning of p , then it may very well be suggested that a man who says that he knows that p but does not admit that he knows that q does not fully understand the import of p in the first place. If so, the apparent failure of (1) to be valid ought to be construed as a failure of the person in question to understand fully his own statement, and not as a failure of the validity of (1). It seems to me that this line of thought might constitute a very good defense of the validity of (1) for the case in which “ p 2 q” is a surface tautology.

3. Another statement in Castaiieda’s review which I have to comment on is his statement that in my system free individual

’ See my papers “Are Logical Truths Tautologies?” and “Kant Vindicated’, forthcoming in Exisrenz und Analytizitiit. 4 Forschungsgespriick des inter- nationalen Forschungszentrums fur Grundfragen der Wissensckafren, ed. by P. Weingartner, Pustet, Salzburg, 1966.

4 JAAKKO HINTIKKA

variables occuring within the scope of the operator “a knows that” (or of its companion operator “it is possible, for all that a knows, that”) have to range over individuals whose identity is known to a. This statement is not completely unambiguous. What is true about it is that a restriction is imposed on the ranges of those variables which occur within the scope of the two operators but which are bound to quantifiers outside them (quantification into an epistemic context). Individual terms not bound to any quantifier have the same range no matter where they occur.

Castaiieda says that the conditions I am imposing on the ranges of variables make it impossible to formulate, without contradiction, statements like “There exists an object of which a does not know that it exists”. He is right in pointing out that the translation of this statement into my symbolism, that is,

(2) “(Ex) -K,(Ey) (x=y)”

is indefensible (inconsistent). He may also be right in implying that this result is undesirable. In fact, I have independently come to consider it as undesirable myself.” However, Castaiieda is not correct in considering the indefensibility of (2) as an inevitable consequence of my approach to quantification in epistemic con- texts. As one can easily verify, an argument which shows the indefensibility of (2) has to make use of either (C .EK=EK= *) and (C.EK=) or else (C.EK=”). Now I have come to think that the conditions (C .EK=) and (C .EK= ’) cannot be ac- cepted universally in any case. It is tempting to say, as I said in Knowledge and Belief (p. 160), that “if you know who does something you ips0 facto know tha t someone does it”. However, there seems to be a perfectly good sense of knowing who a certain person is which does not commit one to holding that the person in question is known to exist. In colloquial language, we might express this somewhat as follows: “If there is such a person as b, I know who he is” or “I know who b is, provided

It was also criticized on different grounds by Dr. P. Weingartner in his contribution to the symposium whose proceedings are referred to in note 2.

KNOWING ONESELF 5

that he really exists”. It is also desirable to take this sense of knowing who (or knowing what) as the basic sense, since the stronger sense in which my self-quotation is true can then be expressed as a conjunction of knowing who b is (in the weaker sense) and knowing the existence of b.

This change of mind occasions a revision of pp. 160-164 of my book. The requisite changes are perfectly straightforward ones, however; and what is said in the book can in any case be reinstated by inserting the additional clauses into the statements discussed there that are needed to express the strong sense of “knowing who” in terms of the weak sense. Thus Castaiieda is again right in pointing out that a change is desirable in my dis- cussion, but he is not right in his diagnosis of the nature of the difficulty .

4. The major novelty in Castafieda’s review is that he calls atten- tion to the interesting difference which there is between the two expressions (i) “a knows that he . . .” and (ii) ”a knows that a . . He suggests in effect that this distinction is neglected in Knowledge and Belief; according to him, the rule (C.KK*) pre- supposes in its qualified form that we construe knowledge about oneself in sense (i) whereas ( C . =K) presupposes that we con- strue it in sense (ii).

The problems to which Castaiieda is here calling our attention are interesting and important, partly because of their intrinsic significance and partly because they seem to constitute a special case of certain more general problems. These more general prob- lems pertain to the obvious difference that there is between speaking, on one hand, of the different individuals to which a term-say b-may refer in the different “possible worlds” that are compatible with what someone knows, and speaking, on the other hand, of that unique individual which as a matter of fact happens to be referred to by b. This difference is illustrated by the difference between knowing, on one hand, something of the

’ A closely related distinction was discussed by Peter Geach in his note, “On Beliefs about Oneself’, Analysis, vol. 18 (1957-58), pp. 23-24.

6 JAAKKO HINTIKKA

next president of Brazil, without knowing who he is or will be (e.g. that he will have to be a member of a certain party), and knowing, on the other hand, something about the man who in fact will turn out to be the next president, without knowing that he will be elected. Speaking somewhat loosely, it seems to me that the difference in which Castaiieda is interested, viz. between “a’s knowing something about a” and “a’s knowing something about himself”, is nothing but a special case of this distinction. Someone, say the person referred to by b, knows something about the referent of b if the (normally) different individuals to which b may refer, for all that he knows, are known by him to have something in common. On the other hand, he knows something about himself if he knows something of that unique individual that b in fact refers to. For instance, the next president of Brazil may know quite a few things about him- self which he does not know to be true of the next president of Brazil-assuming, of course, that he does not know that he will be elected.

The general difference, and a fortiori the special one, can be formulated in the system presented in Knowledge and Belief. Normally, what we say in terms of a free singular term, say b, pertains to the different individuals to which it refers in the dif- ferent possible worlds we are considering, e.g. in the different possible worlds compatible with what someone knows. Hence (ii) is the normal basic way of understanding a statement which says that the referent of b knows something about the referent of b. For instance, Castaiieda is obviously right in saying that this reading is presupposed in my condition (C - =K).

However, from a statement so construed we can always obtain a statement which pertains to the unique individual in fact re- ferred to by b. This transformation is illustrated by the step which takes us from, say,

(31 to

(41

“K, R (b, c) ”

“(Ex) (x=b & K , R(x , c) j”,

KNOWING ONESELF 7

For instance, (3) might say, “a knows that the next president of Brazil belongs to the party c”. The transformed statement (4) will then say, “a knows of the man who in fact will be the next president of Brazil that he belongs to the party c”. This explana- tion incidentally illustrates one of the ways in which the dif- ference between (3) and (4) is occasionally signalled in or- dinary usage: We sometimes distinguish between “knowing that . . . b . . .” and “knowing of b that . . . he . . .”.

Another, perhaps even clearer way of making the same distinc- tion more or less idiomatically would be to follow Quine who has for essentially the same purposes distinguished between “be- lieving that” and “believing to”.” Thus one might try to render (4) as follows: “The next president of Brazil is known by a to belong to the party c”, as contrasted to: “a knows that the next president of Brazil belongs to c”. (In fact, (4) illustrates exactly Quine’s transparent sense of propositional attitudes as distin- guished from the opaque one. However, we have seen that the two “senses” are not independent: the former can be defined in terms of the latter, as one can see from (3) and (4).)

In the type of case Castaiieda is contemplating, we might similarly have as an instance of a’s knowing something about u

“K, R (a, c) ”

and as an instance of a’s knowing something about himself

(6) “(EX) (x=u & K , R(x, c ) ) ” .

The difference between such sentences as (5) and (6) is con- nected with the fact that only bindable (bound) variables, but not free symbols, occurring within the scope of an epistemic operator range over individuals known to the person whose knowledge we are discussing. For the difference between (5) and (6) is due precisely to the fact that in (6) it is the bindable variable x and not the free individual term a that occurs within the scope of K,; for this reason, (6) is about a unique individual

W. V. 0. Quine, Word and Object, Cambridge, Mass., 1960, pp. 149- 150. Cf. Knowledge and Belief, section 6.11.

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known to a, whereas (5) is about several “epistemically possible individuals”.

A similar analysis applies to the difference between “a believes that . . . a . . .” and “a believes that . . . he . . .”. This analysis can be put to test by considering the following ingenious puzzle which was given to me by Peter Geach: One can easily see that a sentence like

“the richest man in the world believes that the richest man in the world believes that he is stingy”

is ambiguous because of the uncertainty of the reference of “he”. The problem is to resolve this ambiguity in terms of a precise symbolic notation.

In terms of our symbolic language, the situation is perfectly clear. We may start from the sentence

”the richest man in the world believes that the richest man in the world believes that the richest man in the world is stingy”,

symbolically

“B,B, ( r is stingy) ”, and ask how the ambiguous sentence is obtained from it by transforming it into a statement which is not any more about the several men who may be richest in the world, for whatever the richest man in the world knows, but about a unique individual. But about which individual? Since there are two belief-operators in the offing here, there are two ways of carrying out the trans- formation. They yield, respectively,

“ ( E x ) ( X = Y & B,B,(x is stingy))”

and

“B, ( E x ) ( x = r 81. B, (x is stingy))”.

Thus the ambiguity is resolved automatically when one tries to translate the ambiguous verbal formulation into our symbolic language, for neither of the two statements just quoted implies

KNOWING ONESELF 9

the other. It is also seen at once where the ambiguity lies: It is essentially due to the fact that in the original verbal formulation the order of the quantifier “(Ex) ” and the belief-operator “Br” was not indicated.

Informally, the difference may be brought out by such for- mulations as

“the richest man in the world believes that he is believed by the richest man in the world to be stingy”

and “the richest man in the world believes that the richest man in the world believes of himself that he is stingy”.

An example may perhaps illustrate the distinction involved. Sup- pose Mr. Paul Getty believes that Mr. Gulbenkian is the richest man in the world. Then the former statement is true if Mr. Getty, who is the richest man in the world, believes that Mr. Gulbenkian believes that Mr. Getty is stingy, whereas the latter is true if Mr. Getty believes that Mr. Gulbenkian believes that he, i.e. Mr. Gulbenkian, is stingy. Of course, this is just part of what our symbolic formulation brings out more concisely.

There is also a closely related but different statement which verbally might be expressed somewhat as follows :

“the richest man in the world believes that he himself believes that he is stingy”.

This is clearly obtained likewise from “BrBr(r is stingy)” by taking both the instances of belief to pertain to a definite in- dividual. This gives us

“(Ex) ( x = r & B , ( ( E y ) (x=y & B x ( y is stingy))))”,

which again accurately catches the desired nuances of the verbal formulation.

These examples illustrate the power and flexibility of the sym- bolic notation used in my book.

5. Turning back to Castafieda’s comments on my book, we can now simply state that what he calls construction (i) is definable

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in terms of ( i i ) . Now what happens in my condition (C.KK’), in its qualified form, is that I require as an additional assumption the presence of the premise

“ (Ex) K , ( x = a) ”

which says that the person referred to by a knows to whom (=to which uniquely determined individual) the term a refers. On this additional assumption, the other premise-let it say that “K,R(a, c)”-may indeed be taken to express a’s knowledge about a unique individual, to wit, about himself. In other words, the presence of the additional premise “(Ex) K , ( x = a)” has to some extent the same effect as the switch from (5) to (6), at least on the relatively trivial further assumption that a is not empty.

Formally, this is shown by the fact that “K,R(a, c)” (= (S)), “(Ex)K,(a=x)”, and “(Ex) (x=a]” together (virtually) imply “(Ex) (x=a & K,R(x, c))” (= (6)). This is easily shown by means of the rules of Knowledge and Belief, and more general results are easily proved in the same way.

Intuitively speaking, it might seem that the implication from “a’s knowing something about a while knowing who a is” to “a’s knowing something about himself” is not unproblematic in that it is based on an implicit assumption. If the assumption is made explicit, however, it is easy to see that it is in fact satisfied. What one has to assume is, loosely speaking, that if someone knows who a is, and if he is in fact himself a, then he knows that he is himself a. This, it seems to me, is surely the case.

The fact that a statement like (4) or (6) is about the in- dividual in question, and not just the several epistemically pos- sible referents of a term, is brought out by the fact that contexts of this kind are subject to the principle of the substitutivity of identicals. For instance, (6) does not depend on the way the person referred to by a happens to be thinking of himself, for (6) together with “a=b” virtually implies “ ( E x ) ( x = b & K,R(x, c))” as well as “ ( E x ) ( x = b & KbR(x, c))”. In more general terms, for any expression F which contains x but does

KNOWING ONESELF 11

not contain any epistemic operators we can prove a virtual implication from

(7) to

(8) “ ( E x ) ( x = c & K,F)”.

This result reflects the intuitively obvious idea that a statement about an individual does not depend on the way this individual happens to be specified.

Our result incidentally also serves to justify Peter Geach’s happy intuition to the effect that the substitutivity of identity obtains in contexts in which we use the locution “himself”.6

Thus Castaiieda is right in saying that in (C .KK’) (qualified) I have to assume that the referent a knows something about himself, not just about the referent of a (i.e. about himself merely as the referent of this term). However, the satisfaction of this assumption is guaranteed by the very presence of the qualifying extra premise “ ( E x ) K , ( x = a ) ” .

Castaiieda claims that this formula is still ambiguous, and that I “propose different readings for it”. It is true that it can be rendered verbally in different ways and that I have done so, but this is the case merely because the different senses that one can try to give to it coincide to some extent in this special case. Intuitively, the point is that in occurring within the scope of “K”, x ranges over definite individuals known to a, and thus the identity “x=a” occurring in “(Ex)K , (x=a)” makes it a state- ment about a himself (the individual), not just about whoever happens to be referred to by the term.

6. That there is some confusion in Castaiieda is shown by his comment that in my system “there just seems to be no way of consistently formulating, e.g., the contingent statement (A) ‘There is a person x such that a knows that a=x but a does not know that he himself =x’.”

On the interpretation which I have argued for, however, (A)

“ ( E x ) ( x = b & K,F) & (b=d)”

’ Geach, op. ci t .

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is clearly inconsistent (indefensible). Since the bound variable x occurs in (A) within the scope of the epistemic operator ”K,“, it has to range over such individuals only as are known to the referent of a-known not merely in the sense that he knows that they exist but also in the stronger sense that their identity is known to him. But if the person in question knows that a refers to one definite individual of this sort, and if this individual is in fact himself, how can he possibly fail to know that a in fact refers to himself? Thus (A) is intuitively inconsistent, and this fact is reflected by its being inconsistent on the assumptions which are made in Knowledge and Belief.

How awkward it is to claim that (A) is contingent is brought out by observing that in (A) it is in effect said of one and the same individual (one of the values of x ) that he both is and is not known by the referent of a to be identical with the referent of a. This we can say only when the individual in question is referred to in different ways. In order to have our bindable variables ranging over individuals, however, such a variable (in the present case x) has to be neutral with respect to the dif- ferent ways of referring to the same individual. The difference Castaiieda has in mind has to be brought out by the use of free individual symbols, not of bindable variables.

What Castaiieda’s remark amounts to is therefore to point out, not the consistency of (A), which would be translated into my notation as’

(9) “(Ex) (K,(a=x) & -(Ez) ( z=a & K,(z=x)) & ( E z ) ( z=x ) ) ”

but rather the consistency of something like

(10) “a knows that (a= b ) but a does not know that he himself = b”.

Now (10) is to be translated into my system as

’ The last clause of (9) is occasioned by my rejection of (C. EK=) (cf. section 3 above).

KNOWING ONESELF 13

(11) “K,(a=b) & - ( E z ) ( z = a & K,(z=b)) & ( E z ) ( z = b ) ”

Here we see that Castafieda does have a point: (11) is indeed contingent, although (9) is not.” However, I suspect that his neglect of the difference between bindable and free symbols has led him to speak of (9) instead of (11). In any case, his point does not tell against my way of treating epistemic logic.

7. I should also like to correct a minor oversight in the formula- tion of the condition (C - =K) in Knowledge and Belief. To- gether with (C. =P), it is (roughly speaking) designed to ex- tend the substitutivity of identicals to the subscripts of epistemic operators which do not occur within the scope of other epistemic operators. In order to secure this end, (C-=K) has to be ex- tended so as to require that “(Ex)KbF” e ,LL whenever “(Ex)K,F“ e ,LL and ”(a=b)” e p.

That the inconsistency (indefensibility) of (9) has little to do with Castafieda’s intentions is also suggested by the fact that (9) remains in- defensible even if we remove the first K , and the clause ( E z ) ( z = x ) from it.