knowledge and true belief in hintikka's epistemic logic

MICHAEL E. BYRD

KNOWLEDGE AND TRUE BELIEF IN

HINTIKKA’S EPISTEMIC LOGIC

A long tradition in philosophy holds that true belief is not equivalent to knowledge. Some philosophers have maintained one half of the equi- valence, that knowledge entails true belief. (For example, see Chisholm [3], p. 16 and Ayer [2], p. 34.) The converse, that true belief entails knowledge, has been rejected by and large. In this paper, I shall investigate the connection between knowledge and true belief in the system of epistemic logic developed by Jaakko Hintikka in his book Knowledge and Belief [4]. Specifically, I will show that if knowledge and true belief are considered in isolation from each other, then, in two important cases, there is no difference between the logics of the two notions. I also show how, in these same cases, differences between the logic of knowledge and the logic of true belief emerge, when the two notions are considered together.

I

Hintikka claims that the logic presented in Knowledge and Belief suggests an answer to the question, “How do knowledge and true belief differ?” According to him, knowledge and true belief have different logics. (Hintikka [5], pp. 82-83.) Where then is the difference between the logics of knowledge and true belief reflected in Hintikka’s epistemic logic? In Knowledge and Belief, he suggests that the difference is reflected in the difference between the version of the epistemic operator governed by the condition A.PKK* and a version governed only by the weaker condition A.PK*. The condition A.PKK* states that if a set 3 is defensible and K,PIE~, Kpz~fJ-3 . . . . K,p,@, P,q&, then the set {K,p,, K,p, ,..., K,p,, q} is defensible. On the same condition, the rule A.PK* only requires that the set (pl, pz, . . . , p., q} be defensible. According to Hintikka, the verb ‘to know’ in its most typical sense obeys A.PKK*.l (Hintikka [4], p. 19). So the logic of knowledge may be represented by the logic of the strong operator governed by A.PKK*. But, Hintikka notes, the verb ‘to know’ is sometimes used in such a way that it means something

Journal of Philosophical Logic 2 (1973) 181-192. AN Rights Reserved Copyright (9 1973 by D. Reidel Publishing Company, Dordrecht-Holland

182 MICHAEL E. BYRD

like ‘is aware’ or ‘rightly believes’. The logic of this sense of knowledge (i.e., true belief) is represented by the weak epistemic operator governed by A.PK*. (Hintikka [4], pp. 18-19.)

The claim that Hintikka locates the difference between knowledge and true belief in the way just outlined is reinforced by remarks in his paper ‘The Modes of Modality’. (Hintikka [5], pp. 71-86.) In it, he says that the logic of knowledge and the logic of true belief are different and that the difference is reflected formally in the difference between an epistemic operator governed by CM & NN+, a condition analogous to A.PKK*, and an operator governed by C.M & N+, a weaker condition analogous to A.PK *. Hintikka says that when we are dealing with genuine knowledge, and not just true opinion, the condition C.M & NN+ must be fulfilled. On the other hand, mere true opinion does not obey C.M & NN+, but only the weaker CM & N+. (Hintikka [5], pp. 83-84. These claims are further reinforced by Hintikka [5], p. 10 and Hintikka [6], p. 144.)

Yet, on Hintikka’s own grounds, one would expect to find the differences between knowledge and true belief reflected in other features of his epistemic logic as well. This expectation is evoked by his metho- dological remarks in ‘Epistemic Logic and the Methods of Philosophical Analysis’ (Hintikka [5], pp. 3-19) taken in conjunction with his views about the adequacy of the epistemic logic that he presents.

In ‘Epistemic Logic and the Methods of Philosophical Analysis’, Hintikka suggests that the meaning which an expression has in a formal logic should be regarded as a basic meaning of that expression. If the basic meaning selected is ‘an appropriate one’, then, given an under- standing of the pragmatic features involved in a particular situation, it should be possible to explain how the expression is used in ordinary language. (Hintikka [5], pp. 6-7.) Such explanations are clearly very important in the theoretical study of natural language.

The basic meanings of the verb ‘to know’ incorporated in Hintikka’s logic, for example, are the strong and weak versions of the epistemic operator. The basic meanings of ‘to believe’ are the strong and weak versions of the doxastic operator.

Hintikka claims that the results of his epistemic logic agree by and large with the way in which the verbs ‘to know’ and ‘to believe’ are naturally used. (Hintikka [4], p. 10.) Moreover, he feels that there are

KNOWLEDGE AND TRUE BELIEF 183

acceptable ways of handling counterexamples. If Hintikka’s claim is correct, then the various versions of the epistemic and doxastic operators qualify as basic meanings of ‘to know’ and ‘to believe’. Consequently, these basic meanings, together with the pragmatic aspects of a situation, should make it possible to explain features of the ordinary use of ‘to know’ and ‘to believe’. One articulate, considered feature of this use is the contrast, mentioned above, that philosophers have drawn between knowledge and true belief. So, the epistemic operators, the basic meanings of true belief defined in terms of the doxastic operators, and the pragmatic features of a given situation should make it possible to explain the contrast between knowledge and true belief.

It might be argued that this contrast can be adequately explained solely in terms of the pragmatic features of a situation, and hence that there need be no difference between the basic meanings of knowledge and true belief. (Austin’s comments on ‘to know’ in [l] might be inter- preted in this way.) Hintikka appears to deny that the contrast can be explained solely in these terms. As noted earlier, he claims that the contrast between knowledge and true belief is reflected in the difference between the two epistemic operators. In this case, the contrast is explained by a difference in basic meanings, not merely by pragmatic factors.

But, if the contrast cannot be adequately explained in terms of pragmatic features, then it should be reflected in differences between the basic meanings of knowledge and true belief. Hence, in Hintikka’s logic, one would expect to find a contrast between the basic meanings of knowledge and true belief.

II

However, in two important cases, these expectations are not fulfilled. Knowledge in the sense of the strong epistemic operator and true belief in the sense of the strong doxastic operator have the same logic when the two notions are considered in isolation. Similarly, knowledge in the sense of the weak epistemic operator and true belief in the sense of the weak doxastic operator have the same internal logic.

More precisely, letting TB,p abbreviate B,,p &p, what can be shown in the two cases under consideration is that

0 A set 97 whose only non-truth-functional connectives are

184 MICHAEL E. BYRD

epistemic operators is indefensible if and only if the set Z is indefensible, where I’ differs from 3 only by containing ‘TB’ at every place where 3 has ‘K’ and ‘N TB- ’ at every place where 3 has ‘P’.

(I) establishes that knowledge and true belief, in the appropriate senses, have the same internal logic in view of the definability of all the other concepts of Hintikka’s epistemic logic in terms of indefensibility.

A detailed proof of (I) is presented in the Appendix.

III

So, in two important cases, there is no difference between the internal logics of knowledge and true belief. However, in two other cases, there is a difference between the internal logics of these concepts.

First, suppose that knowledge has the sense of the strong epistemic operator and true belief, that of the weak doxastic operator. Then there are indefensible sets, containing only truth-functions and epistemic operators, that are rendered defensible when ‘K’ is replaced by ‘TB’ throughout. For example, the set {K,p, -K,K,p} is indefensible if ‘K’ is the strong epistemic operator. But the set { TB,p, -TB,TB,p} is defensible, if ‘B’ is the weak doxastic operator.

Second, suppose that knowledge has the sense of the weak epistemic operator and true belief, that of the strong doxastic operator. Then, there are defensible sets, containing only truth-functions and epistemic operators, that are rendered indefensible when ‘K’ is replaced by ‘TB’ throughout. For example, in this case, the set {K,p, -K,K,p) is defensible, whereas the set { TB,p, N TB,,TB,p) is not.

IV

There are several plausible responses to what is established in Sections II and III. I will discuss three of them. The first two are incompatible with Hintikka’s views. It seems to me that the third response is the one that Hintikka should adopt.

(1) The fact that, in some cases, ‘K’ and ‘TB’ have the same internal logic accurately reflects the fact that some senses of knowledge and true


belief have the same logic. Consider, for example, the strong version of the doxastic operator. This version of the doxastic operator is appropriate when the beliefs in question are reasoned or considered beliefs. (Hintikka [4], pp. 26, 29.) So the logic of the strong operator ‘TB’ represents the logic of considered or reasoned true belief. There is a philosophic tradition which maintains that knowledge may be detined as reasoned or justified true belief. (Ayer [2], p. 34; Chisholm [3], p. 16.) If this tradition is correct, then there should be no difference between the logics of the strong operators ‘K’ and ‘TB’. Indeed, the strong operator ‘K’ should be defined in terms of ‘TB’.

This response is incompatible with Hintikka’s position. If the strong operator ‘K’ could be defined in terms of ‘TB’, then replacement of ‘K’ by ‘TB’ in any set of sentences would preserve defensibility. However, in Hintikka’s logic, such replacement need not preserve defensibility, if sets containing both epistemic and doxastic operators are allowed. For example, the set {N K,,p, TB,,p) is defensible, whereas the set ( TB,p, - TB,,p} is not.

(2) There is a difference between the logics of knowledge and true belief in each of the senses of ‘to know’ and ‘to believe’ that Hintikka’s logic is intended to capture. Furthermore, these differences should be reflected in the internal logics of the respective operators. Since, in Hintikka’s system, the differences are not always so reflected, it must be admitted that these operators do not accurately reflect the logics of the notions that they were intended to capture. Hence, Hintikka’s epistemic logic is inadequate.

This response is flatly incompatible with Hintikka’s view that his epistemic logic is adequate. Where should Hintikka disagree with this response? In his logic, there are differences between the basic meanings of knowledge and true belief in all cases, if sets containing both epistemic and doxastic operators are permitted. But, as shown, there are not always differences in the internal logics of these basic meanings. So Hintikka must deny that differences in basic meanings necessitate differences in internal logics.

(3) There should be differences between the logics of knowledge and true belief in each of the senses of ‘to know’ and ‘to believe’ that Hintikka’s logic attempts to capture. But, in some cases, these differences are not reflected in the internal logics of the respective operators. Instead, the

186 MICHAEL E. BYRD

differences come to light when knowledge and true belief are contrasted with each other or some third notion, Formally, this kind of contrast is revealed if sets containing epistemic, doxastic, and perhaps some third kind of operator are allowed. Then some sets will change their logical properties (defensibility, indefensibility) if ‘K’ is replaced by ‘TB’ throughout.

It seems to me that Hintikka should adopt this response. For suppose. that sets containing both epistemic and doxastic operators are permitted. Then, in Hintikka’s logic, differences emerge even in those cases where the internal logics are the same. The simplest example is 9, = ( TBap, - K,p}.

Y, is defensible in Hintikka’s system, regardless of which version of ‘K’ or ‘B’ is used. Replacement of ‘K’ by ‘TB’ renders the set trivially indefensible. The defensibility of Y, incorporates in Hintikka’s system the claim that true belief does not amount to knowledge in any of the senses of ‘to know’ and ‘to believe’ that Hintikka’s logic is intended to capture.

There are other more interesting contrasts between knowledge and true belief in their strong senses, if mixed sets are allowed. The set

9, = {B,P, N K&p) is defensible, whereas the set 9, = (B,,p, N TB,$,p) is not. The defensibility ofY, implies that ‘B,PD K,B,p’ is not self- sustaining. In Knowledge and Belief, Hintikka offers arguments against the self-sustenance of this implication. (Hintikka [4], pp. 50-54.) The indefensibility of Y, implies that ‘B,,p 3 TB,B,p’ is self-sustaining. Hintikka gives no direct argument for this principle. However, he has indicated that the closely related principle ‘B,,px B,,B,,p’ can be de- fended in much the same way as ‘K,,pz K,K,,p’. (Hintikka [6], p. 159.) It seems to me that investigation of principles like ‘B,,p =I K,B,p’ and ‘B,p 3 TB,B& and Hintikka’s arguments concerning them, will reveal what further light, if any, Hintikka’s logic has to shed on the question of the connection between knowledge and true belief.2

APPENDIX

In Section II of the paper, it is claimed that the strong senses of knowledge and true belief have the same internal logic, and similarly for the weak senses of these same notions. Theorem I shows that this claim is true.

THEOREM I. A set % whose only non-truth-functional connectives are


epistemic operators is indefensible if and only if the set s’ is indefensible, where s’ differs from % only by containing ‘TB’ at every place where X has ‘K’ and ’ - TB- ’ at every place where 3 has ‘P’.

Theorem I will be proved for the case in which knowledge and true belief have their strong senses. The proof for the other case is parallel.

There are three simple lemmas governing truth-functions needed in the course of the proof. They are here stated without proof.

LEMMA 1. If a set 2’ is defensible and ‘P’EZ and ‘q’c3, then the set obtained from 5’ by replacing ‘p’ and ‘q’ by ‘p 8z q’ is defensible.

LEMMA 2. If a set 9 is defensible and ‘~‘EE, then the set obtained from 9. by replacing ‘p’ by ‘p v q’ is defensible.

LEMMA 3. If a set E is defensible and ‘B’E I, then the set 9+ + {Sub”,“p(B)} is defensible.

First, I show

LEMMA 4. A set 9Y whose only non-truth-functional connectives are epistemic operators is defensible only if the set 5Y is defensible, where 3’ is as in Theorem I.

Suppose a proof of the indefensibility of 9J takes the following form:

(1) Assume the set X is defensible. (2) Then the set GY is defensible.

(n) ‘*’ Then the set 9 = {. . . , p, . . . , -p, . . .} is defensible. (n + 1) But the set Z= {. . ., p, . . ., -p, . . .} is indefensible. (n+2) So the set S is indefensible.

Consider the sequence of lines l’, 2’, . . ., (II+ 2)‘, where line i’ is like line i, except that it contains ‘TB’ where line i has ‘K’ and ‘-TB-‘, where line i has ‘P’. The lemma will be proved if it can be shown that each of the lines 1' to (n+2)’ can be justified by means of the A-rules. Lines 1’ and (n+l)’ can be justified as they stand; line (n+l)‘, by the rule A.-. If lines 2’ to II’ can be justified by means of the A-rules, then line (n+Z)’ can be justified by ordinary reductio reasoning.

188 MICHAEL E. BYRD

That these lines can be so justified is shown by a strong induction on the number of lines.

INDUCTIVE HYPOTHESIS. For every line i’, 2 <i< j< n, line l’ can be justified by means of the A-rules.

It will be shown that this hypothesis holds for all lines i’, i < j. The cases that must be considered correspond to the ways in which line j in the original proof could have been justified. Since the set I contains no doxastic operators, line j could have been justified only by rules governing truth-functions and epistemic operators.

(1) Suppose line j was justified by a rule governing truth-functions. Then line j’ can be justified by reference to the same rule. The proof of this fact is left to the reader.

(2) If line j follows from a line k by one of the rules governing epistemic operators, then, in order to justify line j’, other rules are needed.

(a) Line j follows from a line k by A.PKK* :

(A.PKK*) If a set 9’ is defensible and ‘K,P~‘E 3, ‘KJI~‘E~‘, . . ., ‘K,,~,‘E%O, ‘P=q’ES’, then the set (K,p,, K,,p2, . . . . K,p,, q} is defensible.

Line k is of the form: ‘The set ?Y= { . . ., K,p,, K,p,, . . ., K,p,, P,q, . . .} is defensible.’ So line k’ is of the form: ‘The set ?Y’ = (. . ., 73, (PI)‘, TB, (pz)‘, . . . . TB~SP,)‘, IY TB, - (q)‘,. . .} is defensible.’ By the inductive hypothesis, this line can be justified by means of the A-rules. Using the definition of ‘TB’, the rule A. -B, and the rules governing truth-functions, it follows that the set (...,B,(~,)‘,P,‘,B,(P,)‘,P~‘,...,B,C~,)’,P~’, G--(q)‘vq’,...) is defensible. By A. v , either the set W = {. .., B,(p,)‘, p1’, B,(p,)‘, pz’, . . . , Bdp,)‘,~,‘, G-~q’,... }orthesetY’=( . . . . B,(p,)‘,p,‘,B,(p,)‘,pz’, . . . . B,(p,)‘, pn’, q’, . . .} is defensible.

Suppose W is defensible. Then by A.CBB* and A. - -, the set &API)‘, PI’, B,(Pz)‘, ~2’3 ‘e.9 B&,)‘,p,‘, q’) is defensible. By Lemma 1 and the definition of ‘TB’, it follows that the set { TBa(pl)‘, TB,,(p,)‘, . .., TB,(p,,)‘, q’} is defensible. Hence line j’ can be justified by means of the A-rules.

Suppose Y’ is defensible. Every subset of a defensible set is defensible. So the set {B,,(p,)‘, pi’, Ba(pz)‘, pz’, . . ., B,(pJ’, p,,‘, q’} is defensible. By Lemma 1 and the definition of ‘TB’, the set {TB,(p,)‘, TB, &)I,. . . ,


T.&&J’, q’} is defensible. Hence line j’ can be justified by means of the A-rules.

So, line j’ can be justified, if line j is justified by A.PKK*. (b) Line j follows from a line k by A.K:

(AK) If a set fz” is defensible and ‘K,,P’E Z?‘, then Z’+ {p> is defensible.

Line k has the form: ‘The set %” = {. . ., K,p, . . .} is defensible.’ So line k has the form: ‘The set %“‘= {. .., T&(p)‘, . . .} is defensible.’ By the inductive hypothesis, line k’ can be justified by means of the A-rules. Line j’ can be justified by using the definition of ‘TB’, A. &, and Lemma 1.

(c) Line j follows from a line k by A. N K :

(A. N K) If a set ZZ’ is defensible and ‘N K,p’e 3, then the set %” + + {P,, -p} is defensible.

Here line k has the form : ‘The set 22’= {. . ., N K,p, . . .} is defensible.’ So line k’ has the form: ‘The set 3” = { . . ., -TB,p’, . ..} is defensible.’ By hypothesis, this line can be justified. By Lemma 3, it follows that the set %“‘+(-TB,, N -p’} is defensible. Hence line j’ can be justified, as required.

(d) Line j follows from a line k by A. NP:

(A.P) If a set 22’ is defensible and ‘NP~P’E~?@, then the set g+ + (K,, -p} is defensible.

Line k has the form: ‘The set I= {. . ., -Pap,. . .} is defensible.’ So the line k’ has the form : ‘The set 8’ = {. . ., N N TB, N (p)‘, . . .} is defensible.’ By hypothesis, this line can be justified. Using A. N N and Lemma 3, it follows that the set 2” + {TB,, N (p)‘} is defensible. Hence line j’ can be justified, as required.

This completes the proof of Lemma 4.

LEMMA 5. A set 3 whose only non-truth-functional connectives are epistemic operators is indefensible if the set I’ is indefensible, where I is as in Theorem I.

The proof of Lemma 5 proceeds in two steps.

LEMMA 6. If a set 3 whose only non-truth-functional connectives are

190 MICHAEL E. BYRD

doxastic operators is indefensible, then the set s’ is indefensible, where r differs from S? only by containing ‘K’ at every place where X has ‘B’ and ‘P’ at every place where 3 has ‘C’.

The method of proof used is the same as in Lemma 4. Suppose the sequence of lines 1,2,. . ., (n+2) is a proof of the indefensibility of 3. Let the sequence l’, 2,. . ., (n + 2)’ be obtained from the original sequence by replacing ‘B’ by ‘K’ and ‘C’ by ‘P’ throughout. Again, the main problem is to show that the lines 2’ to n’ can be justified by means of the A-rules. This is done by a similar induction on the number of lines. The cases to be considered correspond to the ways in which line j in the original proof could have been justified. Since the set 3 contains only truth-functional connectives and doxastic operators, line j was justified by either a rule governing truth-functions or a rule governing doxastic operators.

(1) If line j follows from a line k by a rule governing truth-functions, then the cases are treated exactly as in Lemma 4.

(2) Suppose line j follows from a line k by a rule governing doxastic operators.

(a) Line j follows from a line k by A.CBB*:

(A.CBB*) If a set Zis defensible and ‘Bapl’~ Z’, ‘B,,p2’c 3, . . ., ‘B,p,eS, ‘C,q’E~, then the set {B,P,, pl, B,P,, Pi,..., B,P,, P., q) is defensible.

Here line k is of the form: ‘The set %= {. .., B,p,, B,p,, . .., B,,p,,, C,,q} is defensible.’ So line k’ has the form: ‘The set %‘=(K&,)‘, Ka(p2)‘,..., K,(pJ’, P,(q)‘, . ..I is defensible.’ By the inductive hypothesis, this line can be justified by means of the A-rules. Applying the rules A.PKK* and A.K to line k’, it follows that the set {Ka(pl)‘, pi’, Ka(p2)‘,p2’, . .., K&3’, p,,‘, q’} is defensible. Hence line k’ can be justified, as required.

(b) Line j follows from a line k by A. N B:

(A. “B) If a set 52” is defensible and ‘N B,p’e 3, then the set Z’+ + (Ca -p} is defensible.

Here line k’ has the form : ‘The set %’ = {. . ., N K,(p)‘, . . .> is defensible.’ By the inductive hypothesis, this line can be justified. By applying the rule A. -K to line k’, line j’ can also be justified.

(c) Line j follows from a line k by A. NC:


(A. NC) If a set I is defensible and ‘ N C,,p’~3, then the set 3? + + {B,-p} is defensible.

This proof is exactly parallel to (b). This completes the proof of Lemma 6.

LEMMA 7. If a set 5? whose only non-truth-functional connectives are epistemic operators is indefensible, then the set s’ is indefensible, where S’ is like %, except for containing Subxzb:,, (B), where S has B.

That is, if an indefensible set 3 whose only non-truth-functional operators are epistemic ones has some elements with subformulas of the form ‘K,p &p’, then the set obtained by replacing ‘Kbp &p’ in those elements by ‘K,p’ is indefensible. Suppose the sequence of lines 1,2,. . ., n+2 is a proof of the indefensibility of % of the usual form. (See the proof of Lemma 4.) Let l’, 2’,..., (n+2)’ be a sequence of lines that differs from the original one only by having Sub,2;, (B), where B occurs in the original sequence. The problem is to show that each of the lines in the new sequence can be justified. Lines 1’ and (n+l)’ can be justified as they stand. So, once again, it is necessary to show that lines 2’ to II’ can be justified by means of the A-rules. This is accomplished by means of a strong induction on the number of lines.

INDUCTIVE HYPOTHESIS. For each line i’, 2< i<j<n, line i’ can be justified by means of the A-rules.

The cases that have to be considered correspond to the ways in which line j could have been justified. Since we are supposing that S has no doxastic operators, line j could only have been justified by one of the rules governing truth-functions or epistemic operators.

(1) In case line j follows from a line k by a truth-functional rule or by one of the rules (A. ,P), (A. N K) or (A.K), the proof is straightforward and is left to the reader.

(2) Line j follows from a line k by A.PKK*. Then line k is of the form : ‘The set ?V = { . . ., Kzl, Kg2, . . . , KJI”, P,,q, . . . } is defensible.’

(a) None of the K& nor P,q contains a subformula of the form ‘K,p dp’. Then j’ follows from k’ by the rule A.PKK*.

(b) Some of the K,p, or P,q has subformulas of the form ‘Kbp &p’. In this case, Subx$!i p (K,& = K, (Subx$i,(p,)) and Sub,$i$s =

192 MICHAEL E. BYRD

=P. (Sub K$‘gp(q)). So line j’ may be derived from line k’ by A.PKK*. Hence line j’ can be justified, as required.

This completes the proof of Lemma 7. In Lemma 5 and Theorem I, we are concerned with only those sets whose sole doxastic operator is ‘TB’. For example, let 3 = { . . . , TB,,p, . . .> be such a set. If S is indefensible, then by definition the set {. . ., B,,p & p, . . .> is indefensible. By Lemma 6, the set { . . . . K,p &p ,... } is indefensible. Lemma 7 shows that ‘KJI & p’ can be replaced by ‘Kap’. Thus the set { . . . . KJI, . ..I is indefensible. More generally, Lemma 6 and 7 together show that if a set whose sole non-truth-functional operators are true belief operators is indefensible, then so is the set obtained from it by replacing ‘TB’ throughout with ‘K’. This proves Lemma 5. Lemmas 4 and 5 entail Theorem I.

University of Wisconsin

NOTES

1 Hintikka no longer believes that the strong epistemic operator represents the most typical sense of knowledge. (See Hintikka [6], pp. 148-149.) However, he still holds that the sense of knowledge represented by this operator is an important sense in which knowledge is contrasted with true belief. (See Hintikka [6], p. 144.) 2 The criticism and suggestions of Rare1 Lambert and Peter Woodruff were of great help in the preparation of this paper.

BIBLIOGRAPHY

[l] J. L. Austin, ‘Other Minds’, Proc. Aristotelian Sot. Suppl. 20 (1946), 14&187. [2] A. J. Ayer, The Problem of Knowledge, London, Macmillan, 1956. [3] Roderick M. Chisholm, Perceiving: A Philosophical Study, Ithaca, Cornell

University Press, 1957. [4] Jaakko Hintikka, KnowZedge and Belief, Ithaca, Cornell University Press, 1962. [5] Jaakko Hintikka, Models for Modalities, New York, Humanities Press, 1969. [6] Jaakko Hintikka, “Knowing that One Knows’ Reviewed’, Synthese 21 (1970).

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knowledge and true belief in hintikka's epistemic logic

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