In the preface to Our Knowledge of the External World, Russell famously maintains:

the logical-analytic method in philosophy … has gradually, in the course of actual research, increasingly forced itself upon me as something perfectly definite, capable of embodiment in maxims, and adequate, in all branches of philosophy, to yield whatever objective scientific knowledge it is possible to obtain. Most of the methods hitherto practised have professed to lead to more ambitious results than any that logical analysis can claim to reach, but unfortunately these results have always been such as many competent philosophers considered inadmissible. Regarded merely as hypotheses and as aids to imagination, the great systems of the past serve a very useful purpose, and are abundantly worthy of study. But something different is required if philosophy is to become a science. (Russell 1914, v)

Dorothy Wrinch encountered the method already as a student at Girton College, in 1914, when she “had the pleasure” of attending Russell’s “lectures at Trinity on ‘our Knowledge of the External world’” and found it so interesting that she took “the great liberty” of writing to him to ask where she could read further about his points “about the psychological interpretation of Newton’s Laws of Motion” (Wrinch 1914).¹ Within that method in philosophy, the “function of logic” is famously “all-important” (Russell 1914, 8). In this paper, we will see, through the lenses of Wrinch’s reflections on logic at the intersection among logic, epistemology and metaphysics, how Wrinch explicitly believed, in her published papers in philosophy, in that “something different” that could lead to a scientific philosophy (§1). We will also see how she still believed, later on in her career, when she turned to philosophy of science and then to biology, in the “all-important” role played by logic in our knowledge of the external world (§2). This does not mean that Wrinch blindly believed in everything her teacher maintained, though. Quite the contrary, we will see that while she full-heartedly endorsed the Russellian project, in her letters to Russell, she in fact also criticised him on some important issues (§3). Wrinch’s reflections, as the reflections of a young British scholar within the Russellian circle, will testify to how, in 1918, one could endorse the Russellian overarching project as presented in Our Knowledge of the External World while holding that we can have kinds of pieces of knowledge of the external world that go beyond those Russell was happy to admit.

1 Logic and Philosophy, Published Papers

In 1917, Wrinch started to publish a series of articles aimed at defending or extending Russell’s views in a variety of topics in philosophy and mathematics. The first, published in Mind, is a defence exactly of Russell’s claims in Our Knowledge of the External World from the objections raised by L. P. Saunders who, in another paper published in Mind, attempted nothing less than

to show that Mr. Russell’s most recent account of our knowledge of the external world is, on purely general grounds, of little or no philosophical value. … his purely ‘logical’ method of solving difficulties is epistemologically unsound. (Saunders 1917, 29)

In her reply, Wrinch defends Russell by using Russell’s own resources. Let’s see one of Saunders’s points. He objects:

Consider his meagre list of ‘certain’ or ‘hard’ data. These are (1) the ‘Laws of Logic,’ (2) sense-data, and whatever is of the same type or order … (1) Mr. Russell does not give a list of the ‘Laws’ he has in mind. (Saunders 1917, 47)

By the time of Saunders’s paper, Wrinch has carefully worked on Russell’s publications, and, together with Lenzen, Nicod and Armstrong, has studied the Principia with Russell himself in London (Lenzen 1971), and spent a period in Oxford, during which, as she stated, “Mr. Russell gave [them] some new stuff of his, developing some of his points in the Lowell lectures” (Wrinch 1916b). She is then quick to offer a reply to Saunders:

Mr. Russell did not, of course, give a list of the Laws of Logic in his Lowell Lectures but a good idea of the nature of one important sub-class of them is given in Principia Mathematica (Wrinch 1917, 448, n. 2)

Saunders continues in his point:

But whatever they are they must either be believed or known. If believed they are not certain; and on any other important sense of certainty, the ‘Laws of Logic’ are no more certain than a great many other beliefs. (Saunders 1917, 47)

Wrinch replies, by quoting Russell directly:

It appears to be a fact that, on reflexion, it is much more difficult to doubt some kinds of propositions than others, and to doubt the existence of some kinds of things than others. … the very hardest propositions to doubt seem to be the Laws of Logic … Thus there is an obvious sense in which the collection of data that Mr. Russell has specified can be called ‘hard’ data— ‘data which resist the solvent influence of critical reflexion,’ and they can quite properly, I think, be called comparatively certain. It seems too, as far as one person can judge, that our own sense-data and the Laws of Logic are the hardest of these hard data and that it is a fact that ‘the more we reflect upon these, the more we realise exactly what they are, and exactly what a doubt concerning them really means, the more luminously certain do they become’. (Wrinch 1917, 448–49; quoting Russell 1914, 71)

As she continues, “Mr. Russell has explicitly said that he is going to assume the Laws of Logic to be certain, having given his reasons for considering such an assumption justified” (Wrinch 1917, 449).

Wrinch believed in those reasons. For her, mathematics and logic are not different disciplines:

It is the degree of complication in certain logical deductions which makes it convenient to shut some of them off from the rest of logic and call them mathematics. There is no fundamental difference whatsoever. (Wrinch 1920–1921, 195).

For complicated logical and mathematical structures there cannot be for Wrinch “crude … intuitions to help us,” exactly because “the intuition is powerless before” structures that are too complicated (Wrinch 1917, 451). Still, as she writes in another piece, published in 1919,

Some of our beliefs in logical proposition [sic!] are not based on anything. Our knowledge of the law of contradiction is, probably, direct. Anyhow our knowledge of some of the propositions of logic is direct. (Wrinch 1919a, 473).

Similarly, in Our Knowledge of the External World, Russell does state that we have “some kind of knowledge of logical forms” (Russell 1914, 44) and that a proposition in pure logic such as “If anything has a certain property, and whatever has this property has a certain other property, then the thing in question has the other property” is “quite self-evident” (Russell 1914, 57). Wrinch surely inherited from Russell the idea that in the case of the Laws of logic “[r]eal doubt” would “be pathological” (Russell 1914, 71). For her, mathematics is “but the child of common sense” (Wrinch 1922b, 382).

Wrinch moreover inherited from Russell his enthusiasm for what logic could do in philosophy. For Russell, having realized that “[u]niversal scepticism, though logically irrefutable, is practically barren” (Russell 1914, 67), we should exclusively take it to “give a certain flavour of hesitancy to our beliefs … [i]f we are to continue philosophising” (Russell 1914, 67–71). For Wrinch, similarly, we should admit that “[w]e have no reply to the professional sceptic” (Morris and Wrinch 1924, 62) but this does not mean that we cannot endeavour to obtain any knowledge, it exclusively means that we should bear in mind that we “stand for ever on sand” (Morris and Wrinch 1924, 63). Although from a “strange, embarrassing position” (Morris and Wrinch 1924, 63), we can still philosophize and to do that we should use, for Wrinch, “Mr. Russell’s brilliant method of logical constructions” (Wrinch 1919b, 145). For the 23 year old Wrinch, epistemology seems “to be the most thrilling part of philosophy.—(Logic being considered as a separate science)” (Wrinch 1918a) and the method of logical constructions “indicates the direction in which the solutions of many epistemological questions of fundamental importance may be considerably advanced” (Wrinch 1919b, 145).² Among those she counts questions about existence:

It would perhaps be advisable to state shortly what appears to be the essence of this method. The problem for the solution of which this method is to be used is as follows. Certain things are given in experience—sense-particulars of various kinds and facts. We then wish to find other terms, such that in analyzing any proposition in which they occur, they themselves do not occur, but only the things which are given in experience. At the same time, these terms are to have certain definite properties. Then although a term a (say) appears in a proposition ϕa yet it will be possible to analyze ϕa into a proposition not containing a if a stand [sic!] for a logical construction. (Wrinch 1919b, 142)

For Wrinch it is “of extreme importance that we should be able to do this” (Wrinch 1919b, 142). For otherwise we would have to maintain that what is asserted by the true sentence “the round square does not exist”, is “that there is an entity which is round and square, and it has the property of not existing” (Wrinch 1919b, 142)³. While “round square” “appears to be a constituent of the original proposition and in the proposition some property seems to be predicated of this fictitious object” (Wrinch 1919b, 142), thanks to Russell’s analysis of the sentence, we can see that “the object disappears” (Wrinch 1919b, 143). The round square is an impossible object. But the method can of course also be applied to possible ones, such as “points” (Wrinch 1919b, 143). Concerning them, together with Russell (1914, 146–47), Wrinch is careful to maintain that the method of logical constructions allows us not to assert the existence of those objects, but does not lead to us asserting that they do not exist:

Occam’s razor states that ‘entities are not to be multiplied without necessity.’ … The essence of the principle lies in postulating primary existence of the fewest possible number of kinds of things necessary to explain the facts: then the method of logical constructions will allow us apparently to use entities in propositions without in advance giving them primary existence. … Of course it would be just as great an error of logical taste to assert dogmatically that points etc., have not got primary existences, as to assert dogmatically that they have: it would in fact be a logical mistake fundamentally of the same kind. Occam’s razor does not advocate that. [O]ne does not assert the existence of points, as there does not seem to be any necessity for such an assertion for a satisfactory explanation of the given proposition: next, of course, one does not assert the non-existence of these things, but further one would assert that the given material is insufficient for such an assertion (Wrinch 1919b, 144–45)

Occam’s razor and the method of logical constructions, both for Wrinch and for Russell, suggest caution with assertion of existence for all those entities that are not found necessary to explain the facts. Wrinch continues:

those who make such an assertion either have some private channel of knowledge, to which we have not got access, or are making the assertion on insufficient grounds. (Wrinch 1919b, 145),

And in the same vein, in a summary, written in 1918, of an “interesting article” (Wrinch 1918d, 623) by Raphael Demos on negative propositions (Demos 1917), she states:

He assumes that there are no negative facts and deduces that negative propositions must have reference to the world of positive facts. … It seems, however, to the reviewer a little unwise to base a theory on such a disputable point as the non-existence of negative facts. (Wrinch 1918d, 623)

While the nature of propositions and the existence or non-existence of negative facts are a purely philosophical matter, points are at the core of physics and in various articles, which she wrote in the 1920s on scientific method,⁴ she is also clear that the method of logical constructions and the consequent suspension of judgement on the existence of the relevant entities applies to all scientific concepts. Concerning science, for example, Wrinch maintains that the method allows us to realize “that in view of the fact that a concept is an empty thing—simply a bundle of formal properties— … our electrons, however much we may like to talk of them, are merely phantoms—or using Eddinigton’s phrase … ‘just a dummy’” (Wrinch 1929a, 122). Similarly, the “Space” of Einstein, Weyl, and Eddington is “a description” (Wrinch 1922a, 200), where “description” is “used in the sense explained by Whitehead and Russell in Principia Mathematica, vol. i.” (Wrinch 1922a, 200, n. 1). So, again in agreement with Russell, logic is crucial not just to philosophy, but also to science. This is not just because it is thanks to a purely logical process that we derive all consequences from the postulates (Wrinch and Jeffreys 1921, 369; Cf. Russell 1914, 211), or because logic helps us compare different sets of postulates and understand, for example, whether some postulates are truly primitive, or should be taken as themselves derived from others (Wrinch 1926–1927, 48–60; Cf. Russell 1914, 42). Logic is crucial to science also because, as the study of logical forms, it provides the realm of possibilities. It is true that for Wrinch “many of the most notable advances in pure abstract thought come about through the stimulus of the amazing outside world” (Wrinch 1929a, 118–19). But it is also true that logic then enlarges our imagination by providing us with possibilities. In order for logic to provide them all, Wrinch, again in perfect agreement with Russell (1914, 45–59), maintains that logic should be the new logic, the one of “the illuminating work of Whitehead and Russell in the domain of pure logic” (Wrinch 1920–1921, 194), the one stemming from “the important work of the Mathematical Logicians who have quietly and definitely constructed a new branch of knowledge out of a nucleus of isolated ideas which have hovered about since the times of the Greeks” (Wrinch 1925, 507). For, as she remarks, only the new logic is suitable, given the complexity of the sciences:

Any theory … which offers any catalogue of logical relations so inadequate and incomplete as those of the older logicians is at once to be rejected. … We therefore want a wider view and a freer imagination. … Examples of simple deductions occur in systematic botany when the elementary theory of classification by means of mutually exclusive and exhaustive classes is used. In this only simple logical propositions, such as the law of contradiction and the syllogism in Barbara, are in question. But take Bohr’s atomic theory and very complicated deductions are made which, owing to their complicated nature, can only be made with the help of elaborate mathematical technique. (Wrinch 1920–1921, 189–95)

For Russell logic provides the realm of possibilities primarily in the sense that it allows us to add to our stock “hypotheses … which only logic would have suggested” (Russell 1914, 59) and then give thought wings. Wrinch agrees on that, but also states, more explicitly than Russell, that the modern logic also restricts the domain of options by ruling out what is impossible:

In the external world whatever happens must be possible logically. If logic says that two properties, ϕ and ψ, cannot co-exist, we shall not find in the world a case of ϕ and ψ being properties of the same thing. The characteristics of being logically possible is a necessary condition of actual happenings. (Wrinch 1920–1921, 203–04)

So, “all physical possibilities must necessarily be mathematical possibilities” (Nicholson, Wrinch, Lindemann, and Carr 1924, 28). Thus for Wrinch the modern logic is crucial to science because, in both restricting the realm only to what is possible and in providing all that is possible, it is “the repository of the possibilities before us in our attempt to understand the world” (Wrinch 1920–1921, 195).

Wrinch believed that the modern logic was more suitable than the old one also in philosophy, and exploited it, so much so that in her philosophical papers she adopted a style that was sometimes more formal than Russell’s own style, and used Russell’s notation so pervasively that one understands the reaction of the old readers Passmore mentions:

In the epistemological articles of Dorothy Wrinch, one begins to find Russellian logical symbolism employed in the discussion of such traditional issues as the complexity of judgments (1919) and their role in remembering (1920). Very gradually, Mind set off on that path which converted it, so some of its older readers thought, into a journal of ‘ϕ-losophy’ and ‘ψ-chology’. (Passmore 1976, 34–35)

So let’s take stock. We just saw that for Wrinch logic provides us with pieces of knowledge that are as firm as they can be. The new logic, applied to the study of facts carried out both in philosophy and in the sciences, also provides us with the ability to get rid of impossible objects and to suspend judgement on there being possible objects that are not necessary to explain the facts. Furthermore, logic provides us with the realm of all possibilities. This is all very Russellian. Now, by reading the passages by Wrinch we just saw and Passmore’s point, it is easy to jump to the conclusion that Wrinch was a mere follower of Russell, who was simply repeating his points by using, even more than he himself did, his notation. This would be an incorrect conclusion, though. She went beyond Russell’s views. First, she extended them. For example, in one of the two papers Passmore mentions, she deals with Russell’s multiple relation theory of judgement, and in it the aim is to go beyond Russell, to, as she herself maintains, “offer suggestions as to the ways in which his idea … can be extended” (Wrinch 1919d, 319). Second, she discussed themes that Russell did not deal with. For example, she discusses how in science we rely on a number of irrelevance postulates, from taking “the time of day at which the experiments are performed to be irrelevant” (Wrinch 1923, 37), to more abstract assumptions about various properties being irrelevant to each other. Still, even though Wrinch was definitely not a mere follower of Russell, she surely was a “follower of Russell” (Grattan-Guinness 2000, 435). The passages we saw, from papers Wrinch published from 1917 to 1929, do indeed support the claim, as Elkind phrased it, that Wrinch “thoroughly embraced [Russell’s] philosophical program and practiced it throughout her intellectual contributions in mathematics, biochemistry, and philosophy. In short, her practice matched Russell’s creed” (Elkind 2019, 36).

2 Science

As Elkind’s quote mentions, Wrinch was not only a philosopher. However unusual it might be for somebody who wrote about primary existence, one could indeed see her “working on a problem to do with gravity, sunlight and the constant distance from the ground of the first branch on chestnut trees” (Blackwell 1969). After having spent a period she herself described as “the application of Scientific Method to the material in hand, whatever it happened to be” (Wrinch 1930a, 6), and in particular to materials such as development, business psychology, and sociology Wrinch (1930b), she then famously turned to focus her attention to biology and in particular to the structure of proteins and, relatedly, to the analysis of crystals. As Elkind maintains, Wrinch practiced Russell’s philosophical programme also through her intellectual contributions in biochemistry and, as Senechal also stresses, her work in science “rested on, and drew on, her ideas of scientific method and the mathematics she’d been doing” (Senechal 2013, 120). As a scientist, Wrinch explicitly stated that logic and mathematics had all-important roles. Her moving to focusing on science happened guided by a clear plan—“Take two abs. different standpoints … I will be a LOGICIAN—PHYSICIST” (Wrinch, n.d.-a)—which took her, for example, to become a founding member of the Theoretical Biology Club, whose main founder, Joseph Woodger, would be remembered by Popper in this way:

I met Woodger first in Paris, in August or September 1935, at the ‘First International Congress for the Unity of Science’, organised by Otto Neurath. The great man of this congress was, no doubt, Bertrand Russell. And it was Woodger’s contribution to the congress, more than anybody else’s, which showed Russell’s merits as a philosopher of science: Woodger described his search for the tools needed to analyse the difficulties he had met in his attempts to construct a theoretical biology, and he described the help he had obtained from Whitehead’s and Russell’s Principia Mathematica. (Popper 1981, 329)

The new logic and mathematics were certainly for Wrinch a useful tool that could help, for example, in individuating curves that, she hoped, were suitable for the wing profiles for airplanes (Wrinch 1924), or that could help D’Arcy Thompson in his research on “cartilaginous vertebrae of a shank” (Thompson 1924a) and “protozoan shells” (Thompson 1924b). But this is not the fundamental help one can obtain from Principia Mathematica. The “aesthetic glories” of a logical “treatment of the world” (Wrinch 1920–1921, 193) such as the one promoted by D’Arcy Thompson⁵, do not come for Wrinch from using mathematics as a tool and the application of mathematics to the sciences is in fact to be made carefully. As she stressed, we should be wary of “over-mathematizations” (Wrinch 1943, 232), we should be careful not to “create a mathematical fog behind which the original facts become intangible” (Wrinch 1943, 232). The explicit all-important roles logic played for Wrinch-the-scientist are those that Wrinch-the-philosopher-of-science-influenced-by-Russell highlighted. As we saw, for Wrinch, as the Russellian philosopher of science, logic had two fundamental roles: first, logic can help with the organisation and comparison of postulates, and with deducing all consequences from them; second, logic provides the realm of possibilities. Wrinch-the-scientist is explicit on both roles. For example, to explain her own vision to the students of Smith College, she used a quote from Newton’s Introduction to his Principia: “Gloriatur Geometria quod tam paucis principiis, tam multa praestet (It is the glory of geometry that from so few principles so much can be derived)” (Senechal 2013, 206). Concerning the second role, Wrinch went back to it as a scientist several times. In a typescript which is not dated, but which she must have written at the turn of the 1930s and 1940s, as she was then at John Hopkins, Wrinch writes:

It seems plain that biology is now at the stage at which it is useful to introduce the type of theoretical scientist who has already contributed greatly to the progress of physics. … During the last years, I have found myself impelled to try to work on biological problems in this way, specifically on the problem of protein structure. The basic problem of protein structure … is the construction of all geometrically permissible polycondensation products or amino acid and amino acid molecules. It has proved possible to find some structures satisfying the chemical and stereochemical specifications. … We do not yet know whether these structures are correct. Whether they are or not, they indicate methods (Wrinch, n.d.-b, 1–2)

Relatedly, in 1941, she states that “No molecules with thousands of atoms and well defined structures can be dealt with except by taking in turn all possible structures, theoretically constructed” (Wrinch 1941). Perhaps not the one of the Principia, but mathematics provides also Wrinch-the-scientist with the needed wider view and freer imagination. As late as in 1960s, when her main topic of study looks so far away from those of the early papers on Russell’s Lowell Lectures or on existence or the nature of judgment, still Russell’s imprint can be clearly detected. In 1963, for example, in a private note concerning the structure of a peptide, she writes:

I should hold fast all through to saying that what I am after is mapping out the possibilities … using a geometrical viewpoint to give in orderly fashion a description of whole arrays of possibilities. (Wrinch 1963)

In another typescript, we moreover can see how for Wrinch as a biologist mathematics can provide science not just with that “novelty in hypotheses” Russell was speaking about (Russell 1914, 242), but also with those constraints she was highlighting as a philosopher of science,

In calling geometry to the service of this branch of megachemistry, a new point of view is necessarily introduced which has as its objective the formulation of all possible types of structure. The gradual development of this point of view has brought many results, surprising or perhaps only novel, notably the realization that geometrical considerations alone can go far toward determining characteristics which chemical structure must have. (Wrinch, n.d.-c, 2)

While she did not follow completely her plan to become a “logician—physicist”, as she dedicated her efforts to biology rather than to physics, throughout her whole career as a scientist, she surely followed the part of the plan that concerned being a logician. She furthermore kept being a Russellian also in another sense. In his final claim in the closing of Our Knowledge of the External World, where he is discussing “the prospect of progress in philosophy” and is admitting that “it would be rash to speak with confidence,” Russell writes:

The one and only condition, I believe, which is necessary in order to secure for philosophy in the near future an achievement surpassing all that has hitherto been accomplished by philosophers, is the creation of a school of men with scientific training and philosophical interests, unhampered by the traditions of the past, and not misled by the literary methods of those who copy the ancients in all except their merits. (Russell 1914, 242)

Although Wrinch did not publish papers in philosophy after 1930, even in the later stages of her career, she was a perfect candidate as a member for that school, as she surely had philosophical interests also then. This is not only because she was working, in science, on structural problems. For example, in the Dorothy Wrinch Papers⁶ we can find a typed copy of P. B. Medawar’s “A Note on ‘The Scientific Method’” as it appeared in The Uniqueness of the Individual (Medawar 1957), which was published as late as in 1957, and we can even find a note, dated 1954, where she proves to be still thinking about the relation between language and what is really there, or, as she would have said 35 years before, what has primary existence: “Language not properly used gets in your eyes. You see things the way you say them. Words prevent one from seeing what is really there” (Wrinch 1954).

3 Logic and Philosophy, Letters

As we just saw, both as a philosopher and as a scientist, Wrinch embraced Russell’s philosophical program. But this should not be taken to imply that she blindly believed everything Russell maintained. Now, it is extremely rare for Wrinch, in her published papers, to disagree, at least explicitly, with Russell.⁷ His claims are usually presented as something to be relied on, not to disagree with. We already read about “the illuminating work of Whitehead and Russell in the domain of pure logic” and Russell’s “brilliant method of logical constructions”, and we can also for example read about crucial questions having been “clearly put by Russell” (Wrinch 1929b, 242). But the situation in her letters to him is different. While not even in the letters she ever called into question Russell’s overarching philosophical project as presented in Our Knowledge of the External World, she still criticised him on various important aspects of the development of logic he suggested and on its all-important role.⁸ To see an example, we can have a look at a letter Wrinch wrote to Russell in August 1918, while he was in prison.⁹ The 3-page letter is packed with remarks, issues and questions on a variety of topics, but some concern the topic we already encountered of the relationship between logic, possibilities and impossibilities, and we can focus on those. The letter also employs an unfamiliar notation, but the points that matter to us, given our aims, still emerge quite clearly.

One of them is the following:

the kind of implication between propos p and q which is interesting is not ‘it is not the case that p and not-q’ but ‘it can’t be the case that p and not-q’ … p⊃_Iq (internal implication as opposed to E, external imp.) … I do NOT see that a kind of implication which subsists between any false prop and any prop is the kind wanted in logic … in Arithmetic (e.g.) … it is deduction from certain props that seems interesting and not deduction from such a prop as “Lloyd George is a wonderful man” (Wrinch 1918c)

Wrinch of course knew that the example she uses—“Lloyd George is a wonderful man”—was false for Russell, and also knew Russell knew that she knew that. Russell had strong words for Lloyd George, also publicly; for example, already in 1916, in an article in The Tribunal, Russell criticised Lloyd George’s words by claiming that “There is a manly note of primitive ferocity about these words” (Russell 1916). Apart from Lloyd George’s role in the fines and bans Russell was subjected to, and then his imprisonment, this alone is not compatible with taking Lloyd George to be a wonderful man. The point Wrinch is making here is then the following. If we account for p ⊃ q as in accordance with the Principia—“‘implies’ as used here expresses nothing else than the connection between p and q also expressed by the disjunction ‘not-p or q.’ The symbol employed for ‘p implies q,’ i.e. for ‘ ∼ p ∨ q,’ is ‘p ⊃ q.’” (Whitehead and Russell 1910, 7)—it is true when p is false, and then we obtain that, for example, “Lloyd George is a wonderful man” implies “2 + 2 = 4”. But this is not the kind of deduction we are interested in, in logic. Even though she does not mention anybody in the letter, Wrinch is of course not the only one, or the first, to have raised this issue, which is one of those which usually go under the label of the paradoxes of material implication. Famously, the point was raised by C. I. Lewis in two papers published in Mind (Lewis 1912, 1914) and Russell did reply to this point in 1919, presented as a point by Lewis and “some authors” (Russell 1919, 153), in his Introduction to Mathematical Philosophy:

In order that it may be valid to infer q from p, it is only necessary that p should be true and that the proposition ‘not-p or q’ should be true. Whenever this is the case, it is clear that q must be true. But inference will only in fact take place when the proposition ‘not-p or q’ is known otherwise than through knowledge of not-p or knowledge of q. Whenever p is false, ‘not-p or q’ is true, but is useless for inference, which requires that p should be true. Whenever q is already known to be true, ‘not-p or q’ is of course also known to be true, but is again useless for inference, since q is already known, and therefore does not need to be inferred. In fact, inference only arises when ‘not-p or q’ can be known without our knowing already which of the two alternatives it is that makes the disjunction true. Now, the circumstances under which this occurs are those in which certain relations of form exist between p and q. … But this formal relation is only required in order that we may be able to know that either the premiss is false or the conclusion is true. It is the truth of ‘not-p or q’ that is required for the validity of the inference; what is required further is only required for the practical feasibility of the inference. (Russell 1919, 153–54)

So, let’s take again Wrinch’s example. As Barker maintains, “[r]ather than reading the horseshoe symbol as ‘implies,’ Russell would have done better to read it as ‘only if’” (Barker 2006, 15). While “Lloyd George is a wonderful man implies 2 + 2 = 4” is true, that is, while it is true that Lloyd George is a wonderful man only if 2+2=4, it is not the case, Russell maintains, that we deduce that 2 + 2 = 4 from “Lloyd George is a wonderful man”. In fact, it is exactly because we already deduced that 2 + 2 = 4 or we already know that Lloyd George is not a wonderful man that we can maintain that “Lloyd George is a wonderful man only if 2 + 2 = 4” is true.

This of Wrinch’s points might then be due to a confusion, at least caused by Russell himself with his choice of “implies”, but still a confusion. She also makes a more general point, though:

With regard to ⊃_I: In Principia all props deal with E-functions, but they are true for I-functions … does it not seem that it is the I-functions not the E-functions which are important in logic? (Wrinch 1918c)¹⁰

Wrinch is speaking about “Arithmetic (e.g.)” (Wrinch 1918c) and, again not alone, is raising the very general question as to whether, for subjects such as mathematics, we should not move to a modal logic. While the question is not simply addressed by detecting a confusion between deduction and truth-functional connectives, Russell thought the question sorted by detecting the confusion, and brushes off the issue by simply closing his passage above with: “I conclude, therefore, that there is no need to admit as a fundamental notion any form of implication not expressible as a truth-function” (Russell 1919, 154). Hence, Russell would have concluded, had he used Wrinch’s terminology, there is no need to introduce any I-function. Now, during the period in which Russell was in prison Wrinch was acting as his secretary and in a message to her from the end of July he asks her to “[t]ell Miss Kyle to hurry up with Introduction to mathematical philosophy — she has had it quite long enough.” (Russell 1918b). So, by the time of her letter, Wrinch might have had seen his reply in the Introduction and still pressed the point because she was not convinced. Or maybe she did not see his reply. We do not know. But what we know is that Wrinch raises a further point, which, although related, is quite different and concerns neither the paradoxes of material implications, nor the necessity of a modal logic for the study of domains such as arithmetic, but, in fact, the facts of our external world, the classic domain for which non-modal logic might seem perfectly adequate.

To see the point, let’s consider two remarks by Russell. First, the following point he puts forward in Our Knowledge of the External World:

If we knew all atomic facts, and also knew that there were none except those we knew, we should, theoretically, be able to infer all truths of whatever form. Thus logic would then supply us with the whole of the apparatus required. (Russell 1914, 53)¹¹

Second, a remark he made in a passage from a paper he wrote in 1910, in reply to Bradley, who also raised some issues for Russell’s notion of implication:

I do not myself admit necessity and possibility as fundamental notions: it appears to me that fundamentally truths are merely true in fact, … I can see many ways of defining necessity which will account for its common uses: we may call a proposition necessary when it follows from a proposition known to be true, or when it can be known without empirical evidence, or when what is affirmed would be equally true of any other subject. … But none of the above meanings of necessity … justify the traditional doctrines as to modality, or the objection which philosophers are apt to feel to a ‘mere fact’. (Russell 1910, 374)

Russell was contrasting what he took to be “some desire to show that every truth is ‘necessary’” (Russell 1910, 374). This is not the point by Wrinch we will see now. In her point, Wrinch shows some confusion. She thinks that since the truth-value of p⊃_Iq is not a function of the truth-values of p and q, then we should take it as atomic rather than molecular. This, as she herself recognises, is “apparently, paradoxically enough” (Wrinch 1918c). If we free her point from this confusion, her point is that, by following what Russell maintains in the second passage above, we are missing some of the pieces of knowledge about the external world, some of the truths he is speaking about in the first passage above. For example, we are missing truths of the kind “it can’t be the case that p and not-q” (Wrinch 1918c) where “if one was given p, the laws of logic would involve one in q” (Wrinch 1918c). Wrinch does not use Lewis’s notation or his label “strict implication”, but her point goes in a similar direction as his point that

Strict implication … admits of the distinction of true and necessary, of false and meaningless. (Lewis 1914, 241),

even though she does not venture into anything like his claim that “the system of material implication is false” (Lewis 1914, 241), as for her p⊃_Eq is indeed a perfectly admissible logically derived fact (Wrinch 1918c). Apart from truths involving connectives, which also Lewis is concerned with, she also goes beyond the connectives and then beyond what Lewis discusses in 1914:

if there were facts ϕ_a; ϕ_b; ϕ_c; ϕ_d … where a b c d … were all the things in the world, then there would be a log. der. fact (x)_E.ϕx; and so too if ϕ_a then a log. deriv. fact (∃x)_E.ϕx. But such facts as ϕ_a, ϕ_b, ϕ_c, ϕ_d even if indeed a b c d … were all the things in the world are not sufficient to give a log. der. fact (x)_I.ϕx—an internal universal. … Thus a der. fact can be inferrible from one or more primitive facts but not such a … fact as (x)_I.ϕx from any number of primitive facts. ∃x.ϕx is more difficult than x.ϕx. Using ∃x.ϕx=_Df ∼ (x. ∼ ϕx). it seems necessary to recognise ∼_E and ∼_I so that there are the following combinations ∼_I((x)_I.ϕx) : ∼_I((x)_E.ϕx) : ∼_E((x)_I.ϕx) : ∼_E((x)_E.ϕx)

it can’t be the case that everything must be ϕ: it can’t be the case that everything is ϕ [:] [it] isn’t [the case that everything must be ϕ: [it] isn’t [the case that everything is] ϕ. (Wrinch 1918c)

Wrinch does not separate, as we are used to doing today, between modal operators on the one hand and truth-functional connectives and quantifiers on the other. But she has in mind facts of the kind everything must be …, it is not the case that everything must be … , and even of the kind it can’t be the case that everything must be … . At no point does Wrinch provide us with an example of facts of these sorts. But we can get an idea of the kind of thing she must have had in mind by considering, first, that these are truths whose truth is due exclusively to logic and, second, that in the very letter we are considering, Wrinch maintains that “logic is a matter of the constitution of the world” (Wrinch 1918c). Hence, she must have had in mind some structural truths, and even though she probably could not, we can indeed, for example, think about something like the necessity of identity—(x)_I.x = x—as distinguished from (x)_E.x = x. Albeit both are true of the external world, only the latter is a generalisation from the atomic facts of our world. Since both are true, also the former needs to be added to the list of the truths that logic can provide us with. Contrary to Russell, there are not just “mere facts”.

Conclusion

Towards the end of the letter, Wrinch states:

Please let me know whether it seems to you absurd or ridiculous to bring in necessity and contingency, I and E because it is a problem which seems to stop me whenever I think about the nature of facts (Wrinch 1918c)

We know from another letter she wrote to Russell in 1916 that when she took “the great liberty” of writing to him in 1914 concerning Our Knowledge of the External World, he was “so good to answer” (Wrinch 1916a) and Russell kept answering Wrinch, and also while in prison. But, as in the case of others, his reply to the points made by Wrinch in the letter, if existent, is not extant. We know that Russell wrote a message to Wrinch on the 26^th of August, but unfortunately the message is not extant, and it is anyway “about books” (Russell 1918b). Nor can we know what he replied, if he did, from Wrinch’s letters. Russell surely did have something to say about those of her points in her letter which we saw, as we know from Wrinch’s letter itself that by the time she wrote it, they had already discussed the topic and he did say something about it—“I remember putting this stuff about the two kinds of implication to you before” (Wrinch 1918c)—but, she continues, “I expect because I did not understand you, I was not satisfied, and cannot now remember your answer” (Wrinch 1918c). Neither the letter she wrote to him on the 20^th of August (Wrinch 1918a), nor the one she sent on the 23^rd (Wrinch 1918b) mentions the topic and nor does any other piece of writing. But in the letter of August the 23^rd—even though the topic is different, as she speaks about “pin-point” subjects—she does claim: “I drink in all you say about logic” (Wrinch 1918b). Maybe she did drink also that what she was criticising him on was instead correct and then that there are no modal structural truths about the external world logic provides us with. If that is true, it is a shame. As Baldwin stresses, Russell

never provides a logical construction of possibilities which shows how to get beyond his complaint that ‘possibility always marks insufficient analysis’ so that ‘when analysis is completed, only the actual can be relevant, for the simple reason that there is only the actual, and that the merely possible is nothing’. One can only conclude that on this topic, regrettably, Russell left a fundamental gap in the execution of his ‘logical-analytic method of philosophy’. (Baldwin 2017, 168)

In her letter, Wrinch does urge: “I know this does not advance it much” (Wrinch 1918c). Still, she attempted at starting to fill that fundamental gap. She was not alone to do that at that time. Lewis, for example, also attempted at filling the gap. But she was doing that while fully endorsing the general project of Our Knowledge of the External World, as we saw in §§1-2. So she showed that within that very project, it might be neither absurd nor ridiculous to bring in necessity and contingency. For Wrinch logic does not exclusively provide us with the ability to avoid some statements of existence, with the consequences of our postulates, and with possibilities “which only logic would have suggested”. Logic can surely do all of that and enlarge our imagination when it comes, for example, to understanding the structure of proteins. But for Wrinch its role is all-important also because logic can provide us with some truths that go beyond those Russell was happy to admit, that is, structural truths about the constitution of the world, which are indeed pieces of knowledge of the external world.

Acknowledgements

Thanks to the British Society for the History of Philosophy for their support of the archival research conducted for the preparation of this paper. This paper was presented at the TiLPS History of Analytic Philosophy Workshop: The Global Reception of Russell’s Scientific Philosophy and at the 2024 British Society of the History of Philosophy Annual Conference. Thanks to the audiences for their stimulating questions. Thanks to Sébastien Gandon, Ties van Gemert, and an anonymous referee for their helpful comments on an early draft of this paper.