Russell and the Stopped Clock

Many philosophers today appear to regard Russell’s example of the stopped clock as the really hard Gettier counterexample, the one to beat — or believe that it can be tweaked slightly to defeat most current solutions. In the last post I sought to counter this view by describing my own truthmaker solution and showing that it defuses this example in the same way as the others. Here I want to say a little bit more about this example in the context of Russell’s writing on epistemology.

The first thing that needs repeating is that Russell himself did not put up the stopped clock example as a Gettier counterexample — i.e. he did not see it as an example of a justified true belief that yet wasn’t knowledge. Rather he saw it as an example of a true belief that yet wasn’t knowledge because the proper justification was here lacking. Here is the full quote from Human Knowledge: Its Scope and Limits, pp. 170—1.

It is clear that knowledge is a sub-class of true beliefs: every case of knowledge is a case of true belief, but not vice versa. It is very easy to give examples of true beliefs that are not knowledge. There is the man who looks at a clock which is not going, though he thinks it is, and who happens to look at it at the moment when it is right; this man acquires a true belief as to the time of day, but cannot be said to have knowledge. There is the man who believes, truly, that the last name of the Prime Minister in 1906 began with a B, but who believes this because he thinks that Balfour was Prime Minister then, whereas in fact it was Campbell-Bannerman. There is the lucky optimist who, having bought a ticket for a lottery, has an unshakeable conviction that he will win, and, being lucky, does win. Such instances can be multiplied indefinitely, and show that you cannot claim to have known merely because you turned out to be right.

If you have a true belief just from dumb luck you do not have knowledge: knowledge requires that you have come by your belief legitimately, as the result of honest toil. That was Russell’s point. But Gettier went further: his idea was that you could have acquired the belief as the result of toil and yet still have been merely lucky that it was true, because its truth wasn’t guaranteed by the evidence that you obtained and that provided you your justification. But would Russell have agreed with this? Or would he have thought that this was a pseudo-problem, generated by a false view of how beliefs are justified? I think, the latter.

First let us note that, for Russell, it is, in the first instance, beliefs that are true or false, and that sentences only become true or false derivatively in expressing beliefs. Also — and most importantly — beliefs are far richer for Russell than simple sentential reports. One can have beliefs without language; Russell notes that many animals have beliefs, but no language. But even for us: beliefs may go far beyond what we are able to express, or ever actually express. As he says: ‘the belief is richer in detail and context than the sentence, which picks out only certain salient features’.

You say “I shall see him soon”, but you think “I shall see him smiling, but looking older, friendly, but shy, with his hair untidy and his shoes muddy” — and so on, through the endless variety of details of which you may be only half aware.

Given this, I think we can see that Russell would have had very little patience with Gettier’s first counterexample, about Smith and his belief that the man who will get the job has six coins in his pocket. For Russell, the content of this belief fixes the referent as Jones, because it is Jones who Smith believes will get the job. Thus, since it is Smith who ends up getting the job, Smith’s belief is simply false. It is irrelevant that Smith happens to have six coins in his own pocket — that is just dumb luck — what matters is that Smith’s belief was a belief about Jones. Thus, for Russell, this would be merely a case of a false belief — and not knowledge simply because it is false.

What Gettier did — his trick, as it were — was to make beliefs thin: delimited by the content of a proposition. But then there was a second trick: the believed proposition was expressed in words (as though the belief was merely a string of words in Smith’s head) and thus was capable of implying other sentences and of being treated as separate from Smith’s intentions — in fact capable of being ambiguous as to its referent. Thus, in the proposition the man who will get the job has six coins in his pocket, Gettier can construe the referent of the descriptive term the man who will the job as whomever is picked out by that description, as a matter of objective fact. And that man is Smith himself, since, unbeknownst to himself, he will get the job.

But the two steps that Gettier takes are steps that Russell explicitly warned against. Beliefs are thick, for Russell, and it is they that are true or false. And it is also they that have evidential support. There is no reason to believe that the evidential support for a thick belief will descend to a proposition believed in Gettier’s thin sense — because so much content is removed. Thus, for Russell, Gettier’s target proposition the man who will get the job has six coins in his pocket has no evidential support if the descriptive phrase the man who will get the job does not specifically refer to Jones. It is void.
For Russell, Gettier’s first counterexample would simply be a mistake in analysis.

But in fact the second counterexample would fail for similar, though not exactly the same, reasons.

By treating Jones’s belief (remember that was Brown owns a Ford) as a proposition in a thin sense — one that has the structure of a sentence — Gettier is able to use the elementary Propositional Calculus to form another proposition that will be true if that original proposition is true (remember: that new proposition is Either Brown owns a Ford or he is in Brest-Litovsk). If Jones is going to believe this disjunctive proposition it would be because he has additional beliefs about the truth functional character of the inclusive disjunction and the validity of the rule of Addition (Or-Introduction). His evidence that Brown owns a Ford would not be enough, alone, for him to believe the disjunction. But the situation is even worse than this.

Truth functional implications apply to propositions in logical derivations — not to beliefs. It isn’t the Belief-Functional Calculus. From the belief Brown owns a Ford Jones could not move to the disjunctive proposition but only to the more complex conditional proposition

If Brown owns a Ford then either Brown owns a Ford or he is in Brest-Litovsk.

With his beliefs about truth-functionality, Jones could well believe this conditional. He should. But one cannot say, on Jones’s behalf, that since Jones believes Brown owns a Ford he must then believe that either Brown owns a Ford or Brown is in Brest-Litovsk. Because Jones should protest: Only if my belief is true will that disjunction be guaranteed to be true; if it is false there is no such guarantee, and my evidence must then be misleading; and my believing something to be true does not mean that it is true. So I do not, as a matter of fact, believe the disjunction: my belief in this case is void. (This argument seems to me to be very similar to an argument that John Broome has given in Ratio — but I don’t have the reference available.)

In other words Jones should believe

  1. If p is true then p or q is true for all q,

but there is no good reason for him to believe

2.    If I believe p then I must believe p or q for all q.

One can see this perhaps more clearly if we note that Jones, believing p, may give up p when it is pointed out to him that it implies some other proposition r.

The upshot is that there is no reason to suppose that Jones should believe Gettier’s disjunctive proposition. And I think — though of course one cannot be sure — that this is what Russell would say about the example. For Russell evidence and belief are fused together rather like parts of a single organic process. The evidence once received is turned into the belief; the thickness of the belief is a function of the thickness of the evidence. The belief is a mental state in a rich sense, more resembling, in its complexity, a painting than it resembles a thin proposition. This suggests that evidence need not be transferable — from a belief to a proposition that is logically implied by it (in Gettier’s sense). And if it is not transferable then Gettier’s disjunction need have no evidential support. In the case of the stopped clock I think Russell would say that merely glancing at a clock does not give you sufficient evidence to believe that it is accurate. (But I emphasise that with respect to Russell, I am speculating as to what he would say to Gettier.)

It is this transferability of evidence that does the heavy lifting in Gettier’s argument and opens up the gap between a justified true belief and knowledge. Once we allow it we must find a way to close it again. This is where the truthmaker account comes in. By insisting that the state of affairs that is the truthmaker be the same as the state of affairs that the evidence is evidence of, the gap is closed.

It is worth noting that, in Human Knowledge, Russell describes a truthmaker theory. I think that most students of philosophy do not understand this — almost certainly because they have never been taught it. They might be aware that Russell outlines something resembling a truthmaker account in the Lectures on Logical Atomism, from 1918. But there seems to be a general idea that this is something he gave up. He didn’t. In his works of the 1940’s — An Enquiry into Meaning and Truth and Human Knowledge — he holds to it. It is true that Russell doesn’t use the word ‘truthmaker’ but he does certainly speak of facts making beliefs and sentences true. Thus he says

In like manner we want a description of the fact or facts which, if they exist, make a belief true. Such fact or facts I call the “verifier” of the belief. (p. 166)

So a verifier is a truthmaker. The whole theory of truthmakers is just the development and continuation of the view initiated by Russell and Wittgenstein before and during WWI. It is a great shame — and a lost opportunity that lasted for nearly a full century — that this view was not taken up and developed as it needed to be. And even today the theory is treated as something odd and unfashionable — as though philosophers were peacocks that can only stand to be seen in the latest dress!


Postscript for bibliophiles.

The copy of Human Knowledge that I have is a first edition — I bought the book second hand many years ago. It comes from the library of T. D. Kellaway, who was Professor of Pathology at Cambridge University. But what is particularly interesting is that the dust jacket of the book is printed on the back of a WWII map — since paper was still in short supply in Britain through the 1940s and rationing was still firmly in place. Images below.


The Truthmaking Solution to the Gettier Problem

In 2003 I set out a new solution to the Gettier problem that had bedevilled the theory of knowledge. Here I want to describe that solution and make some personal remarks as to its genesis and subsequent history. (All the relevant papers can be found on

The idea is easily described. Let us take the second of Gettier’s own counterexamples to the idea that knowledge is justified true belief (and no more than that) — that is, the idea that in having a justified true belief one has satisfied the sufficient conditions for knowledge. Jones has good reasons to believe that Brown drives a Ford, in the form of Brown’s own testimony and his (Jones’s) own observations. Jones picks a set of cities that begin with the letter ‘B’: Barcelona, Brest-Litovsk, Boston, say, and then forms propositions for each such city, of the kind:

Either Brown owns a Ford or he is in Barcelona,

Either Brown owns a Ford or he is in Brest-Litovsk…

Jones believes each of these propositions simply because he believes on good evidence that Brown owns a Ford, and this is the first disjunct of each of these sentences. (It must be emphasised that the disjunction here must be understood as inclusive disjunction.) It turns out however that Brown does not own a Ford but that he is in Brest-Litovsk. Thus this sentence is true and believed on good grounds, and hence is justified — despite the fact that Jones has no reason at all to believe that Brown is in Brest-Litovsk. It must be emphasised that this problem rests on us sharing Gettier’s intuition that under this circumstance Jones does not know that Either Brown owns a Ford or he is in Brest-Litovsk but that his belief is both true and justified. Most people have shared this intuition — but if one doesn’t then one will not see a problem here. At any rate I do share Gettier’s intuition.

The truthmaker solution to this and similar Gettier problems is that the truthmaker for the sentence — the state of affairs that makes it true — is not, as it normally would be, the state of affairs of which Jones has evidence. Instead we have two different states of affairs, one of which is the truthmaker, and the other of which the evidence for the belief is evidence of. Thus all of these classical Gettier cases involve a doubling of the states of affairs that are involved. Thus I proposed an analysis of knowledge that goes as follows:

The Knowledge-conditions. X knows A iff  X believes A; X is justified in believing A; A is true; and the evidence that X has which constitutes the justification is evidence of the very state of affairs that makes A true.

In addition to what I would call the classic Gettier problems — an example of which is above — there is another set called the Ginet-Goldman cases, also called the fake barn cases. An example is as follows. Along a particular stretch of road there are many fake barn facades so skilfully done as to be indistinguishable, to the casual motorist, from the real thing. Driving along the road you form a sequence of beliefs that this is a barn, that is a barn, this next one is a barn and so on. However, one of these barn facades, call it R, is the facade of a real barn. Passing R you form the belief that is a barn. This is a true belief. Moreover you have good evidence that it is true. So it would seem that you know it is a barn. Some distinguished philosophers however have wanted to claim that it is not knowledge — simply because you also formed similar beliefs when driving past all of the fake barn facades — and in those cases your belief was false. One might say that your belief about R was true, but so embedded in a series of false beliefs that you cannot take credit for it. So it is not knowledge.

This also depends on our sharing the intuition — that the motorist here has a justified true belief but does not have knowledge. But how plausible is that intuition in this case? Certainly it must be agreed that the motorist has many beliefs that are false and that therefore do not count as knowledge. But in the case where the motorists belief is true, where the evidence is of a genuine barn nothing has gone wrong. Thus my intuition in these cases is that the motorist does have knowledge after all. And note that in this case there is no doubling of the states of affairs that are responsible for the attribution of knowledge to the motorist. So my truthmaker account deals with this view automatically.

I put this solution forward in 2003 at a conference on epistemology — in fact at the last session of the conference, when many had gone home. However the attendance at the paper was still respectable, and the conference had attracted most of the people in Australia who were centrally concerned with the theory of knowledge: Tim Oakley, John Bigelow, Anne Newstead, Michaelis Michael, etc. Along with these there were many overseas visitors. The paper was finalised by the time of the conference because I had been testing out this solution and refining it for two years, 2000 and 2001, in my epistemology classes — lectures which were then written up as the first four chapters of my book A Primer on Knowledge. It’s also worth noting that I’d been working on truthmaker theory itself, on and off, since 1995 when Kevin Mulligan was visiting the Sydney department — one result of which came out in Logique et Analyse in 2003. (It was this visit by Mulligan that propelled David Armstrong’s interest in truthmaker theory.) The point of saying this is to note that this solution did not just come out of nowhere — there was a prepared ground.

The paper, ‘Truthmaking and the Gettier Problem’ came out in 2006 in a volume of papers from the conference: Aspects of Knowing: Epistemology Essays, published by Elsevier, edited by Stephen Hetherington, who had also been the conference organiser.

In the original paper I took the view that Russell’s stopped clock example — which, historically was the first classic Gettier counterexample in the literature, published in 1948 in Human Knowledge: Its Scope and Limits — was dealt with by my account: it involves a doubling of the relevant states of affairs: the evidence is of a clock which stopped 12 (or some multiple of 12) hours earlier, whereas the truthmaker is the time that actually obtains. However an email from Gonzalo Rodriguez-Pereyra raised a doubt about this. In reply I repeated my belief that it required no more than had already been given. Over time however it seemed to me that the case was interesting enough, and complex enough, that it required further comment — and after all, what matters in philosophy is whether one has said enough to convince everyone, and not simply whether one has convinced oneself: I felt that I had done Rodriguez-Pereyra a disservice by not attempting to make the case in more detail. And so I wrote a short paper for Analysis in which I tried to lay this problem to rest — once and for all (so I hoped). (This was ‘Gettier and the Stopped Clock’, Analysis, 72, 2, 2012. See here)

The solution was this: Firstly the actual time depends on governmental stipulations and decisions, so the truthmaker in this case is partially constituted by such decisions. This does not make such facts any the less objective. Once the stipulations are in place it is an objective fact that the time is now 11.10 am. Clocks represent the correct time to us by being synchronised causally with an authoritative original clock, and with being constructed so as to “keep time”. But when you look at a clock (in keeping with Russell I am assuming that clocks are the usual 12 hour analog clocks) you do not know the time just from that looking, you use a wealth of contextual information — the position of the sun in the sky, the sound of traffic, the sounds of birds and people — to check whether the  clock is accurate or no longer running. Merely glancing at clock in the absence of this contextual information is not sufficient for forming a justified belief. Neverthelesss when the clock is running and has been properly set the evidence you acquire from looking at the clock plus this contextual information is evidence of the correct time. When the clock is stopped or slow (or fast) the evidence you acquire is of a different state of affairs — even if it happened to stop 12, or a multiple of 12, hours earlier. There is a doubling of the relevant states of affairs — so my account deals with it by adding the extra condition, as above. All is good.

Thus let us suppose that a clock stopped at 2.00 am but then suddenly started working properly at 2.00 pm of the next day — and let us suppose that no one looked at the clock in the 12 hours during which it was stopped. Then, I say, it is still not giving one a knowledge of the correct time — instead it is indicating a time that is 12 hours earlier. No one may be aware of this: but still it is so, ex hypothesi.

It should be noted that Russell himself did not claim that in looking at the stopped clock you acquired a justified true belief (that yet wasn’t knowledge). His example was meant to show merely that this was an example of a true belief that wasn’t knowledge. So my view about the stopped clock case is consistent with Russell’s own view, whereas those who believe that merely looking at a clock is enough to give you a justified belief as to the time go considerably beyond what Russell was prepared to assert. The onus is on them to make their case.

As a personal aside I might note that my grandmother worked for Russell at the end of the 1940’s and through the 1950’s — and thus through the time when he was writing Human Knowledge: Its Scope and Limits. (In fact she claimed to have saved Russell’s then wife, Patricia Spence, from committing suicide by jumping off the balcony: they were going through a particularly unpleasant divorce at the time. My aunt was Conrad Russell’s best friend in childhood — indeed one has to say his only friend.)

It should be noted that Russell not only gave the stopped clock example he also gave an example that is logically equivalent to Gettier’s first counterexample (the one not given above). He is given scant credit for his contributions to the theory of knowledge, particularly those found in his later works. The recent treatment of Russell amounts to a scandalous neglect, in my opinion.

I’ve been asked a few times how David Armstrong reacted to my solution to the Gettier problem. After all, it could be said to vindicate his idea — expressed in many later books — that truthmakers were the key to understanding Truth and many metaphysical problems. The answer is that he didn’t know of it — as far as I know. During 2003 and the time around it Armstrong was teaching in America and even when he returned I didn’t bump into him until much later. By that time the Gettier paper was old news for me and I was thinking about other things. So I don’t ever remember mentioning it to him. Which is a pity: I think he would have been chuffed.

Since the Analysis paper I’ve written a number of other papers on the topic, which are here here and here — which were all attempts at stopping the view from being misunderstood. But the misunderstandings, though unpleasant and unnecessary, were the least unpleasant aspect of a rather unpleasant subsequent development. A number of unscrupulous philosophers — four to my certain knowledge, but there may be more that I haven’t been alerted to — have attempted to take my work, reword it slightly, and claim it as their own. They have wanted to claim that they have solved the Gettier problem — though years later, in fact.  Plagiarism is NOT too strong a word for this. I think this speaks volumes about the current state of academic philosophy — but such plagiarism requires a blog post of its own.