A theorem of Epistemic Logic

by George N Schlesinger, December 15, 1982

In an important paper, ‘Is epistemic logic possible’, Max 0. Hocutt has
raised the question whether epistemic logic consists of logical truths in which
epistemic terms occur essentially. Admittedly there are theorems which
follow simply from the definition of ‘a knows that p’. However, this does
not amount to a unique feature of the concept of knowledge that would
explain why a specialogic should be constructed around it.

I shall try to show that epistemic logic is definitely ‘possible’ since there
are unquestionably statements expressing logical truths in which epistemic
terms occur essentially. On flrst thought it may appear that

(a) (p -+ q) & JBsp – JBsq

is a good example of such a statement. Unlike Bsq the question whether
JBsq holds does not seem to depend on s’s logical acumen or state of mind.
Even if s fails actually to grasp the connection between p and q, since objectively
speaking q is ‘included’ logically in p which s is justifled in believing,
there would seem no reason to deny (a).

Admittedly (a) has been denied recently, but I believe, on the basis of
faulty argument. Steven Levy in a paper entitled ‘Do you know everything
you know?’ questions the validity of

(b) Ksp & (p – q) – RBsq

where ‘RBsq’ stands for ‘it is rational for s to believe that q’. He says:
Suppose that s knows that p, that p entails q and that s correctly infers q from p. We
may, perhaps, be tempted to approve that any belief formed by s that q, as a result of
this process is a rational belief without inquiring further into the matter. But to do
so would be a mistake. Suppose that s also has a firm belief that r (it makes no difference
whether or not this belief is rational). Suppose further, that r entails not -q and that s
correctly infers not -q from r. In such a situation it would hardly be rational of s to
believe that q. Rather, s should be required to escape the dilemma either by further
inquiry or by suspending any belief that q or not -q. Thus, even though the antecedent
of (a) is satisfied, given s’s other beliefs it would not be rational for him to believe q.2
This argument is erroneous. After all there are two possibilities:

(1) r is not sufficiently well established. Thus in the view of the fact that
p, which is known by s to entail q is strongly enough supported, s is in fact
not justified in holding r (which he has correctly inferred as being inconsistent
with q).

(2) s is objectively justified in continuing to subscribe to r since r has been
firrmly enough established.

Now in case (1), s either adopts the attitude that is required by the standards
of rationality or he does not. Should he choose the first alternative
there is certainly no problem; s in fact rejects r and can therefore hold q without
any reservations. Undoubtedly therefore we may insist that the consequent
(b) is as true as its antecedent. Should he adopt the second attitude,
(b) still remains true since how the antecedent, and not only the consequent,
is false! Given that s believes r and thus -q and assuming that he knows that
– ,p, it follows that -Bsq and thus -Bsp.

In case (2) of course it follows that q has to be withdrawn because of its
incompatibility with the well established r. Thus it can no longer be the case
that RBsq nor that RBsp and therefore -Ksp. Once more therefore (b)
remains true since its antecedent is false.

To summarize: Levy’s example is harmless, since no matter what the
circumstances are, either RBsq remains true, or if not, Ksp is false too.
However, (a) may be rejected on other, legitimate grounds. Many philosophers
hold that q is not justified in believing that p even if p can be established
on the basis of what s knows, as long as a himself is not aware of the
way this is to be done. Suppose that Goldbach’s conjecture is true and I
believe this to be so and there exist a rigorous proof showing this to be so
however I am ignorant of this proof and have not been assured of the truth
of Goldbach’s conjecture by any expert who is familiar with the proof, then
I can neither be said to know that Goldbach’s conjecture is true nor that I
am justified in believing this to be so. In other words, the question whether
‘JBsp. is true or false is not determined solely by such objective factors as the
evidence s possesses and the logical relationship between it and p. Admittedly
it is a necessary condition that there be adequate evidence but unless s also
realized this, ‘JBsp’ is not true.

Let me therefore introduce the symbol ‘JB*sp’ to mean ‘objectively speaking
p can be established on the basis of information in s’s possession’ which of
course amounts to less than ‘JBsp’ that stands for ‘s is actually justifiled in
believing that p’. It may seem then that

(c) (p +q)&JBsp ?JB*sq

represents a logical truth. It turns out however that even (c) will not do.
Suppose that s has adequate evidence that p is true and thus JBsp even though
in fact p is false. Also suppose that s is incapable of inferring q from p but can
clearly see that q is false. In this case JB*sq is false in spite of the fact that
the antecedent of (c) is true.3

However (c) may be amended in a number of ways to yield a valid expression.
It is possible for instance to substitute ‘K’ for ‘JB’ and obtain
(d1) (pt q) & Ksp – JB *sq

The previous objection does not apply to (dl) since no longer can p be false
given that s knows that p. Now (dl) is not of much use because when we find
its antecedent true than though we are entitled to assert the consequent,
JB*sq, this has no practical significance. It does not matter whose name s
might be, since even the best flesh and blood individual cannot be depended
on not having occasional mental blocks and thus failing to make even the
most obvious inference JBsq does not necessarily follow from JB *sq.
However (dl) is logically equivalent to
(d2) (p – q) & -JBB*sq -+ -Ksp4

The last expression is useful and I propose to provide an important illustration
of its application.

Mark Steiner in discussing Descartes’ dream argument points out that
Descartes’ premises do not seem to entail his conclusion. Steiner claims that
the premises with which Descartes starts out are:

(pi): -K-D (I do not know that I am not dreaming)


(P2): D -+ -KS (If I am dreaming I do not know that I am
standing up)
and his conclusion
(g): -KS

Now since Descartes has not asserted that D it is not clear how (g) may be
derived from (Pi) and (p2). Steiner claims that (g) may be derived from the
given premises provided we add a third premise not mentioned by Descartes,

(*) If one is committed to -KP (P for any sentence) then it is irrational
for him to assert P.
His demonstration then proceeds as follows:
(1) KS – -D from (p2) by counterposition

(2) K(KS -+ -D) (1) Necessitation rule of epistemic

(3) KKS -+ K’D (2) Another rule of epistemic logic

(4) -K-D -* -KKS (3) Counterposition

(5) -~-KKS (Pl) and (4) modus ponens

(6) Asserting KS is irrational (5) and (*)

This is ingenious. In fact too much so, thus it is hard to believe that Descartes
‘thought’ of all this in the vaguest sense of that term. I believe that many will
have a strong feeling that there must be some other, shorter way, by which to
arrive at (6). Let me supply an argument showing why this feeling is justified.
We shall suppose that there are two surveyors aand b where a is a highly
intelligent person who among other things is an accomplished logician while b
is a very simple minded person whose numerous limitations include a total
ignorance of formalogic.

CASE 1: Let Da = The calculator used by a is defective
S = The amount of material required to cover the walls
of building B is n.

Let us also suppose that a performs a series of measurements and computa
tions some of which he checks with his calculator and at the end of which he
arrives at the conclusion that S – which happens to be true. It so happens
that a does not know (while b does know) that the computer he is using was
not defective. It is clear then that we have

(pl) -Ka -Da and (p2) Da -*-KaS.

These being parallel statements to those dealt with by Steiner, we need go
into any details in order to show that a will arrive via the steps taken by
Steiner at (5′) -KaKaS.

CASE2: Let us have the previous situation repeated exactly while substituting
b for a and vice versa. In particular b in now using a calculator which
he does not know to be in good order, while a knows this (and soon the role
of this bit of assumption will become clear). Obviously we shall get as far as
(1 “)KbS – -Db, but no further. Steiner has explained that he moved from
(1) to (2) = K(l), on the assumption that if I myself have demonstrated the
truth of (1) then surely not only must (1) be true and I believe it to be true,
but I am justified in believing it to be true, that is, I know that (1) is true i.e.
K(1). Thus the move from (1) to (2) is facilitated not by pure logic alone,
it also requires that I myself should have derived (1) from true premises, that
is from (P2). It follows therefore that for any person who is not aware that
(P2) entails (1) because, for instance, he does not know the rule of logic
called Counterposition, and b is such kind of a person, K(l) is not true and
hence (6) remains underivable. In other words it is impossible to demonstrate
that -KbKbS.

Some may find this result paradoxical. We have seen that a who is a superior
surveyor does definitely not know that S, while b, in spite of the fact
that he is much less likely to do a good job, is in the advantageous position
in which he cannot legitimately be labelled as being ignorant of his knowledge
that S. There will be however philosophers who will not fmd this result intolerable.
In other contexts it has already been remarked that sometimes
“One may know less by knowing more”6 and it is not ultimately so strange
that a sophisticated person should be shrewd enough to discern certain
obstacles that stand in the way of knowing a proposition which may happen
to be true, obstacles to which a more simple minded person remains oblivious.
Hence the former may gain the insight to conclude that he is not entitled
rationally to subscribe to S even though S may well be true while another
person in his blissful ignorance cannot be disqualified from holding that he
knows that S. It was after all Socrates who said that he knew more than
others since he was aware of his total ignorance. It may well be that Socrates
happened to hold a lot of true beliefs,beliefs also held by his fellow Athenians.
Yet he, but not they, disowned the claim that he actually knew that those
beliefs represented the truth.

However from our previous story we may also derive another strange
looking result which cannot so easily be smoothed out. While admittedly
Kb (Kbs — -Db) does not follow from (1 “) the expression (2*) Ka(KbSb -*
-Db) is derivable since a knows enough logic to arrive at (1”). From this,
following Steiner we get

(3 *) KaKbS – Ka -Db

which is the counterpart of (3) and

(4*) -Ka -Db -* -KaKbS

the counterpart of (4). We recall that in the original reasoning Steiner’s next
step was to apply Modus Ponens to (Pl) and (4) to derive

(5) ,KKS

It is however, clearly impossible to make the parallel step here to derive from
(pr) and (4*)

(5*) -KaKbS

since (pr) is -Kb -Db and the antecedent of (4*) is quite a different proposition,
namely, -Ka -Db. Not only are we not given this last proposition but
in fact it was explicitly stipulated that on the contrary Ka -Db for we said
that a knew that the calculator used by b was in working order.

We have thus reached a paradoxical result. We have seen that in Case 1 a
definitely does not know that KaS but in Case 2, which is a mirror image of
Case 1 with the roles of a and b precisely reversed, a himself in spite of all
his knowledge and intelligence is not in the position to declare b to be ignorant
of KbS or that b is ignorant of KbS. The previous explanation does not seem
to be applicable here. After all Socrates who was enlightened enough to hold
that he knew nothing, did not regard his intellectually less accomplished
fellow citizens to be more knowledgeable. On the contrary he was prepared
to declare their ignorance with even greater conviction than his own.
Thus we have added reason to believe that here exists a different deriva
tion of Descartes’ conclusion, in particular, one which is more straight forward
and does not require the troublesome step from (1) to (2), which is not
legitimate, unless the knower himself has established (1). We expect there to
be a briefer route leading to the conclusion one which will also permit the
derivation of -KaKbS no less than -KaKaS.


In my book Metaphysics: Methods and Problems 7 I have demonstrated that by
applying elementary probability notions, Descartes’ conclusion may easily
be derived from (Pi) and (P2). It seems to me now however that Steiner
would be entitled to reject this derivation. He may very well claim that it is
a misunderstanding of the concept of probability to think that it may be
applied in the context of Descartes’ argument. We are certainly not faced
with a situation concerning which we could say that in the past in similar
instances the frequency of cases that turned out to be such that our experiences
have proved to have been generated by genuine external factors was
such and such. It makes therefore no sense to assign any probabilities to
metaphysical statements in general and to the statement ‘the external world
exists’ in particular.

However, now we are in a position to overcome all difficulties and objections
by making use of the epistemic principle we have introduced in the first
section, namely (d2)(P — q) & -JB*sq -* -Ksp. Descartes asserted (Pi)
-K -D. The reason he gives is that all his experiences are insufficient to
imply, indicate or provide rational basis for believing with enough confidence
that what goes on in his mind is generated by corresponding external
factors any more than this is true in the case of a dream. In other words he
holds (p*’) JB* -D. Now substitute into (d2) ‘KS’ for ‘p’ and ‘-D’ for q
and we get

(d*) (KS ‘ -D) & -JB * -D ‘ -KKS.

Conjoining (1) and (p*’) we get the antecedent of (d*) and that together with
(d2) logically implies by modus ponens -KKS.

With the aid of our theorem we thus are able to derive quickly the desired
conclusion without having to make use of concepts like probability whose
applicability in the present context may be questionable. The brevity of our
proof in comparison to Steiner’s makes it into a somewhat more likely can
didate to amount to reconstruction of Descartes’ own reasoning. An additional
important advantage it has of course that it does not involve Steiner’s
problematic move from (1) to (2) and our proof is therefore independent
of the logical acumen of the knower.


1 Notre Dame Journal of Formal Logic (1972), p. 435.
2 Ibid.
3 I owe this important point to the Editor.
4 Another valid expression is
(e) JB*s [p & (p – q)] JB*sq
p may of course be false, but should it be the case that JB*sq then surely there is no
justification to believe both p and p – q which entail q. Obviously also
(f) (p – q) &JBsp & -IEsq – JB*sq
is valid where ‘IEsq’ stands for ‘s has some independent evidence relevant to the credibility
of q’.
5 ‘Cartesian scepticism and epistemic logic’, Analysis (1979).
6 e.g., Carl Ginet, ‘Knowing less by knowing more’, MW Studies in Philosophy (1980),
pp. 151-161.
7 (Oxford, 1983), p. 255.

Department of Philosophy,
University of North Carolina,
Chapel Hill, NC 2 7514,