some varieties of epistemological scepticism

From: Philosophia Vol. I Nos.l-2 Pp. 107-116 January 1971

S O M E V A R I E T I E S O F E P I S T E M O L O G I C A L

S C E P T I C I S M

RICHARD L. PURTILL

In this paper I examine an argument which seems to be at the root of many forms of epistemological scepticism. Although my purpose is not primarily historical, I think that it is true to say that the line of argument which I will examine has provided at least some of the motivation and support for many historically important varieties of epistemological scepticism, and that once the weakness of this line of argument is exposed these positions lose, if not all, at least a great deal of their plausibility.

I will use, although in a very elementary way, certain tech- niques of modal and epistemic logic, and it is well to make clear what these are. First I will use a Hintikka-like notation, writing "Kap" for "a knows that p," "K a ~ p" for "a knows that not-p" and so on. I will also assume without argument that "K a p" logically implies "p." Second, I will employ a system of pro- positional modal logic at least as strong as Lewis' system S.3; especially I will make use of principles Strict Transposition:

(p -+ q) ~- (~ q -+ ~ p )

and Strict Implication of Possibilities:

( p ~ q ) _ - - ( ~ p - , ~ q )

where the "arrow" symbol stands for strict implication, the four bar symbol for strict equivalence, and the diamond for logical possibility. I will also employ the usual equivalences for exchanging

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modal operators, which are based on the equivalence Modal Operator Exchange:

D p - ~ Q ~ p

where the square stands for logical necessity. At any point where I employ any more than this elementary machinery I will discuss any further principle which is involved.

We are now prepared to discuss the argument which seems to support many varieties of epistemological scepticism. From what was said above it is clear that I accept the principle

K a p ~ p

and that this implies, by Strict Transposition:

~ p - ~ ~ K a p

and that this in turn implies by Strict Implication of Possibles:

~ p ~ K a p

In words, what we are saying is that if it is logically possible that a statement may be false then it is logically possible that we do not know it. By a factual statement we mean one such that although it is true it is logically possible that it be false, i.e. a statement of the form

p . ~ p

It then follows that for any factually true statement which we know, it is logically possible that we do not know it. That is, from any statement of the form

K a p . ( p . ~ ~ p)

there follows a statement of the form

K a p - ~ > ~ K a p

All this I think is both true and common-sensical. The mis- take which seems to me to lie behind many forms of epistemological skepticism if put in words goes something like this: "If you know, you can't be wrong: therefore if you may be wrong you can't really know."

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EPISTEMOLOGICAL SCEPTICISM

We might, as a first attempt, try symbolizing "if you may be wrong then you don't know," as

~ ~ p ~ ~ K a p

and interpret the sceptic as claiming that this follows from the principle "if you know you can't be wrong." But if this truism is interpreted as

K a p ~ p

we can show that this does not imply

~ p ~ ~ K a p

We first use Strict Transposition and double negation to show that

< ~ p ~ ~ K a p

is equivalent to

K a p - - * ~ ~ p

and use Modal Operator Exchange to show that this in turn is equivalent to

K a p ~ [ ] p .

The first remark to be made about this is that

K a p ~ [- lp

is neither equivalent to nor implied by

K a p ~ p

Of course, one could interpret "if you know you can't be wrong" as the second of these rather than as the first. But in either case one can see that this variety of epistemological skepticism simply amounts to a determination to regard only necessary truths as suitable objects of knowledge.

Of course some philosophers, for example Plato, have held this position explicitly, for a variety of reasons. But some of these

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reasons were certainly bound up with the confusion I am exam- ining: the idea that knowledge cannot be mistaken and therefore must be confined to some suitably "safe" objects, such as necessary truths. But unless we hold an implausible view of the meaning of ,if you know you can't be wrong" in the first place, or mistakenly try to derive it from the plausible version of "if you know you can't be wrong" then this position is simply a confusion.

The next move which might be made by an epistemological sceptic might be an at tempt to make plausible the principle

C ~ K a p ~ ~ K a p

which in English might be read "If you may not know then you don't know." But this principle seems immensely implausible for a number of reasons. Firstly, simple transposition gives us

~ K a p - - * ~ @ ~ K a p

which is equivalent to

K a p ~ U 1 K a p

which can be read as "If you know then it is a necessary truth that you know." But surely even if I knew only necessary truths, it is logically possible that I should have failed to know some of those t ru ths - - th i s position out-Platos Plato.

An interesting sidelight on this principle is that

K a p --, V-1Ka p

is a statement of the form

p - * E]p

As I have shown in another context 1 for any system of modal logic at least as strong as Lewis' S.5, the following theorem is provable

(p -+ [] p) -+ ( ~ p 2p)

and from this it follows that if it is true that if you may not know then you don't know then it follows that if you may know you do know, that is

@ ~ K a p - - , ~ K a p

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EPISTEMOLOGICAL 'SCEPTICISM

The implausibility of this surely needs no laboring: there are great numbers of things which I may know but which in fact I do not know.

A form of epistemological scepticism which prima facie seems to offer more hope might begin with a closer analysis of "you may be wrong." It might be argued that "a may be wrong about p" is not properly rendered b y " ~ ~ p" or by " ~ ~ K a p" but is a more complex notion. One plausible analysis might use the notion of belief. To be wrong about p is to believe p, where p is in fact false. Using the Hintikka-like notation "B a p" for "a believes p" we can write "a is wrong about p" as "B a p ~ p" and "a may be wrong about p" as " ~ (B a p ' ~ p)". The principle "if you may be wrong you don't know" may then be written as

@ ( B a p . ~ p ) ~ ~ K a p .

As you might expect by now my first countermove is to trans-

pose: ~ ~ K a p ~ ~ @ ( B a p . ~ p )

removing the double negative this is

K a p ~ - . ~ ~ ( B a p . ~ p )

or "If you know then it is impossible that you be mistaken." This sounds enough like the familiar "If you know you can't be wrong" to have some initial plausibility, though even here we might have qualms. However by successive transformations we can go from

@ ( B a p ' ~ p)

[ ] ~ ( B a p - ~ p )

(by modal operator exchange) then to

[ ] ( ~ B a p v p )

(by De Morgan's rule and double negation) then to

[] ( B a p ~ p)

(by definition of material implication) and finally to

B a p ~ p

by definition of strict implication.

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Thus the principle currently under consideration becomes

K a p ~ ( B a p - - , p)

In English this might be "If you know p then believing p logically implies that p is true." Now this gives me, at least, a shock of recognition. At a certain stage of Cartesian skepticism it is precisely statements of which it is true that "believing makes it so" that are proposed as the only genuine objects of knowledge.

The most prominent members of this class of statements, of course, are the cogito and its close relations. If I believe that I exist, I exist. If I believe that 1 am thinking I am thinking, and so on. However, quite significantly any logical truth is also a member of this class of statements. For on any standard account of strict implication a logical truth is strictly implied by any statement and thus afortiori by the statement that some individual believes that it is true. It is fascinating historical speculation that some insight into this fact lay behind Descartes' confused and confusing doctrine of "clear and distinct ideas." For cogito-like statements do indeed share an important quality with logical truths; they are both members of the highly restricted class of statements such that "B a p ~ p" holds.

A slightly more complicated version of the view now under consideration would result from a more complex analysis of "a is mistaken about p." It might be argued that believing something false is not an adequate analysis of being mistaken in the most epistemologically interesting sense. On this view "a is mistaken about p" should be analysed as "a believes that he knows that p, but p is false." In symbols this would be "B a K a p" ~ p" and the principle "if you may be wrong you don't know would be

~ ( B a K a p - ~ p ) - - - , K a p

By a series of transformations similar to that above this is equivalent to

K a p ---, (B a K a p ---, p)

A yet more complex analysis of "a is mistaken about p"

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would be " B a K a p . ~ K a p" which would lead to another form of the principle

K a p ~ (B a K a p ~ K a p )

It should be clear enough that these two versions of the principle in question are successively stronger:

K a p ~ ( B a K a p ~ K a p )

implies

Kap-- - , ( B a K a p ~ p)

via the principle "K a p ~ p" and strict hypothetical syllogism. However the principle

K a p ~ ( B a K a p ~ p)

does not seem to imply

K a p ~ ( B a p ~ p)

even though "B a K a p" implies "B a p." What can we say from a philosophical point of view about

any of these three versions of "if you may be wrong, then you don't know"? The first and perhaps most obvious point is that none of them can play the role which they have sometimes been assigned in "foundation" theories of knowledge such as Descartes'. If we invoke any of these principles to establish some especially favored status for statements such as the cogito we cannot later ignore this principle, and later declare that we have arrived at knowledge of ordinary empirical truths by arguments which do not satisfy the condition that "B a p ~ p". Schopenhauer's gibe against de- fenders of the cosmological argument, that the principle of causality is dismissed, like a cab, after having been used to arrive at a de- sired destination, would apply with equal force to any such use of a version of the principle "if you may be wrong, then you don't know."

Of course, a Cartesian might argue with a genuine epistemological spectic; "Even you will grant that we know certain sorts of statements, those such that if we believe them they are true. Well,

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t h e n . . . " But of course the Cartesian would be invoking not

But its converse

K a p - - , (Ba p ~ p)

(B a p ~ p) ---, K a p

which leads to quite a different argument. The second point which can be made about a sceptical posi-

tion based on one of the various analyses of "if you may be wrong then you don't know" is that surprisingly enough it seems to re- duce to the earlier and less plausible form of epistemological scepticism. The reduction is not a strict one, since it involves some questionable principles, but it is I think, sufficiently con- vincing. To show that anyone who holds the principle

O ( B a p . ~ p ) - - * ~ K a p

ought consistently to hold the principle

~ ~ p ---, ~ K a p

it is necessary to invoke the bridging principle

~ p---, O (B a p ' ~ p)

that is "if a statement is possibly false it is possible to be mistaken about it." The possible counterexamples to this claim would pre- sumably involve statements which are so blatantly false as to be unbelievable, statements for which something of the form " ~ O B a p" held. However the epistemological dogmatist may cheerfully concede the unknowability of the unbelievable. Pre- sumably the interesting cases are those such that

( @ B a p . ~ ~ p ) ~ @ ( B a p - ~ p )

holds, and for the interesting cases, e.g. scientific and historical knowledge it would be plausible to assert " ~ B a p." It is worth noting that in general

( 0 p" ~ q) -* ~ ( p q)

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is not and cannot be a theorem of any system of modal logic for it would involve the unacceptable

(0 P 0 ~ p)--+ 0 (P' ~ P)

However for some specific statements something of this form may hold, just as "p ~ [] p" is not in general true but may be true for some specific statements.

Very similar remarks hold for the derivation of

O ~ p ~ ~ K a p

from either of the more complex versions of the "if you may be wrong you don't know" principle.

The principles needed would be

and

~ p--+ ~ ( B a K a p " --~ p)

@ ~ p - * @ ( B a K a p . ~ K a p )

and these in turn would most plausibly be derived from

( O B a K a p . O H p) -* O ( B a K a p - ~ p)

and

( G B a K a p . �9 ~ K a p ) - + ~ ( B a K a p . ~ K a p )

for the class of cases for which " ~ B a K a p" was plausible. The demonstration that anyone who holds the principle

@ ~ p --+ ~ K a p

ought consistently to hold the principle

@ ( B a p . ~ p)--+ ~ K a p

is more straightforward. The bridge principle involved is

~ ( B a p ' ~ p ) - - + ~ p

and this is a consequence of a familiar distribution rule for modal logic

(p" q) --, (O p" ~ q)

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Equally straightforward is the derivation of the bridge principle

O ( B a K a p . ~ p)--, ,~ ~ p

However the derivation of

O ( B a K a p . ~ K a p ) - , O ~ p

involves the further principle

< ;~ K a p - - - , ~ p

which may be held to be questionable. So far purely formal considerations (or mainly formal con-

siderations) can take us. Beyond this, the truth of any view which restricts our knowledge to necessary truths, or truths such that belief make them so, etc. is open to familiar philosophical refuta- tions. The aim of this paper has been to show that such initially plausible principles such as "if you may be wrong then you don't know," when carefully formulated, do not constitute an indepen- dent argument for any form of epistemological scepticism. Rather they turn out to be exactly equivalent to some such sceptical view. Thus to use such a principle as an argument for such a view is precisely to beg the question.

WESTERN WASHINGTON STATE COLLEGE BELLINGHAM, WASHINGTON, 98225 USA RECEIVED: 22 NOVEMBER, 1970

NOTES

1 "Ontological Modalities" in Review of Metaphysics, Vol. XXI No. 2. December,

1967. To be reprinted in Bar-On. Z. [cdilor) Rr its Onlo/og). (Hebrew University Press, in preparation.)

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some varieties of epistemological scepticism

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