What was Russell’s objection to the coherence theory of truth?

This post is inspired by a fun thread from Michael Bench-Capon:

This is a nice question. In Problems (pages 191-192 in this edition from Archive.org) Russell gives two objections—in typical fashion, he calls them “great difficulties”—against the coherence theory of truth:

The first [objection] is that there is no reason to suppose that only one coherent body of beliefs is possible. […] The other objection to this definition of truth is that it assumes the meaning of “coherence” known, whereas, in fact, “coherence” presupposes the truth of the laws of logic.

I ignore the first argument here because Bench-Capon’s thread does not concern the first argument. Russell next argues for his second claim about coherence theories:

Two propositions are coherent when both may be true, and are incoherent when one at least may be false. Now in order to know whether two propositions can both be true, we must know such truths as the law of contradiction. For example, the two propositions “this tree is a beech” and “this tree is not a beech” are not coherent, because of the law of contradiction. But if the law of contradiction were itself subjected to the test of coherence, we should find that, if we choose to suppose it false, nothing will any longer be incoherent with anything else. Thus the laws of logic supply the skeleton or framework within which the test of coherence applies, and they themselves cannot be established by this test.

The idea here is that one cannot specify what coherence means without invoking the notion of truth. That is, in order to explain what it means for a collection of beliefs to be coherent, we will need to say that they can all be true. But this is just to invoke the notion of truth, or at least the possibility of their all being true, to explain their coherence. And because the coherence theory was supposed to explain what truth is, this is just to explain what truth is by appealing to what truth is.

Russell points out that one will need the logical laws to be true independently of their coherence with any given body of beliefs in order to give any belief-system independent criterion of coherent bodies of beliefs. How does that go? The example of the beech tree indicates what Russell has in mind. Suppose I accept the coherence theory of truth, and that I further think that a collection of beliefs is coherent when all the beliefs in question could be true. But now I say that the collection of beliefs B containing p and not-p is incoherent. And somebody asks, “Why is B incoherent?” Presumably, I will need to explain this by appealing to the impossibility of both p and not-p being true. But now we have completed the circle and arrived back at truth, which was our starting point.

Russell puts the same point in a slightly different way in a 1907 essay, “The Nature of Truth” (pages 156-157 of this edition from Archive.org). He in effect argues that the coherence theorist cannot explain why some collections of beliefs are coherent (somehow correct) and others incoherent (somehow in error) without appealing to the notion of truth:

…we are concerned with the question, not how far a belief in the coherence-theory is the cause of avoidance of error, but how far this theory is able to explain what we mean by error. And the objection to the coherence-theory lies in this, that it presupposes a more usual meaning of truth and falsehood in constructing its coherent whole, and that this more usual meaning, though indispensable to the theory, cannot be explained by means of the theory. The proposition “Bishop Stubbs was hanged for murder” is, we are told, not coherent with the whole of truth, or with experience. But that means, when we examine it, that something is known which is inconsistent with this proposition. Thus what is inconsistent with the proposition must be something true; it may be perfectly possible to construct a coherent whole of false propositions in which “Bishop Stubbs was hanged for murder” would find a place. In a word, the partial truths of which the whole of truth is composed must be such propositions as would commonly be called true, not such as would be commonly called false; there is no explanation, on the coherence-theory, of the distinction commonly expressed by the words true and false, and no evidence that a system of false propositions might not, as in a good novel, be just as coherent as the system which is the whole of truth.

I like this passage because it shows that Russell is concerned with the ability of the coherence theorist to explain truth using coherence. Russell’s challenge is to specify what coherence means independently of the notion of truth. Russell’s view is that this cannot be done.

It might be useful to put it a little differently. Suppose I am a coherence theorist. Let B be a collection of beliefs that is incoherent because, say, it contains p and not-p. In explaining what coherence is, I have two choices: (a) I can say that B is incoherent because p and not-p violate the logical principles included in B, or (b) I can say that B is incoherent because p and not-p violate logical principles not included in B.

Strategy (b) gives up the coherence theorist’s game. Appealing to logical laws that are true independently of their coherence with the body of beliefs B is just to give a test of truth of beliefs contained in B using the impossibility of their both being true. But truth was the very thing that coherence was supposed to explain.

On the other hand, strategy (a) does not explain why some collections of belief are true and others are not. If the only test of coherence is through internally accepted logical principles, then there is no way to say, of a given body of beliefs, that it is coherent or incoherent. One cannot give a belief-independent explanation of coherence at all. There is nothing to which you can appeal in making this judgment besides the internal logical principles of a given collection of beliefs.

Well, why should that be so bad? Assume that all judgments of coherence are, to borrow a Carnapian phrase, internal questions. Russell’s objection still stands: it is just that the coherence theorist is not able to offer an account of what makes a given body of beliefs coherent. Even on strategy (a), to say what coherence is, one would first have to pick a coherent belief system, then explain what coherence means in general. But if all questions about coherence are internal, then this is precisely what we cannot do.

As a side note, the Stanford Encyclopedia of Philosophy article on the Coherence Theory of Truth calls Russell’s objection the specification objection. I do feel that the summary there overlooks that strategy (a) does not answer Russell’s objection of stating what it means in general for a given collection to be false and for another to be true, but the clear distinction of internal and external questions may be, in Russell’s development, a post-Carnap conceptual development that should not be held against the 1907-1912 fellow.

What does the “atomism” in “logical atomism” mean?

On r/askphilosophysomeone asked about the meaning of the “atomism” in Russell’s “logical atomism”:

Today I started reading Bertrand Russell’s “Logical Atomism,” as I’m trying to prepare for reading the Tractatus in the near future. I feel that I have a basic understanding of Russell’s general arguments, and I certainly understand why he chose a title with the word “logic.” As for “atomism”, this is a term I’ve previously only heard in the context of John Dalton and Democritus. Russell’s only explanation of the title that I’ve seen is that “atomism” refers to his belief in multiple perceivable realities as opposed to a single “true” Reality. What is this notion of “atomism” that he’s referring to here? How did it happen to be named in the same way as Dalton and Democritus’ extremely scientific notions? What does atomic theory have to do with perceptions of reality? Are there any other notable members of this atomist school of thought besides Russell? Thanks everyone. -u/officialbobbydunbar

I like this question because it is an easier and different question from “What is logical atomism?” Russell address the “atomism” in his logical atomism explicitly:

The reason that I call my doctrine logical atomism is because the atoms that I wish to arrive at as the sort of last residue in analysis are logical atoms and not physical atoms. Some of them will be what I call “particulars”—such things as little patches of colour or sounds, momentary things—and some of them will be predicates or relations and so on. The point is that the atom I wish to arrive at is the atom of logical analysis, not the atom of physical analysis. –Lecture I, page 3 here

So Russell does not want to merely arrive at physically indivisible entities. They should be logical atoms. What does this mean? Russell tells us in the 1911 essay “Analytic Realism” (the essay wherein he first coins the phrase “logical atomism”):

This philosophy…claims that the existence of the complex depends on the existence of the simple, and not vice-versa, and that the constituent of a complex, taken as a constituent, is absolutely identical with itself as it is when we consider its relations. This philosophy is therefore an atomic philosophy. –Collected Papers Vol. 6, page 133

So what Russell wants from the atoms, whatever they are, is that they are logically separable entities: one logical atom exists and has whatever properties it has independently of the any other atom.

Russell’s view on what the logical atoms in fact are changes over time. In 1924, for example, he says that the world consists of events rather than facts (see “Logical Atomism,” page 148 here) without sense-data as constituents, which is a rejection of most of his 1918 view of particulars. But he never wavers in his view that whatever is a logical atom should be what it is independently of all other atoms, even if it comes into causal relationships with other atoms as a contingent matter of fact.

This is a crucial tenet of logical atomism: in Russell’s view, it underwrites our practice of deploying terms in symbolic logic, even if we realize later that such symbols really designate entities that are logically complex, is underwritten by a claim about the structure of the world: it is structured so that a given entity, if it is a logical atom, is logically independent of all others. This is what underwrites his claim (on pages 143-145 here) that we can know properties of one logical atom without thereby needing to consider the logical properties of other logical atoms. This claim about the structure of the world is part and parcel of his denial of the doctrine of internal relations (which he describes on pages 140-141 here), a doctrine that he attributes to idealists like Bradley.

It is useful to compare Russell’s account of logical atomism with Bradley’s overall view: Bradley held that all entities are necessarily related to one another, so that talk of any one entity was a misleading abstraction from the only genuine entity, the entire cosmos (see Appearance and Reality, Chapters 1-3, e.g., page 23 here). Russell emphatically rejects this view, holding that there are no such necessary relations between things, and that talk of a single entity (logical atoms) is perfectly scientifically and philosophically acceptable. This, and chiefly this, is the underlying purpose of the atomism in his logical atomism.

That time, way back in 1917, when Dorothy Wrinch threw mad shade

Dorothy Wrinch (1894–1976) was a productive and intelligent philosopher, mathematician, and biochemist who contributed to a wide array of fields. She published in economics and probability, x-ray crystallography and protein structures, the cardinal arithmetic of Principia Mathematica‘s second volume, and, of course, philosophy. One of her particular interests was the relation between philosophy and logic, and logic’s role in the scientific method.

Wrinch holds many distinctions. She was the first woman to teach mathematics to men at Cambridge. She was a single mother in academia during a period that was doubtless especially hostile to women in the workplace. In 1930, she also wrote a pseudonymous book, The Retreat from Parenthood, that was no doubt partly informed by this experience, and one that might be viewed as a counterpart to Russell’s 1928 Marriage & Morals. Wrinch wrote a thesis on logic, sadly now lost so far as I know, under Bertrand Russell’s supervision, though this was unofficial because Russell had been sacked due to his pacifist activities during the World War I; officially, G. H. Hardy was Wrinch’s thesis supervisor. She also got into a bitter dispute with Linus Pauling over the structure of proteins, a dispute now known as the “cyclol controversy.” These biographical details are covered in Marjorie Senechal’s superb I Died for Beauty: Dorothy Wrinch and the Cultures of Science.

Here I want to focus on a hilarious episode from 1917, that one time Wrinch threw mad shade at a Professor L. P. Saunders, who was criticizing Russell’s 1914 Lowell Lectures, known to us at Our Knowledge of the External World: As a Field for Scientific Method in Philosophy. Wrinch, then a graduate student studying mathematics at the University College London, was responding to Saunders’ critical piece on Russell’s Our Knowledge.

Saunders argues, somewhat bizarrely, that if facts involving sense-data and the laws of logic are certain, then nothing else is certain:

Mr. Russell, I have to point out, regards the laws of logic and sense-data as the hardest of hard facts; thus, in the end, the other hard facts…are not really “certain”. It is fair, therefore, to say that the only facts that are known, according to Mr. Russell, are sense-data and the laws of logic…

Mr. Russell, like all empiricists, does not take his own position with sufficient seriousness. He tells you, in effect, that sense-data alone are certain facts; and in violent contradiction to this he asks you to accept a great many other statements (viz., the statements constituting his position as such, statements about it, and statements about other philosophies) as true! And yet one would have thought it unnecessary to have to point out that if sense-data are the only really certain (i.e. certain) facts, that then nothing else is certain. Unfortunately, although this is, in one sense, quite clear, it is also, in another sense, not clear, seeing that this statement itself claims to be true. How Mr. Russell came to overlook this is very difficult to understand. (Saunders 1917: 49-50)

Of course, Russell’s view by no means implies that whatever is implied by the laws of logic in conjunction with facts about sense-data are not part of the certain facts. Surely, whatever is logically implied by these is also certain. Wrinch rightly picks up on this point, finding Saunders’ reasoning ridiculous and saying as much in her 1917 reply:

Now it must be evident that in general, it is possible to infer from a given set of premisses which includes a principle of deduction, other propositions not contained in this set.² In the system we are discussing, the Laws of Logic are included in the premisses and yet Prof. Saunders takes exception to the fact that other propositions are asserted, apparently merely because they are different from the premises. […] Since the Laws of Logic are included in this body [of certain propositions], all logical deductions from the premisses can be justifiably asserted in his system. This class of propositions will, I think, cover all Mr. Russell’s statements “constituting his position as such, statements about it, and statements about other philosophies”. (Wrinch 1917: 449)

Ouch. But that was not the shade that inspired this post. It is this little footnote 2, which reads:

Cf. Principia Mathematica in which three volumes of propositions are inferred from a very small number of primitive propositions.

That is a hilarious rejoinder, indeed. “You think that if the laws of logic and facts about sense-data are certain, then nothing else is? Have you even read Principia?”

For Cheap Logic Textbooks

Some time ago, I saw the below tweet about expensive logic texts:

Now I feel rather strongly that assigning expensive textbooks is a horrible practice for students. They already pay quite a lot in tuition and housing, and for meals and medical care. All these things can and should change. But I want to focus on logic texts because  this is something instructors have direct control over. Costly tuition and housing practically require large-scale institutional solution. Textbooks lie in our domain.

I claim that instructors should assign cheap introductory logic textbooks, preferably free or open-source ones. This is because:

  1. Cheap logic textbooks lessen the financial burden on our students.
  2. The introductory logic material is so well-understood that there are plenty of cheap logic textbooks available that cover all the instructor’s desired material.

So the one-two punch is: there are some tremendously good consequences of doing X, and practically doing X is so readily achievable. If you want some specific collection of topics (categorial logic + propositional deduction + propositional tables; propositional deduction & trees + first-order predicate deduction & trees; propositional deduction & trees & tables + first-order deduction; and so on), there is some book out there that covers at least all these topics. A partial list of logic course-ware and texts is given by Richard Zach here. (Check out the Open Logic Project, too, which has a number of builds you can pick to suit your needs!) And you may skip the stuff you do not want. Plus, there are plenty of cheap texts for informal fallacies in case you want something on that, too. (Check out An Illustrated Book of Bad Arguments!)

Now what are the reasons for doing otherwise, for instead assigning an expensive logic text in its 10th edition? One might offer the following:

  1. Most students can afford the expense or have external funds for it.
  2. The assignment and grading software is practically convenient, even practically necessary due to extraordinary enrollments.

So the one-two punch is that costs are neither a big problem, and they are practically convenient, sometimes even practically needed. My suspicion is that many instructors assign an expensive book out of habit or because the software is more convenient. Perhaps there is also a feeling that our job as instructors is not to solve the problem of making higher education affordable. Mostly, I expect the basis for this practice is habit.

Now (1) is false. Even if it were not false, it would not justify choosing an expensive logic textbook over a cheap one. It only suggests that both are permissible. And if both using a cheap textbook and using a costly one are permissible, then given the availability of cheap and adequate alternatives, why use a costly one?

Now (2) is more interesting. But it is still insufficient. Assessment is part of our job, and it can be done without making students pay a cost for it. (Some of the links above include logic quiz generators. Some open-source software even allows for automatic grading of proofs or translations.) And if you really are teaching so many students that assessment takes too much time—there are teachers who have large lecture-style logic courses that are practically infeasible to grade by hand themselves—then it would actually be cheaper to hire a grader using a “student fee” rather than make all your (say) 100+ students pay $100+ for logic software. So if practical convenience or need is the reason, then there is a practically cheaper alternative that does more good. And finally—this is the real kicker—there already is free online software that does grading for you. (Just check out Graham Leach-Krouse’s Carnap project.)

So logic books in my courses are, and will continue to be, freely available, either through my institution’s library or elsewhere. I might add that pretty much all of the above applies equally to introduction to philosophy anthologies.

“Why is it, according to Russell, not the case that every proposition is of the subject-predicate form?”

This very nice question was recently asked on the Reddit forum r/askphilosophy. Here is the (reposted) answer.


Russell’s reasoning is given in the passages immediately preceding that quote. Russell wants to argue that, in the case of symmetrical relations, we can understand them as one-place predicates if we wish. This is because we can, using one-place relations, preserve the usual inferences involving symmetrical relations. For example:

  1. A and B both have property G. [premise]

  2. B and C both have property G. [premise]

  3. Thus, A and C both have property G. [from 1-2]

On a theory involving only one-place relations, this inference can be preserved. This is because A and B, and C, too, all have the same property. So there is no problem of relating A’s color to B’s color, and then to C’s color. They all have the same color. So one could at least see how the one-place relations story could go for symmetric relations.

The case of asymmetric relations is entirely different. Russell’s view is that, if we try to eliminate all relations in favor of one-place relations, then we cannot preserve ordinary inferences involving asymmetric relations. Take Russell’s example:

  1. A is greater than B. [premise]

  2. B is greater than C. [premise]

  3. So, A is greater than C. [from 1-2]

Russell says about this sort of example:

Take for example “A is greater than B”. It is obvious that “A is greater than B” does not consist in A and B having a common predicate, for if it did it would require that B should also be greater than A. It is also obvious that it does not consist merely in their having different predicates, because if A has a different predicate from B, B has a different predicate from A, so that in either case, whether of sameness or difference of predicate, you get a symmetrical relation.

Russell is outlining two strategies for accounting for this clearly valid inference, assuming we only have one-place relations. We can say that A being greater than B consists in them having (a) the same property, or (b) different one-place properties.

Strategy (a) fails. A having the property being greater than B does not imply that B has this same property (indeed, B cannot have this property). So we have to introduce distinct properties for A and B. That is, we must pursue strategy (b). But then we have the problem of relating these one-place relations together.

This problem cannot be overcome. There is no way to get from the first two claims to the third (absent, perhaps, some modal logical principles about tallness). Just consider how that argument would look if we only have one-place relations:

  1. A has the property being greater than B.

  2. B has the property being greater than C.

  3. So, A has the property being greater than C.

This argument gets the form:

  1. F(A)

  2. G(B)

  3. H(A)?

And it won’t help to add one-place relations either, so that A being taller than B consists in their having different predicates. Just consider:

  1. A has the property being greater than B and B has the property being smaller than A.

  2. B has the property being greater than C and C has the property being smaller than B.

  3. So, A has the property being greater than C.

But (3) doesn’t follow still. The argument has the form:

  1. F(A) and P(B)

  2. G(B) and Q(C)

  3. H(A)?

So, in short, if we eliminate all relations in favor of one-place relations, then there is no practicable way to make inferences involving the asymmetric or transitive properties of relations.