knowing that one knows and the classical definition of ... that one knows and the classical...

Knowing That One Knows and the Classical Definition of KnowledgeAuthor(s): Risto HilpinenSource: Synthese, Vol. 21, No. 2 (Jun., 1970), pp. 109-132Published by: SpringerStable URL: http://www.jstor.org/stable/20114716 .

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RISTO HILPINEN

KNOWING THAT ONE KNOWS AND THE CLASSICAL

DEFINITION OF KNOWLEDGE*

I. THE KK-THESIS

In his Essay in Modal Logic, G. H. von Wright suggested that the logic of knowledge

- or epistemic logic - is a branch of modal logic.1 E. J.

Lemmon has recommended a system of epistemic logic based on the

Feys-von Wright modal system M,2 and in Knowledge and Belief Jaakko

Hintikka proposed a stronger system which corresponds to Lewis's S4.3

The distinctive axiom of S4 is 'Np^NNp',* and the epistemic counter

part of this formula is the principle that knowing implies knowing that

one knows, that is,

(KK-thesis) Kap^KaKap,

where 'Ka9 stands for 6a knows that'. I shall call this principle 'the KK

thesis'. In recent literature on the theory of knowledge and epistemic

logic, this principle has been subjected to a great deal of discussion.5 In

this paper, I shall discuss the KK-thesis from the point of view of various

definitions of knowledge.

II. THE CLASSICAL DEFINITION OF KNOWLEDGE

In Perceiving: A Philosophical Study, Roderick M. Chisholm defined the

concept of knowledge in the following way:6

(DKi) 6a knows that /?' means:

(i) a accepts p,

(ii) a has adequate evidence for p, and

(iii) p is true.

This definition is a species of what I shall call the classical definition of

knowledge. According to the classical definition, knowledge is justified true belief, or true opinion combined with reason. A definition of this

Synthese 21 (1970) 109-132. All Rights Reserved Copyright ? 1970 by D. Reidel Publishing Company, Dordrecht-Holland



110 RISTO HILPINEN

type has been discussed by Plato in Theaetetus and in Meno,1 and it

underlies, e.g., Jaakko Hintikka's recent work on epistemic logic.8 Other

versions of the classical definition have been put forward by Roderick

M. Chisholm, David Rynin, and A. J. Ayer, among others.9 In all of

these definitions, the concept of knowledge is defined in terms of three

conditions, of which the first may be termed the condition of acceptance or belief the second, the condition of justification or evidence, and the

third, the condition of truth.

In Rynin's formulation of the classical definition, the first clause of

(DKi) is replaced by10

(DKi2) a believes that p.

(DKii) and (DKi2) can be regarded as equivalent. The second clause of

the classical definition, the requirement of justification, has been for

mulated in many different ways. Rynin has replaced (DK^i) by11

(DKii2) a can give adequate evidence that p9

and in Theory of Knowledge, Chisholm has used the condition12

(DKii3) p is evident for a.

Keith Lehrer has stated this condition in the form13

(DKii4) a is completely justified in believing that p.

There are differences of meaning between the conditions (DK^i^DKi^), but the common function of these conditions is to distinguish knowledge from true opinion.

A. J. Ayer's definition of knowledge is slightly different from the

definitions discussed above. According to Ayer,14

(DK2) a knows that p if and only if

(i) a is sure that p,

(ii) a has the right to be sure that p, and

(iii) p is true.

It seems natural to assume that a has the right to be sure that/7 if and only if a is completely justified in believing that p. Thus (DK2ii) can be taken

as equivalent to (DKii4), and if 'a is completely justified in believing that



KNOWING THAT ONE KNOWS 111

/?' implies 'a has adequate (complete) evidence that /?' (or '/? is evident'),

(DK2ii) implies (DKxii) (or (DKii3)). (DK2i) implies ?DK^) (or (DKi2)): if a is sure that/?, a accepts p (or believes that/?), of course. The converse

implication does not seem to hold: (DKxi) does not assert anything about

the certainty of #'s beliefs. Many philosophers have criticized Ayer's condition of certainty, however, and argued that the requirement of

certainty included in (DK2ii) (or, e.g., (DKii4)) is sufficient.15 If this

view is accepted and Ayer's definition is modified accordingly, it comes

close to other formulations of the classical definition.

III. KK-THESIS AND THE CLASSICAL DEFINITION

If 6a believes that/?' (or 6a accepts/?') is expressed by 6Bap' and the justifi cation requirement of the classical definition by 6Eap\ (DKX) may be

expressed in short as

(DK3) Kap if and only if

(i) BaP, (ii) Eap,

and

(iii) p is true.

Here I shall not choose any specific interpretation for <,Eap> ; it can be

taken to represent any suitable formulation of the justification require

ment, e.g., one of the conditions (CK^iM^CKi^), or some other similar

condition. Various interpretations of 6Ea' will be discussed later on. In

the sequel, I shall discuss only the knowledge of one single person a; thus

the subscript 'a' may be omitted, and (DK3) can be expressed in the sim

plified form16

(DK4) Kp =

Bp8cEp&p.

According to (DK4),

(1) KKp = K(Bp &Ep &p).

In discussion of the logic of knowledge, it is normally assumed that the

operator K distributes over conjunctions, that is,17

(CK&) K{p&q) = Kp&Kq.



112 RISTO HILPINEN

(1) and (CK&) imply

(2) KKp =

KBp &KEp &Kp9

which is equivalent to

(3) KKp = BBp &EBp &Bp &BEp &EEp &Ep&p.

Thus the KK-thesis assumes (after obvious simplifications) the form

(4) Bp &Ep&p=> BBp &EBp &BEp &EEp.

It seems plausible to assume that (4) is not valid, unless the consequent is

implied by Bp&Ep; thus the validity of (4) depends upon the validity of

(5) Bp&Ep 3 BBp &EBp ScBEp &EEp.

The validity of (5) is certainly not obvious. Below, I shall consider it by

breaking (5) to simpler implications, of which the conjunction implies

(5). If these simpler implications are acceptable, (5) can also be accepted as valid, and counter-examples to these more elementary implications will be taken as evidence that (5) is not valid.

Let us assume that a believes that /? and has complete (or adequate) evidence that /? (in other words, a is completely justified in believing that

/?).18 Is a also justified in claiming that he knows that/?, that is, is

(6) Bp&Ep =>EKp

valid? It is not implausible to answer in the affirmative, if a believes that/? and has complete evidence that/?, a knows that/?, if/? is in fact the case.

Thus a's evidence that p is ipso facto evidence that he knows that /?.

According to (DK4), (6) is equivalent to

(7) Bp&Ep => E{Bp &Ep &/?).

If it is assumed that the operator 'jE" is distributive with respect to con

junctions, that is,19

(CE&) E(p&q) = Ep &Eq9

(7) is equivalent to

(8) Bp&Ep id EBp&EEp.

(CE&) is a very plausible assumption, and is accepted by most students of




epistemic logic.20 The validity of (8) may be regarded as a consequence of

two principles,

(9) Bp z> EBp and

(10) EpzDEEp.

The first principle states that, whenever a believes that /?, he is also

completely justified in believing that he believes that /?. This seems fairly

plausible from the intuitive point of view; if a in fact believes that /?, he

need not collect any additional information in order to be in a position to state with full assurance that he believes that p.21 The second principle

expresses an important fact about the methodological role of 'adequate' or 'complete' evidence. Complete evidence must, indeed, be 'complete' in

the sense that it actually terminates the inquiry concerning /?. If a has

complete evidence to justify his belief that /?, no further inquiry (that is, no additional factual information) is needed to ensure that the evidence

is, indeed, complete. This is precisely what is expressed by (10).22 It also

seems plausible to strengthen (10) to an equivalence, i.e.

(11) Ep = EEp.

(11) is especially natural, if the adequacy or completeness of evidence is

thought of as being dependent upon criteria which are similar to logical or conventional criteria.23 If a has complete evidence that /?, then surely he possesses all the information he needs to determine whether he has

complete evidence or not.24 Furthermore, if a concludes that he has

adequate evidence, but this is not the case, a has made an error which is

(from the methodological viewpoint) similar to a logical error, and errors

of this type are presumably never justified. Thus (11) is valid.

On the basis of the argument given above, the validity of (8) seems very

plausible. If both (8) and

(12) Bp&Ep^BBp&BEp

are valid, (5) (and consequently the KK-thesis) is also valid. The simplest

way of studying (12) is to examine the implications

(13) Bp^BBp and

(14) Bp&Ep ^BEp.



114 RISTO HILPINEN

The conjunction of (13) and (14) implies (12). (13) is the doxastic counter

part of the KK-thesis, that is, the principle that believing implies be

lieving that one believes. This is not obvious, but not entirely implausible

either; perhaps (13) can be tentatively accepted.25 Thus the validity of (5)

depends upon (14). Under the most natural interpretation of 'Ep\ (14) seems definitely unacceptable. (14) states that ii a believes that/? and has

complete evidence that/?, a believes that he has complete evidence that/?.

Suppose that 'a has complete (or adequate) evidence that/?' means that a

possesses information which in fact is complete evidence that /?. Of

course, a may believe that /? and possess information which is adequate evidence that/?, without believing that he has adequate evidence; a may fail to recognize his evidence for what it is worth. In such a case the

antecedent of (14) is true, but the consequent false. This counter-example to (14) is also a counter-example to the present version of the KK-thesis.

This failure of the KK-thesis can also be explained as follows : Ac

cording to (DK4), the KK-thesis implies

(15) Kp^EKp&BKp,

that is,

(16) (Kp z> EKp)&(Kp ZD BKp).

In (16), the first conjunct is valid, but the second is not: a may know

something without believing that he knows, i.e., without being aware of

his knowledge, if he fails to recognize the adequacy of his evidence.26

IV. FIRST CRITICISM OF (DK^

The previous argument against 6Kp id KKp9 is based upon a case in which

a believes that/? and has complete evidence that/?, but does not recognize the adequacy of his evidence, and thus fails to believe that he knows that

/?. However, examples of this kind have also been taken as counter

examples to definition (DKJ. For instance, Keith Lehrer has expressed the justification requirement in the form (DKii4), and argued that

(DKxii) does not imply (DKii4): "If a person has evidence to completely

justify his belief he may still fail to be completely justified in believing what he does because his belief is not based on that evidence. What I

mean by saying that a person's belief is not based on certain evidence is




that he would not appeal to that evidence to justify his belief." Further

more, "even if a person has evidence adequate to completely justify his

belief and his belief is based on that evidence, he may still fail to be

completely justified in believing what he does. For he may be unable to

provide any plausible line of reasoning to show how one could reach the

conclusion he believes from the evidence that he has."27 According to

Lehrer, a knows that /? only if his belief that /? is actually based upon

adequate evidence in the sense that he would appeal to this evidence, if

he were demanded to justify his belief; a must recognize his evidence for

what it is worth. It is not obvious how this requirement should be for

malized in the notation employed here. However, it is natural to assume

that a would not appeal to his evidence, if he did not believe he has

evidence adequate to justify his belief that /?; thus it may be taken that

Lehrer's analysis of knowledge implies28

(17) Kp^BEp,

where 'is/?' is interpreted as (DKxii). An obvious way of fulfilling this

requirement is, of course, simply to add the consequent of (17) to the

conditions defining knowledge, and replace (DK4) by

(DK5) Kp =

Bp&Ep &BEp &p.

According to (DK5),

(18) KKp-KBp&KEp&KBEp&Kp,

which implies

(19) KKp = Kp &BBp &EBp &BEBp &EEp &BEEp &BBEp &EBEp&BEBEp.

Thus the KK-thesis assumes the form

(20) Bp &Ep&BEp &p z> BBp &EBp &BEBp &EEp &

BEEp & BBEp & EBEp & BEBEp.

By virtue of (9), (10) and (13), the antecedent of (20) implies

(21) BBp &EBp &EEp & BBEp & EBEp ;

thus (20) is valid, if its antecedent implies

(22) BEEp &BEBp &BEBEp



116 RISTO HILPINEN

as well. This can be proved, if we accept the following principle:

(CBimp) If /? 3 q is valid (logically true), Bp => Bq is valid.

By virtue of this principle, (10) implies

(23) BEp^BEEp;

and (9) implies

(24) BBp^BEBp,

and thus by (13),

(25) BpziBEBp.

By substituting 'jE/?' for '/?' in (25), we obtain

(26) BEp^BEBEp.

(22) can now be proved from (23), (25), (26) and the antecedent of (20) by

propositional logic. Thus (20) (i.e., the present form of the KK-thesis) is

valid.29

Certain qualifications are necessary here. (DK5) is probably not an

accurate formalization of Lehrer's notion of knowledge; thus the argu ment presented above is a persuasive rather than a strict proof of the

Lehrer-type variant of the KK-thesis. Nevertheless, it seems to me that

(DK5) catches certain important aspects of Lehrer's conditions on

knowledge. In fact, Lehrer has presented in another paper, 'Belief and

Knowledge',30 an argument which seems to support the KK-thesis. Some

philosophers have argued that knowledge that /? does not imply belief

that/?. Colin Radford has based this claim upon a case in which a person

gives correct answers to certain questions (and thus, according to

Radford, knows the correct answers), but does not believe that the

answers are correct. In such a case, Radford concludes, the person knows

something he does not believe.31 Lehrer's counter-argument is as fol

lows:32

(27a) If a does not believe that /?, a does not believe that he

knows that /?.

(27b) If a does not believe that he knows that /?, then, even

though a correctly says that/? and knows that he has said

that /?, a does not know that he correctly says that p.




(27c) If, even though a correctly says that/? and knows he has

said that/?, a does not know that he correctly says that/?, then a does not know that /?.

(27d) If a does not believe that /?, a does not know that /?.

This argument seems to have the form

(28a) ~Bpzi~BKp.

(28b) ~BKp=>s.

(28c) s=>~Kp.

(28d) ~Bpz>~Kp.

Conclusion (28d) follows from (28a)-(28c) by propositional logic, and is

equivalent to 6Kp^>Bp\ i.e. the conclusion Lehrer wants to prove. The

conjunction of (28b) and (28c) implies

(29) Kp^BKp.

This is precisely what is denied in the counter-example to the KK-thesis

presented in Section III. However, (29) follows from (DK5) (by (13) and

the doxastic counterpart of (CK&)). (DK5) is in this respect at least in

accord with Lehrer's intuitions.33

V. KK-THESIS AND THE LOGIC OF KNOWLEDGE

All definitions of knowledge discussed above imply

(CK) KpzDp.

In addition, we have assumed that the operator K distributes over con

junctions ((CK&)). The following principles seem plausible also :

(CKeq) If /? = q is valid (logically true), Kp = Kq is valid.

If/? and q are logically equivalent, they have the same truth-conditions

and the same (descriptive) content; thus knowing that /? is just knowing that q.

(CKlog) If/? is valid (logically true), Kp is valid.

A logical truth has no content; it asserts nothing. Knowing nothing but

logical truths represents a minimal amount of (factual) knowledge. Whatever you know, you know that /?, if/? is a truth of logic.34



118 RISTO HILPINEN

The principles (CK), (CK&), (CKeq) and (CKlog) define a system of

epistemic logic which corresponds to von Wright's modal system M. If

'K' is replaced by W, (CK), (CK&), (CKeq) and (CKlog) become

logical truths or valid rules of inference of M. If the KK-thesis is sub

joined to these axioms, we get the epistemic counterpart of S4. Among the consequences of (CK), (CK&), (CKeq) and (CKlog) are, e.g., the

principles

(CK 3) K(/? => q) => (Kp 3 KqY* and

(CKimp) If /? z> q is valid, K/? zd Kq is valid.

It has been claimed that epistemic logics of this type are 'unrealistic'. For

instance, (CKimp) states that a knows all the logical consequences of

whatever he knows. This is unrealistic, it has been argued; in the ordinary sense of 'know', people do not know everything implied by their know

ledge. Even (CK&) has been criticized on these grounds : "On reflection,

[(CK&)], though no doubt in general true, begins to look less like a

logical truth. For may not a particularly stupid person know that p&q and yet not realize that/? is a consequence of what he knows?"36 In the

same vein, it has been suggested that knowing that /? and knowing that q do not imply knowing that p and q, for a very stupid person may fail to

"put two and two together".

However, if we attempt to build our epistemic logic along the lines

suggested by the above criticism, the result is rather meager - our (propo

sitional) epistemic logic boils down to (CK); no other principles seem to

be immune to counter-examples from ordinary language.37 This ap

proach to the logic of knowledge does not appear very interesting. A more interesting approach is to take the principles of epistemic logic

as conditions of a certain type of 'rationality'. According to Hintikka's

interpretation of epistemic logic, statements which can be proved to be

false by means of epistemic principles, are not necessarily inconsistent in

the sense that their truth is a logical impossibility, but 'indefensible'.

Indefensible statements "depend for their truth on somebody's failure

(past, present, or future) to follow the implications of what he knows far

enough." 38

According to this interpretation, defensibility is "immunity to

certain kinds of criticism"39, viz. criticism of 'epistemic negligence'.40 Thus Hintikka restores the epistemic interpretation of modal logic by




reinterpreting the metalogical notions of consistency and inconsistency as defensibility and indefensibility. Statements whose negations are

indefensible are termed 'self-sustaining' ; thus the notion of self-suste nance corresponds to the customary metalogical concept of validity.

Alternatively, we may preserve the customary interpretation of our

metalogical notions and interpret 6K* in an unconventional way. Such

an interpretation is suggested by the observation that the alleged pe culiarities of the logical behavior of 'know' (e.g., the possible failure of

(CKimp)) seem to depend upon the condition of belief included in

definitions of knowledge. For instance, suppose that a knows that /?, and

/? implies q. According to (DKJ, a has complete evidence that /?, /? is

true, and consequently q is true as well. Furthermore, it is natural to

assume that a has complete evidence that q, if a has complete evidence

that/? and/? implies q; thus a has complete evidence that q. Nevertheless,

according to the criticisms mentioned above, a may fail to know that q. Given the definition (DKJ (or some variant thereof), this is, of course,

possible only if a does not believe that q, that is, if a's body of beliefs is

indefensible. And this appears perfectly possible; most philosophers

accept the possibility of people's holding wildly inconsistent, let alone

indefensible, bodies of belief.41 Perhaps the epistemic interpretation of

the system M (and possibly that of S4) can be 'saved', if we drop the

condition of belief from the definition of knowledge, and replace (DK4)

by

(DK6) Kp =

Ep&p.

In fact, it is easy to see that the concept of knowledge defined by (DK6) satisfies (CK), (CK&), (CKeq), and (CKlog), if the notion of complete evidence satisfies (CE&),

(CE) Ep => ~ E~p,

(CEeq) If /? = q is valid, Ep = Eq is valid,

and

(CElog) If/? is valid, Ep is valid.

These conditions seem to me entirely uncontroversial.42 (DK6) can be

said to define 'a is in a position to know that/?' or simply '/? is knowable to

a\ According to this interpretation, 'Kp' means that a knows that/?, if a

believes that p, recognizes the adequacy of his evidence, etc. This notion



120 RISTO HILPINEN

of knowledge is not without interest. In fact, so far as philosophers are

interested, not in the actual 'belief-performance' of persons, but in the

conditions under which people are justified in making knowledge-claims,

they are likely to be more interested in this notion of knowledge than in

the concept defined by (DK4) (or (DK5)).43 According to (DK6),

(30) KKp = EEp &Ep&p;

thus the KK-thesis is equivalent to

(31) p&EpzDEEp,

and follows from (10). The present form of the KK-thesis is obviously

valid, if 'complete evidence' is interpreted in the way described in Sec

tion III.44

In the arguments presented in Sections III and IV, the notions of

validity and consistency may be interpreted as self-sustenance and

defensibility. Thus, if the condition of belief is included in the definition

of knowledge and epistemic logic is interpreted in the way suggested by

Hintikka, the acceptability of the KK-thesis is very sensitive to shifts in

the formulation of the classical definition. However, there is another

sense of 'defensibility' in which the negation of the KK-thesis is clearly indefensible. I shall term this notion 'indefensibility in the generalized sense' or simply 'g-indefensibility'. According to Hintikka's doxastic

logic,

(32) If/? logically implies q, Bp&~ Bq is indefensible.

Suppose now that a believes that/?, and/? is complete evidence that q. p is

complete evidence that q, if q is a logical consequence ojf /? or p gives

'complete' (or adequate) nondemonstrative support to q. In the former

case Bp&~Bq is indefensible (in the customary sense), in the latter case

it is not. However, there is something 'wrong' with a person who refuses

to believe what has been proved 'beyond reasonable doubt'. I shall

express this by calling the belief sets of such persons 'g-indefensible'.

Indefensibility in the customary sense may be termed 'strict indefensi

bility' or 'logical indefensibility'. Thus,

(33) If/? is complete evidence that q,Bp&~ Bq is g-indefensi ble.45




g-defensibility implies strict defensibility, but the converse does not hold:

for instance, the belief sets of extreme sceptics may be strictly defensible, but g-indefensible.

If a has complete evidence that /?, he presumably knows (and thus

believes) something which gives complete evidence that/?. Thus, according to (33),

(34) Ep&~Bp is g-indefensible.

In other words, if a's beliefs are g-defensible,

(35) Ep => Bp

is true, and (DK4) is equivalent to (DK6). Again the KK-thesis follows

from (10). If knowledge is defined by (DK4), the negation of the KK

thesis is g-indefensible, but not strictly indefensible. In other words, the

KK-thesis is, if not 'self-sustaining', at least 'g-self-sustaining'. If the

completeness of evidence is thought of as being defined by inductive

rules of inference, the KK-thesis is true in a world populated by people who are not only 'deductively omniscient', but also 'inductively om

niscient'.46

VI. SECOND CRITICISM OF (DKj

Above, it was suggested that 'a has adequate evidence that/?' means that a possesses information which gives adequate or complete support to /?. In other words, if a has complete evidence that /?, there is an evidential

statement r such that r gives complete support to /?. Of course, a must

know that r, otherwise a could not be said to 'possess' the information.

Thus we have tacitly assumed the following definition of 'Ep' :

(DEX) Ep if and only if there is an evidential statement r such

that Kr and Crp,

where 'Crp' is short for V is complete evidence that/?'. This analysis of 'Ep' makes the definition (DK4) circular. (DEX) is also

circular: 'E' is defined in terms of 'K' and 'K' is defined in terms of 'E\

The classical definition of knowledge implies an infinite regress of

justification: knowledge is defined as true belief justified by knowledge, which again is supported by further knowledge, etc.47

The standard method of escaping this regress is to postulate the



122 RISTO HILPINEN

existence of 'basic knowledge'. Basic knowledge consists of beliefs which

are true and directly evident, and need not be supported by further

knowledge. The regress of justification stops at such directly evident or

'initially credible' beliefs.48

According to this classical view, all our knowledge is ultimately

justified by basic beliefs; thus it is assumed that the relation of complete

support is transitive :49

(CCtrans) Crs => (Csp zd Crp).

'Ep' can now be defined by

(DE2) Ep if and only if (i) Dp, or

(ii) there is a statement r such that

Dr & Crp,

where 'Dr' stands for V is directly evident (to a)'. If (DE2i) or (DE2ii) is

satisfied, a is completely justified in believing that /?. However, if 'Ep' is

defined by (DE2), definitions (DK4)-(DK6) are clearly inadequate

(though they might serve as definitions of basic knowledge, provided that

'E' is replaced by '/)'). It is normally assumed that Ep does not imply p ; one may be completely justified in believing a statement which is in fact

false. Let us assume that r is directly evident, but false,50 and /? is ade

quately supported by r. If/? is true and a believes that/?, then, according to (DK4) and (DE2), a knows that /?. Of course, in this case a does not

know that /?: 'knowledge' derived from false premisses is not really

knowledge. This difficulty does not arise, if 'Ep' is defined by the regress

entailing definition (DE^, as this definition implies that the evidential

statement r is true.

There are many different ways of escaping this difficulty. Above, it was

suggested that a knows that/? on the basis of r, if, apart form the obvious

requirements that /? be true and believed, /? is supported by r, and r is

basic knowledge (i.e., true, believed (by a), and directly evident (to a)). Thus we can define knowledge as a conditional concept, 'a knows that /? on the basis of r', briefly 'Krp', as follows:

(DK7) Krp = Br&Dr&r& Crp &Bp&p.

If V is substituted for '/?', the right-hand side of (DK7) follows from




Br&Dr&r; thus r itself is known on the basis of r, if it is basic knowledge.

(DK7) covers basic as well as non-basic knowledge.51 This definition can again be criticized on the same grounds as (DK4).

It may be argued that a knows that/? only if a's belief that/? is based upon

r, a can show that r supports /?, etc. This criticism will not be considered

here in any detail. It is not even relevant, if our main interest is in the

notion corresponding to (DK6), that is, 'a is in a position to know that /?

(on the basis of r)'. This concept may now be defined by

(DK8) Krp = Dr&r& Crp &p.

According to (DK8),

(36) KrKrp = Dr &r&CrKrp&Krp,

that is,

(37) KrKrp =

Dr&r&Cr(Dr&r& Crp &/?) & Crp &p.

Again it may be assumed that complete support is conjunctive, that is,

(CC&) Cr(/? &q) = Crp & Crq ;

thus (37) is equivalent to

(38) KrKrp =

Krp & Crr & CrDr & CrCrp.

According to (38), the KK-thesis is equivalent to

(39) Dr&r&Crp&rz>Crr& CrDr & CrCrp.

The first conjunct of the consequent of (39) may be regarded as analytic

(or valid); thus (39) is implied by the conjunction of

(40) Dr ZD CrDr

and

(41) Crp 3 CrCrp.

(40) and (41) can be defended by arguments similar to that presented above in support of (10) and (11). (40) states that, if r is directly evident, no further information (i.e., information in excess ofthat provided by r)



124 RISTO HILPINEN

is needed to justify that r is directly evident. This is plausible enough, and

in accord with the traditional view that directly evident statements are of

a special type, e.g., statements about the 'contents of experience' or

statements which contain only 'observational predicates'.52 Recognizing the evidential status of the facts expressed by such statements does not

presuppose any information about the world - except these facts them

selves. (41) can be defended on the basis of similar considerations. Thus

the present form of the KK-thesis may be accepted as valid.53

If knowledge is defined by (DK7), the KK-thesis can be refuted in the

same way as (5) was refuted in Section III. This counter-example consists

in pointing out that, even though p is adequately supported by r, a may fail to believe that /? is so supported, i.e., he may be unaware of Crp (or

Dr). However, if (DK7) is modified in accordance with Lehrer's criticism

of (DKO, i.e., by supplementing the definiens of (DK7) by 'BDr' and

'BCrp', the KK-thesis follows from (40), (41), (13), (CB&), (CBimp), and

(42) BpzDCrBp,

as the reader may verify. (42) is very plausible, if it is assumed that Bp is

directly evident (for a), and thus does not need any justification. It may thus be concluded that the modification of the classical defi

nition discussed above does not affect the acceptability of the KK-thesis.

VII. GETTIER'S COUNTER-EXAMPLE TO THE CLASSICAL

DEFINITION OF KNOWLEDGE

In a recent paper, Edmund Gettier presented an interesting counter

example to the classical definition of knowledge.54 Gettier's example is

relevant to all formulations of the classical definition discussed above; here I shall formalize it as a counter-example to (DK8).

Let us assume that

(43) Er&r&Crq,

and that

(44) /? is logically equivalent to q v s.

If the logic of knowledge is at least as strong as that of M, we have to




impose the following condition upon the notion of complete support:

(CCimp) If /? => q is logically true, Crp zd Crq.

This condition seems entirely uncontroversial. (43), (44) and (CCimp)

imply

(45) Crp.

Furthermore, if

(46) s is true and q is false,

/? is true. According to (43), (45), (46), and (DK8), a knows that/? on the

basis of r (or strictly speaking, a is in a position to know that /?). How

ever, in this case it is strongly counter-intuitive to say that a knows that/?, for a's claim to know that/? can be attacked (and presumably refuted) by

pointing out that the justification of/? involves the acceptance of a false

proposition, viz. q. As was mentioned above, inferential knowledge cannot be derived false premisses, and Gettier's example shows that the

intermediate steps through which p is derived from r must not include

false propositions either. In view of Gettier's example, Keith Lehrer and

Thomas Paxson have supplemented the classical definition by the re

quirement that the justification of /? by r be 'undefeated' (though the

expression 'undefeasible' may be more appropriate).55 In the case con

sidered above, a's claim to know that /? can be defeated by pointing out

that q is false. Thus the expression 'Crp' included in the definiens of

(DK8) must be reinterpreted as representing an 'undefeasible' justifi cation of/? by r. However, if Crp is undefeasible, there is no reason why

CrCrp should be defeasible, and thus (40) and (41) seem acceptable on

this interpretation, too. Hence Gettier's example does not seem very

interesting from the point of view of the KK-thesis.

VIII. CONCLUDING REMARKS

Finally, I wish to discuss certain criticisms of the KK-thesis which, so it

seems to me, are based upon misunderstanding the aims of epistemic

logic in general and the KK-thesis in particular. Two such criticisms are

presented in a recent article by David Rynin.56 First, Rynin argues as

follows: "If one assumes that if one knows that /? then one must know



126 RISTO HILPINEN

that one knows that /?, or be sure that /?, and if it be granted that one

could be mistaken unless one knows that one knows, it would seem to me

that there could be no knowledge, for in my view one could always be

mistaken, i.e., one could never be sure in the intended sense." According to the KK-thesis, the truth-conditions of 'Kp' and 'KKp' are identical.

Thus the KK-thesis implies that, whatever uncertainty there is included

in a's claim to know that p, the same uncertainty is involved in a's claim

that he knows that he knows that /?. If a could be mistaken in his justified claim that /? (in the sense intended by Rynin), though he knows that /?, he could also be mistaken in the claim that Kp, though he knows that he

knows that /?. Rynin's argument rests on a fallacy of equivocation.57

Secondly, Rynin argues that "if it is demonstrable that if one knows then

it follows that one knows that one knows it would seem to me very difficult if not impossible ever to know, for it seems clear to me that

this infinite process soon takes us beyond comprehension." Obviously

enough, no 'infinite processes' are involved here. If the truth-conditions

of 'Kp' and 'KKp' are identical, the latter statement yields no information

in excess of that provided by the former. According to the KK-thesis,

'Kp', and, say, 'KKKKp' are just two different ways of expressing the

same thing in writing (as far as the logic of knowledge is concerned),

though the former expression is usually preferable for obvious reasons.

The difficulty here is on the side of those philosophers rejecting the

KK-thesis, for they have the task of inventing for the latter expression a

meaning which differs from that of the former.58

Arthur C. Danto has presented, against the KK-thesis, an argument of

a different type.59 According to Danto, knowledge implies understanding: a knows that /? implies that a understands /?. It is possible that a knows

that a knows that /?, and consequently understands /?, without under

standing Kp. Thus a may know without knowing that he knows. By

'understanding /?' Danto seems to mean understanding the sentence '/?' ;

thus his argument presupposes that 'K' is treated as a predicate of sen

tences.60 Danto's argument implies that 'Kp' is a sentence in the syntax

language, and epistemic terms are prefixed, not to sentences, but to

names of sentences (i.e., '/?' here is the name of some sentence in a's

object language). However, if epistemic terms are understood in this way, all plausible epistemic interpretations of modal logic become incon

sistent.61 This difficulty does not arise, if 'K' is regarded as a sentential




operator. Thus there are good reasons for not interpreting 'K' in the way Danto does. If this interpretation is rejected, Danto's argument loses its

power.

Department of Philosophy,

University of Helsinki

REFERENCES

* This work has been supported by a grant form the Finnish Cultural Foundation

(Suomen Kulttuurirahasto). 1 An Essay in Modal Logic, North-Holland Publ. Co., Amsterdam, 1951, pp. 29-35.

Cf. also G. H. von Wright, 'Deontic Logic', Mind, N.S. 60 (1951) 1-15. 2 E. J. Lemmon, 'Is There Only One Correct System of Modal Logic?', Proceedings of the Aristotelian Society, Suppl. Vol. 23 (1959), pp. 39-40. 3

Knowledge and Belief : An Introduction to the Logic of the Two Notions, Cornell Uni v.

Press, Ithaca, N.Y., 1962. Cf. also Jaakko Hintikka, 'The Modes of Modality', in

Proceedings of a Colloquium on Modal and Many-Valued Logics (Acta Philosophica Fennica 16), Helsinki 1963, pp. 65-81 ; especially p. 78. 4 This axiom is 'distinctive' in the sense that we obtain S4, if it is added to the axioms

of M. 5

See, e.g., Roderick M. Chisholm, 'The Logic of Knowing', Journal of Philosophy 60

(1963) 773-95; Arthur Danto, 'On Knowing That We Know', in Epistemology: New

Essays in the Theory of Knowledge (ed. by A. Stroll) (Sources in Contemporary

Philosophy; F. A. Tillman, ed.), Harper & Row, New York, 1967, pp. 32-53, and

Analytical Philosophy of Knowledge, Cambridge Univ. Press, Cambridge, 1968, es

pecially pp. 147-58; Jaakko Hintikka, 'Epistemic Logic and the Methods of Philo

sophical Analysis', Australasian Journal of Philosophy 46 (1968) 37-51 ; E. J. Lemmon, 'If I Know, Do I Know That I Know?', in Epistemology (ed. by A. Stroll), pp. 54-82;

Charles Pailthorp, 'Hintikka and Knowing that One Knows', Journal of Philosophy 64

(1967) 487-500; David Rynin, 'Knowledge, Sensation, and Certainty', in Epistemology

(ed. by A. Stroll), pp. 10-31, especially pp. 29-30. 6 Cornell Univ. Press, Ithaca, N.Y., 1957, p. 16. Later Chisholm has rejected this

definition; cf. his Theory of Knowledge, Prentice-Hall, Englewood Cliffs, N.J., 1966,

pp. 6-7, and the discussion below. 7

Theaetetus, 201 c-d; Meno 98. In this classical approach to epistemology, the funda

mental question is : What must be added to true opinion to yield knowlegde? 8 See Knowledge and Belief, especially pp. 20-1, 50. 9 See Roderick M. Chisholm, Theory of Knowledge, p. 23 ; David Rynin, 'Knowledge,

Sensation, and Certainty', p. 10; A. J. Ayer, The Problem of Knowledge, Penguin

Books, Edinburgh 1956, p. 35, and 'Knowledge, Belief, and Evidence', in Danish

Yearbook of Philosophy, Vol. I, Munksgaard, Copenhagen, 1964, especially pp. 18-9; Richard B. Braithwaite, 'The Nature of Believing', Proceedings of the Aristotelian

Society 33 (1932-33) 129-46, especially pp. 129-30. 10 David Rynin, op. cit., p. 10. 11

Rynin, loc. cit. 12 See p. 23. 13

'Knowledge, Truth, and Evidence', Analysis 25 (1965) 168-75, see p. 168.



128 RISTO HILPINEN

14 The Problem of Knowledge, p. 35. 15

See, e.g., E. J. Lemmon, 'If I Know, Do I Know That I Know?', pp. 70-1 ; cf. also

A. D. Woozley, 'Knowing and Not Knowing', Proceedings of the Aristotelian Society 53

(1952-3) 151-72. 16 In what follows, I shall adopt the customary conventions regarding the combining force of various logical connectives, that is,

~ is the strongest connective, & is stronger than V, V than ==>, and => is stronger than =. 17 If the logic of knowledge is assumed to be similar to the system M (or S4), (CK&) is logically true. 18

Here, the expressions '? has complete (or adequate) evidence that /?' and 'a is com

pletely justified in believing that/?' are used interchangeably. It may be argued, however, that they are not synonymous; cf. Section IV. 19 For an intuitive justification of this principle, see, e.g., Carl G. Hempel, 'Deductive

Nomological versus Statistical Explanation', in Scientific Explanantion, Space and

Time (Minnesota Studies in the Philosophy of Science, Vol. 3) (ed. by H. Feigl and G.

Maxwell), Univ. of Minnesota Press, Minneapolis, 1962, especially pp. 150-1 ((CE&) is implied by Hempel's condition (CR1)), and Risto Hilpinen, Rules of Acceptance and

Inductive Logic (Acta Philosophica Fennica 22), North-Holland Publ. Co., Amsterdam,

1968, pp. 37-9, 44-6. 20

Henry E. Kyburg, Jr., is an exception. See, e.g., his 'Probability, Rationality, and a

Rule of Detachment', m Proceedings of the 1964 Congress for Logic, Methodology, and

Philosophy of Science (ed. by Y. Bar-Hillel), North-Holland Publ. Co., Amsterdam,

1965, pp. 301-10. Kyburg's rejection of (CE&) is motivated by the fact that no purely

probabilistic definition of 'complete evidence' satisfies (CE&). For a discussion of this

issue, see Jaakko Hintikka and Risto Hilpinen, 'Knowledge, Acceptance and Inductive

Logic', in Aspects of Inductive Logic (ed. by. J. Hintikka and P. Suppes), North

Holland Publ. Co., Amsterdam, 1966, pp. 1-20, and Risto Hilpinen, Rules of Ac

ceptance and Inductive Logic. 21 This view is accepted by Chisholm. According to Chisholm, believing is a 'self

presenting' state: "If I do believe that Socrates is mortal, then, ipso facto, it is evident

to me that I believe that Socrates is mortal" (Theory of Knowledge, p. 28). This is

precisely what (9) expresses. In this context, the expression 'evident' is, of course, more

natural than 'has adequate evidence'. For other proponents of this view, see Chisholm,

op. cit., p. 28n. 22 This view of the methodological role of complete (or conclusive) evidence is defended

and discussed in detail by Douglas Arner in 'On Knowing', Philosophical Review 68

(1959) 84-92. Cf. also Jaakko Hintikka, 'Epistemic Logic and the Methods of Philo

sophical Analysis', pp. 48-9. 23

According to Arner, "what counts as conclusive evidence is a matter of tacit,

continuing agreement among the users of language", (op. cit., p. 87). Cf. also S?ren

Halld?n, 'A Pragmatic Approach to Model Theory' in Proceedings of a Colloquium on

Modal and Many-Valued Logics (Acta Philosophica Fennica 16), Helsinki 1963, pp. 56-8.

Halld?n's arguments concern the notion of logical truth, but they can also be applied to the notion of complete evidence. 24 'Information' here refers to factual information, i.e., information about the world. 25 If (13) is accepted, can it be strengthened to an equivalence, i.e., is

(i) BBp =5 Bp

also acceptable? According to Nicholas Rescher, (i) is ,"from a strictly intuitive point of




view, a feasible candidate for acceptance by a logical theory of belief" ('The Logic of

Belief Statements', in Topics in Philosophical Logic (by N. Rescher), D. Reidel Publ.

Co., Dordrecht, 1968, p. 48). Nevertheless, Rescher is not ready to accept (i), for it

implies, if subjoined to certain other seemingly acceptable doxastic principles,

(ii) Bp^p,

which is clearly unacceptable. However, it is possible to construct a plausible logical

theory of belief in which (i), but not (ii), is valid. In fact, such a theory is already available: If the semantical theory of deontic modalities presented by Jaakko Hintikka

in 'Deontic Logic and Its Philosophical Morals' (in Jaakko Hintikka, Models for

Modalities, Selected Essays, D. Reidel Publishing Co., Dordrecht, 1969) is reinter

preted as logic of belief, (i) is valid, but no untoward consequences (like (ii)) are forth

coming. The crucial semantical conditions are:

(iii) If BpepeQ, and if v e Q is a doxastic alternative to //, then Bp e v.

(iv) If Bp eveQ, and if v is a doxastic alternative to at least one fieQ,

thenpev.

Condition (iii) yields (13), and (iv) yields (i). For the terminology employed in (iii) and (iv), see Hintikka, Knowledge and Belief, Ch. 3. It may be pointed out, however, that in

Knowledge and Belief, Jaakko Hintikka rejects (i) (p. 123). These observations are also relevant to (11). If (iii) and (iv) are modified in such a

way that they concern adequate evidence (or justification of belief), they yield (11). In

fact, on this interpretation (iii) and (iv) seem more plausible than in the case of the

notion of belief. 26 Cf. Chisholm, Theory of Knowledge, p. 6. This failure of the KK-thesis is described

by Hintikka as a case in which "somebody is in a position to know something without

being aware of it" (Knowledge and Belief, p. 118n). 27

'Knowledge, Truth, and Evidence', p. 169. 28 This formalization is especially natural, if belief that/? is interpreted as a disposition to act as if p were true. For surely justifying one's belief by appealing to certain

evidence is to act as if the evidence were adequate to justify the belief. 29 (9), (13), and (25) imply

(i) Bp 3 EBp &BBp ScBEBp ;

thus, according to (DK5), the principles accepted here imply

(ii) Bp => KBp. In Knowledge and Belief, Jaakko Hintikka rejects (ii), although he accepts (13) and the

KK-thesis (pp. 51-3). This view can be reconciled with the present analysis of 'know'

for instance by rejecting (9), and accepting instead only the weaker principle

(iii) Bp&Ep=>EBp. A detailed discussion of the implications of (9) and (ii) cannot be attempted here.

30 Philosophical Review 76 (1968) 491-9.

31 'Knowledge

- By Examples', Analysis 27 (1966) 1-11.

32 Lehrer, op. cit., p. 498.

33 The sentence s is somewhat puzzling. It cannot be analyzed by propositional logic in such a way that Lehrer's argument would be acceptable (i.e., that all premisses

would be seemingly true and imply the conclusion), s seems to be a subjunctive con



130 RISTO HILPINEN

ditional 'If a should correctly say that/? and know that the has said that/?, he would not

know that he correctly says that /?'. Lehrer's explanation of the meaning of 'a knows that he correctly says that/?' (p. 498n).

cannot be correct, as it makes s inconsistent. If *Ap9 stands for 'a says that /?', then 'a

correctly says that/?', 'a knows that he says that/?' and 'a knows that he correctly says

that/?' are expressed by 'Ap &/?', 6KAp\ and 6K(Ap &/?)', respectively. 34 These arguments are plausible in so far as we are speaking of factual knowledge,

i.e., knowledge that something is the case, and they are especially natural, if the ex

pression '/?' occurring in the scope of a modal operator is taken to represent, not the

statement/?, but the descriptive content of /?, i.e., in Wittgenstein's terminology, the

sentence radical Cf. Erik Stenius, Wittgenstein s 'Tractatus', Basil Blackwell, Oxford,

1960, Ch. IX; 'The Principles of a Logic of Normative Systems', in Proceedings of a

Colloquium on Modal and Many- Valued Logics (Acta Philosophica Fennica 16), Hel

sinki 1963, pp. 247-60; 'Mood and Language-Game', Synthese 17 (1967) 304-33;

reprinted in Philosophical Logic (ed. by J. W. Davis, D. J. Hockney, and W. K. Wilson), D. Reidel Publ. Co., Dordrecht, 1969, pp. 251-71. 35 An axiom system consisting of (CK), (CK^) and (CKlog) is also sufficient for the

epistemic counterpart of M. Cf. Robert Feys, Modal Logics (ed. by J. Dopp), Gauthier

Villars, Paris, 1965, p. 124. 36 E. J. Lemmon, 'If I Know, Do I Know That I Know?', p. 77. 37 Cf. E. J. Lemmon, 'Is There Only One Correct System of Modal Logic', p. 38. 38 Jaakko Hintikka, Knowledge and Belief, p. 32. 39

Hintikka, op. cit., p. 31. 40 This expression has been used by R. M. Chisholm in 'The Logic of Knowing' ; see

p. 784. 41 In 'The Logic of Belief Statements', Nicholas Rescher has emphasized the

" 'irra

tional' or 'illogical' nature of belief" (p. 42 in Topics in Philosophical Logic). It may be

argued, however, that philosophical logicians should not be too much concerned

about this 'irrationality'; cf. Alan Ross Anderson, 'Comments on von Wright's

"Logic and Ontology of Norms" ', in Philosophical Logic (ed. by J. W. Davis et al),

p. 110. 42 Cf. Risto Hilpinen, Rules of Acceptance and Inductive Logic, pp. 38-9. 43 Cf. Hilpinen, op. cit., p. 28. 44 In 'If I Know, Do I Know That I Know?', E. J. Lemmon has presented an example

which he takes to be a counter-example to the present form of the KK-thesis (pp. 69-70).

However, Lemmon's example is concerned with much too weak senses of 'a knows'

and 'a has adequate evidence' to count as a genuine counter-example to the present

form of the KK-thesis. 45 Of course, this condition cannot be formulated in terms of knowledge, as complete evidence does not imply truth. 46 The expression 'logically omniscient' has been used by Chisholm; see 'The Logic of

Knowing', p. 781. 47 For a detailed discussion of this regress and various reactions to it, see S. G. O'Hair,

Foundations in Epistemology, Cambridge Univ. Press (forthcoming). 48 An up-to-date version of this Cartesian view has been presented by Roderick

Chisholm in Theory of Knowledge; see especially pp. 24-9. 49 The requirement of transitivity is not obvious in the case of inductive support. For

instance, Isaac Levi has proposed an inductive acceptance rule which does not satisfy this condition (See Gambling with Truth, Alfred A. Knopf, New York, 1967, pp. 151-2).




The transitivity requirement has been defended by Risto Hilpinen in Rules of Acceptance and Inductive Logic, pp. 102-4. 50 This is not possible, if it is assumed that directly evident propositions are 'absolutely certain' or 'incorrigible'. This view of the basis of knowledge encounters many difficul

ties, however, and is not accepted here. Cf. O'Hair, op. cit., especially Ch. VII. 51 Here we assume that Crr is valid. 52

See, e.g., C. I. Lewis, An Analysis of Knowledge and Valuation, Open Court Publ.

Co., La Salle, 111., 1946, Ch. VII; Carl G. Hempel, 'Fundamentals of Concept Formation in Empirical Science', in International Encyclopedia of Unified Science, Vol. II: 7, University of Chicago Press, Chicago, 1952, pp. 20-3, and Alfred J. Ayer, The Foundations of Empirical Knowledge, Macmillan, London, 1940.

J. L. Austin has critized the view that there are basic statements in the present sense

(Sense and Sensibilia, Oxford University Press, Oxford, 1962, pp. 114-5). Austin

speaks of 'incorrigibility', but a similar criticism may be applied to the wider notion of

'directly evident'. According to Austin, no type of sentence is incorrigible as such, some

statements are incorrigible in fact under certain circumstances. Statements which are

incorrigible only in this sense cannot serve as the 'basis' of knowledge, for, in order to

be in a position to assert that, say, r is incorrigible (in Austin's sense), we have to know

that the circumstances are appropriate, and this presupposes factual information not

conveyed by r. (Cf. O'Hair, op cit. Ch. V.) But this is just another way of saying that (40) is not satisfied.

If the absolute notion of 'directly evident' is rejected, we may still give an interesting

interpretation to (D8). 'Dr' can be taken to mean that r is perfectly free from doubt, or

that no justification for r is demanded. The resulting sense of 'knowledge' is genuinely conditional or 'local' : In so far as r is taken for granted, a knows that /?. For this local

sense of knowledge and of justification, see C. S. Peirce, 'The Fixation of Belief, in

Collected Papers of Charles Peirce, Vol. 5 (ed. by C. Hartshorne and P. Weiss), Har

vard Univ. Press, Cambridge, Mass., 1932, p. 233, and Isaac Levi, Gambling with Truth, Alfred A. Knopf, New York, 1967, pp. 3-6. 53 In view of (DK8), we could now define '?/?' by

Ep=Dr&r&Crp.

Given (i), (DK?) and (DKs) are equivalent (apart from the fact that Ep and Kp must

now be taken as conditional on r). 54 'Is Justified True Belief Knowledge?', Analysis 23 (1963) 121-3, Gettier presents two examples; the formalization below corresponds to the second example. The first

example involves a definite description, and is more problematic than the

second. 55 Keith Lehrer and Thomas Paxson, Jr., 'Knowledge: Undefeated Justified True

Belief, Journal of Philosophy 66 (1969) 225-7; see especially pp. 225-7. 56 David Rynin, 'Knowledge, Sensation, and Certainty', p. 29. 57 This fallacy is natural here, for '? is certain that he knows' is one of the 'residual

meanings' of 'a knows that he knows'; cf. Hintikka, Knowledge and Belief, p. 116. 58 Cf. also Jaakko Hintikka, 'Epistemic Logic and the Methods of Philosophical

Analysis', p. 49. This difficulty does not arise in the case of the counter-example to the

KK-thesis discussed in Section II; according to (DK4) and the principles of doxastic

logic accepted here,

(i) KKp&~KKKp



132 RISTO HILPINEN

is indefensible, as the reader may easily verify. Thus all instances of the schema

(ii) KKp=K...Kp

are valid (or self-sustaining). 59 'On Knowing That We Know', pp. 49-50, and Analytical Philosophy of Knowledge, pp. 147-52. 60 See especially Analytical Philosophy of Knowledge, p. 148. 61 Cf. Richard Montague, 'Syntactical Treatments of Modality, with Corollaries on

Reflexion Principles and Finite Axiomatizability', in Proceedings of a Colloquium on

Modal and Many-Valued Logics (Acta Philosophica Fennica 16), Helsinki 1963, pp.

153-67; see especially pp. 158-61. This inconsistency result follows, if the language under consideration includes arithmetic.



knowing that one knows and the classical definition of ... that one knows and the classical...

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