Conditionals and Lewis Carroll’s Barber Shop Paradox

In the 1880s Lewis Carroll, of Alice in Wonderland fame, disputed logical principles about conditionals with the (then) celebrated Oxford philosopher John Cook Wilson. Carroll published a paper with a short story about a barbershop, calling it  “A Logical Paradox”.  A dozen or so of his learned colleagues offered confusing replies.  Some twenty years later, after Lewis Carroll’s death, Bertrand Russell dismissed it wittily; Cook Wilson dismissed it scathingly that same year.  

Cook Wilson and Russell were both wrong.  That is hindsight: the logic of counterfactuals did not properly take off for another fifty years.

The Argument                                                      page 1

Two questions to be answered                             page 1

The problems with conditionals                           page 2

A    Questioning the rule of Conditional Proof    page 2

B     Questioning Import-Export                           page 3

Pragmatics of Conditionals                                   page 4

Appendix.  John Cook Wilson’s note about the paradox  page 4

1. The Argument

Allen, Brown, and Carr have a barbershop.  

(a) At least one of them must be in, to receive clients. 

(b) If Allen is out, so is Brown, who is required to accompany the ailing Allen.

Lewis Carroll’s protagonist argues that Carr is sure to be in.  Here is his argument, in summary, with present day annotations:

1. Assume Carr is out
2. Assume that Allen is out
3. Brown is in                                         ….  by 1, 2, and (a)
4. If Allen is out then Brown is in          …. 2-3: Conditional Proof
5. If Allen is out then Brown is out        ….  (b)
6. Carr is not out                                     …. 1, 4, 5 Reductio ad absurdum

That clearly can’t be right, for Allen might, from the kindness of his heart, stay in when Carr needs to be out.  

• Two questions to be answered

First, if the argument is so obviously fallacious, how can it have been so seductive as to confute Carroll’s dozen or so philosopher friends?  And secondly, just what are the wrong or questionable moves in the argument?

I suppose that initially the argument appeals because individual steps seem to be warranted by familiar valid inferences. But that cannot be the reason why the appeal seems to persist when that appearance is overcome. It is rather, I think, that there is another valid argument just nearby, in the neighborhood so to speak. And even a genuine problem about just how we should understand reasoning under supposition.

Notice that the two premises have a modal character: they are about what must be.  What if we give the same necessity to what is assumed as suppositions in the argument?  

1*.  Assume that Carr must be out

2*.  Assume then also that Allen must be out

The latter brings in train, via (b) that Brown must be out as well. So all three must be out, but (a) says that at least one must be in.  A straightforward impossibility.

What did Lewis Carroll’s readers, attempting to follow the argument with charity, think they had to do? Just what is it, to “assume” or “suppose”?  Reasoning under supposition, and its probability version, conditionalization, have their pitfalls.  If you ask me to assume Allen is out, am I not meant to mentally remove all other possibilities?  If not, if I am meant to do something less radical, just what is that?  

It seems plausible to me that inherent difficulties with reasoning under supposition that accounts for the puzzled attention the ‘paradox’ received. The connections between supposing, updating one’s beliefs (whether for real or for the sake of argument), and the (at that time as yet unborn) logic of conditionals remain a subject for puzzles to this day.

• The problems with conditionals

The first, and obvious, reply to Lewis Carroll is that the conjunction of 4 and 5, “If Allen is out then Brown is in and if Allen is out then Brown is out”, is not a self-contradiction.  This was clearly stated by Miss E. E. C. Jones (1905). Her view of the nature of conditionals seems close to two views I’ll mention below, C. I. Lewis and Arthur Burks, and she was sharply rebuked by Cook Wilson (of which more in the Appendix below). 

Russell offers the same reply, that same year, but on the basis of a quite different view. He gives it as an argument on behalf of the thesis that “if … then” is truth-functional, that the conditional is just the material conditional.  Therefore 4. and 5. together amount to “Either Allen is in or Brown is either in or out”.  So 1-5 establishes only that either Carr is in or Allen is in. “The odd part of this conclusion is that it is the one which common sense would have drawn” (for recap see Jourdain 1918: 39).

But that the conditional is the material conditional is not a conclusion which common sense would have drawn.  What we can say, more modestly, is this:

if Modus Ponens holds for the conditional – and this may be criterion for what it is to be a conditional at all – then 4. and 5. cannot both be true if Allen is out.  So together they imply that Allen is in.  

And that suffices for that point.

Nevertheless we are not home safe, for the argument appears to have several ellipses.  Even if we grant that 4 and 5 together imply only that Allen is not out, there is still the question of how we arrived at 4 and 5, and how we go from there to conclude Carr’s absence.  

There appear to be two ways, and both involve forms of argument that are invalid in general in reasoning with conditionals.  One is an unquestioning reliance on Conditional Proof, and the other is a reliance on another rule, Import-Export.

1. Questioning the rule of Conditional Proof

The rule of Conditional Proof applied in an argument with conditionals raises an eyebrow, due to the puzzles about counterfactuals which Goodman and Chisholm posed in the 1950s.

The paradigm example of a problem with conditionals at that time was that the following two conditionals are compatible:

(c)  If this match be struck it will light

(d) If this match be moistened and struck it will not light

So Weakening of the antecedent is not an admissible rule for conditionals in general.  But then Conditional Proof isn’t either, for witness how it leads at once to a derivation of the rule of Weakening of the antecedent:

1. P → Q       … premise
2. P & R        … assume
3. P                ….   from [2]
4. Q                 …..   [1], [3] Modus Ponens
5. (P & R) → Q …. [2]-[4] Conditional Proof

So that rule of Conditional Proof, used in the paradox argument, is at best suspect.  It is suspect in modal contexts in general. Frederick Fitch introduced limitations to its use when necessity and possibility are in play, and later Richmond Thomason did the same for when conditionals are involved.  

2. Questioning Import-Export

Lewis Carroll’s Barber Paradox did not disappear after Russell’s witty dismissal or Cook Wilson’s acrimonious note.  The logic of conditionals evolved slowly between Goodman (1947) and Stalnaker (1970), but we witness an important step in the discussion of the paradox by Arthur Burks and Irvin Copi (1950).  This discussion is from the point of view of Burks’ theory of causal conditionals.

In their summary of the argument they restate the premises as follows:

Letting ‘A’ abbreviate “Allen is out”, and similarly with ‘B’ and ‘C’,  we can state these two rules as

(1) if (C and A) then not-B

(2) If A then B

and they diagnose the argument as proceeding implicitly from these by the step from (1) to 

(3) If C then (if A then not-B)

They call it ‘exporting’ and it is one-half of what we now call the principle of Import-Export which equates (1) and (3).

After this move, the rule of Modus Tollens leads from (2) and (3) to not-C, provided (if A then B) and (if A then not-B) are contraries.

That they are indeed contraries, rightly understood, Burks and Copi insist.  For they read these conditionals as asserting that the antecedent provides a sufficient reason for the truth of the consequent.  And it cannot be the case that this antecedent, a contingent proposition about Allen’s presence, could give sufficient reasons both for Brown’s presence and Brown’s absence.

But the argument is nevertheless invalid, they submit, for the move from (1) to (3) is invalid. Import-Export fails for causal conditionals.

Violations of Import-Export appear in the likeliest of places.  It is a valid principle for the material conditional, but what isn’t?  It does not hold generally in modal contexts. 

This was already pointed out by Lewis and Langford (1932: 146) They define the conditional of  strict implication as follows:  A ==> B is true exactly if it is necessary that either not-A or B.  Import-Export fails for ==>; for example: 

• It is necessary that if the cupola is both square and round then there are gryphons in it.  But if in fact the cupola is round it does not follow, neither necessarily or in any other way, that necessarily, if the cupola is square there are gryphons in it. 

Burks and Copi submit that the ‘exporting’ inference does not hold for causal conditionals either, and the clear fallaciousness of the barber paradox argument shows that.

We can add to this today that we have still other examples, of different sorts, for failures of Import-Export in reasoning with conditionals.

Pragmatics of conditionals

There is a different way to approach reasoning about the barbershop.  What if I ask you “Will Carr be out?  What if Allen is out?”

The “if” question asks you to consider a situation that may or may not be actual.  Doing so, you will keep fixed some of the things you know or believe, but set aside some that would stymie the question.  Now you have to make a choice.  If you keep fixed that at least one must be in, then you say “Carr may be out; in that case, if Allen is out, Brown is in”.  If on the other hand you keep fixed that Brown is out when Allen is out, you answer, “Then Carr cannot be out, for if Allen were out then Allen and Brown would both be out”.  And finally, you certainly do not keep both those bits of your belief fixed, together with the supposition that both Carr and Allen are out, for that would stymie the question.  

So, by the natural way in which we deal with “What if?” questions, selecting (whether consciously or unconsciously) what we keep fixed and what we leave open, we avoid the self-contradictions in the barber shop paradox and elsewhere.

There are various versions of the logic of conditionals whose design begins with the intuition that the speaker assumes the antecedent while keeping some things open and some things fixed.  The phrase other things being equal is taken seriously there, and taken to have a certain kind of content.  All the principles of the logic in question follow from what the author takes that content to be like in general.  

Appendix.  John Cook Wilson’s note about the paradox

The notes about the paradox in Mind 1905 signed “W.” appear to be by Cook Wilson (see Marion 2022).  W. begins with his own take on the fallacies in the paradox, and then proceeds to castigate Miss Jones’ article which, he submits, “fails in all points”, and he ends with an elephantine bit of irony at her expense.  

So let us examine his own response.  Like Burks and Copi he submits that the paradox generating argument really “starts from the proposition

(v.) ‘If Carr is out, then if Allen is out, Brown is in.’”

This he follows at once with an explanation that (v.) means

‘If Carr is out and Allen is out, Brown is in’

unaware that the rule of Import-Export is exactly one of the principles put in question when it comes to the logic of conditionals.  

There are further familiar difficulties.  Cook Wilson held that all propositions are categorical, and ‘hyptheticals’ are not propositions at all, but codes for rules of inference (‘inference tickets’ as Ryle would later say).  So when W. decides to get to the heart of the matter, he tries hard to present the paradox in ‘the right way’:

But the proposition ‘ If Allen is out Brown is in’ is a universal proposition which if valid at any time is valid at all times; it represents a rule which is always valid. It cannot, therefore, be a consequence of Carr’s being out, necessarily valid only at the time when Carr is out. In fact the proposed interpretation amounts to the absurd statement: “The rule ‘If Allen is out Brown is in,’ which if valid at all must be valid at all times is only necessarily valid at the times when Carr is out”.

Cook Wilson’s view is not in the least plausible.  Imagine this dialogue:  “Is your brother in”? “Look in the hall.  If his hat is there he is in.” “Oh, is that always the case?” “No, not at all, but it is today because he went out wearing his hat.”

Note also that on Cook Wilson’s view of conditionals, it does not seem that they can be meaningfully embedded as parts of statements.  It needs the bits of meta-language that he inserts.  This too flies in the face of common examples (such as, in fact, his line (v.) above).

As for Cook Wilson’s responses, then, there is no need to add a final bit of irony .

PS. Why should I refer to E. E. C. Jones as “Miss Jones”, and not to Cook Wilson as “Mr. Cook Wilson”?   Here I follow the style on page viii of Mind 1905’s table of contents.  It is clear that the editor did not want the unusual to go unnoticed.  

Jones offered a spirited reply to W. in Mind’s last issue of that year.

Emily Elisabeth Constance Jones, portrait by John Lavery

References

Burks, Arthur W. (1951)  “The logic of causal propositions”.  Mind 60: 363-382.

Burks, Arthur W. and Irving M. Copi (1950) “Lewis Carroll’s barber shop paradox”.  Journal of Symbolic Logic 15 (03): 219-222.

Carroll, Lewis (Charles Dodgson) (1884) “A Logical Paradox”.  Mind N. S. 3 (11): 436-438.

Jones, E. E. C. (1905) “Lewis Carroll’s Logical Paradox”.  Mind N. S. 14 (56): 146-148 and 576-578.

Jourdain, Philip E. B. (2011) The Philosophy of Mr. B*rtr*nd R*ss*ll. London: George Allen & Unwin. Downloadable on Project Gutenberg.

Lewis, Clarence I. and C. H. Langford (1932/1959) Symbolic Logic. New York: Dover.

Marion, Mathieu (2022) “John Cook Wilson”. The Stanford Encyclopedia of Philosophy. Edward N. Zalta (ed.).

Moktefi, A.  (2007) “Lewis Carroll and the British nineteenth-century logicians on the barber shop problem”.  Proceedings of The Canadian Society for the History and Philosophy of Mathematics’ Annual Meeting (Concordia University, Montréal, July 27-29, 2007).  Ed. Antonella Cupillari.  http://www.cshpm.org/archives/proceedings/proceedings_2007.pdf. Article text available on ResearchGate.

W.  (John Cook Wilson) (1905)  ‘Lewis Carroll’s Logical Paradox’. Mind N. S. 14: 292–293. With correction, Mind N. S. 14: 439.