Following a couple of brief notes on what I call the “canonical form” of a paradox and on how the Paradox of Inference – the problem derived from Lewis Carroll’s dialogue between Achilles and the Tortoise – has been received in the literature, I offer what seems to me a philosophically useful version of the paradox. This version of the problem places it more clearly into the epistemologist’s hands than does the original version.
1. What is a paradox?
I’m afraid there isn’t good philosophy in the lack of a consensual, theoretically neutral answer to that question. I would have thought that we are at least very close to having such a consensual answer. But I still hear some bewildering remarks on the issue from our experts.
I would have thought that no serious objection could be raised against Mark Sainsbury’s characterization of the philosophical notion of a paradox in the following passage of his 1995 book:
“This is what I understand by a paradox: an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable. So, generally, we have a choice: either the conclusion is not really unacceptable, or else the starting point, or the reasoning, has some non-obvious flaw.”
But Roy Sorensen (2003) calls attention to four potential problems here (although Sainsbury is explicitly targeted for only two of them, and only those two points are acknowledged by him as problematic for a tenable definition of “paradox”).
First, according to Sorensen, “Sainsbury identifies the paradox with the unacceptable conclusion of an argument that has acceptable premises and an acceptable inference pattern”. Even though Sorensen prefaces that claim by noting that “[m]ost philosophers agree arguments play an essential role in paradox”, that hardly softens the blow of what I would have regarded as an extremely unsympathetic interpretation of Sainsbury’s definition. “Surely, that’s not what Sainsbury meant!” was how I reacted at first. “It is extravagant, to put it mildly, to claim that a single statement, the conclusion of an argument, is the paradox!” But then I come across the following characterization by Gregory Moore (1998) in a Routledge Encyclopedia article:
“Etymologically, a paradox is something ‘against’ (‘para’) ‘[common] opinion’ (‘dox’). Nowadays it means a claim that seems absurd but has an argument to sustain it. A paradox appears ‘paradoxical’ when one is uncertain which premise to abandon.” (Emphasis added.)
Things have just gotten a whole lot worse: We are now allowed to believe that any old argument can back up a paradoxical conclusion!
My worries turn into a full-blown headache when I seek James Cargile’s (2005) help in another dictionary entry. Noting that “[o]ne interpretation of ‘paradox’ is ‘statement conflicting with received opinion’, Cargile goes on to dignify the single-claim account, implicitly deeming it intelligible, possibly as legitimate as that meaning of the term that makes it synonymous with “antinomy”, a term which, according to him, “applies not to a statement which conflicts [with received opinion], but to the conflict itself”. And this is about as close as Cargile comes to using the word “argument” in explaining what a paradox is. There is, of course, much to like in what he writes about the notion. But the concept of an argument never takes pride of place in his general account of the matter.
It gets worse. Relying on both Sainsbury and Quine (1961), this is Jonathan Vogel (1992) in another dictionary entry:
“Somewhat loosely, a paradox is a compelling argument from unexceptionable premisses to an unacceptable conclusion; more strictly speaking, a paradox is specified to be a sentence that is true if and only if it is false.”
But no explanation is given for how the “strict”, single-statement account evolves from the “loose”, argument-based account! 
Moreover, we don’t have do be so literal when interpreting Quine, do we? Here’s the misleading passage from Quine that seems to be the source of all evils:
“May we say in general, then, that a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it? In the end I think this account stands up pretty well. But it leaves much unsaid. The argument that sustains a paradox may expose the absurdity of a buried premise or of some preconception previously reckoned as central to physical theory, to mathematics, or to the thinking process.”
But it takes only a pinch of charity – and a few more lines in Quine’s paper – to see that he is just as comfortable calling the whole paradoxical argument “the paradox”. Speaking of the Barber, he says: “The paradox is simply a proof that no village can contain a man who shaves all and only those men in it who do not shave themselves.”
As for Sorensen’s positive account of the philosophically relevant meaning of “paradox”, I haven’t found any more than this: “I take paradoxes to be a species of riddle.” There follows a charming display of erudition where even perceptual illusions are deemed relevant to an understanding of the notion of a paradox. “I think philosophers exaggerate the role of arguments in paradoxes”, he notes, with welcome clarity.
Having determined that this is all backward and unhelpful for my philosophical needs, I turn to Benson Mates (1981) and, sure enough, my Sainsbury-induced appreciation of the role of arguments in paradoxes is rewarded: “[paradoxes] may be defined as seemingly valid arguments that obtain implausible conclusions from plausible premises (or that obtain, from given premises, conclusions seemingly independent of those premises).”
Now, I don’t think I need validity (or even the appearance of it) creeping into my definition of “paradox”. Doris Olin (2003), a fellow enthusiast of the role of arguments in paradoxes, thinks that I do need validity. This is what she calls a “traditional definition”: “A paradox is an argument in which there appears to be correct reasoning from true premises to a false conclusion.”  Her analysis of the concept wraps this definition in a number of qualifications and, most importantly, replaces “correct reasoning” with “valid reasoning”. I can’t pursue the matter of whether that is a well-advised move here (I don’t think it is). For present purposes, suffice it to note that Olin joins Mates and Sainsbury as regards the essential features of paradoxes.
Count John Etchemendy among the “traditionalists”. His Cambridge Dictionary entry on “paradox” begins as follows:
“Paradox, a seemingly sound piece of reasoning based on seemingly true assumptions that leads to a contradiction (or other obviously false conclusion). A paradox reveals that either the principles of reasoning or the assumptions on which it is based are faulty.”
Taken literally, there is an angle from which this is only a slight conceivable improvement over Olin’s offering, if, as I would claim, validity is really not an essential element. (Again, I’m not discussing Olin’s interesting argument for validity as a requirement, but, in any case, we’re striving for easily attainable convergence here, not the ultimate resolution of our toughest disagreements.) For, like in Mates’ definition, only the appearance of soundness is required.  But I see no improvement on Sainsbury’s characterization (if we don’t interpret him as Sorensen has). Unlike Olin, on the other hand, and most unfortunately, Etchemendy leaves no room for the acceptance of the paradoxical conclusion in a satisfactory analysis of a paradox. 
So, I’m ready to take the following as a Sainsbury-inspired working definition of “paradox”:
(WD) A paradox is an argument the conclusion of which is both unacceptable and obtained by seemingly good inference from seemingly irresistible premises.
We can now turn to the three remaining problems from Sorensen’s discussion of Sainsbury’s definition, for all three apply to WD. (His avowed objections to Sainsbury are in this group.) Luckily, they can all be disposed of in one fell swoop.
Sorensen explicitly objects that “[a] believer in classical logic can simultaneously perceive an argument as sound and as a positive instance of the paradox of strict implication”.  This gives him the opportunity to note that
“The paradox can be in how you prove something rather than in what you prove. This point causes indigestion for those who say that all paradoxes feature unacceptable conclusions. Their accounts are too narrow.” (p. 106)
His second objection – my third problem – is derived from an ingenious collaboration between the kind of reasoning that gives rise to the Preface Paradox and the classical theory of validity. Here’s the main passage:
“In addition to being modest about whether all my beliefs are true, I should be modest about whether all my beliefs are consistent. The more I say, the more opportunities I have to contradict myself. I say very much in this book and so believe that the assertions in this text (even apart from those in the preface) are jointly inconsistent. Take the first 10,001 assertions I make in this book. I believe that any conjunction of 10,000 of them is inconsistent. Now consider any argument that takes 10,000 of my 10,001 assertions as the premises and takes the negation of the remaining assertion as the conclusion. This jumble argument would fit R. M. Sainsbury’s definition of a paradox…The conclusion of the jumble argument is unacceptable to me because I sincerely assert its negation in my book. Each premise of the jumble argument is acceptable because I sincerely assert it in my book. The reasoning in the jumble argument is acceptable to me because I think the argument is deductively valid: Any argument with jointly inconsistent premises is automatically valid…But since jumble arguments are not really paradoxes, they show that Sainsbury’s definition is too broad.” (pp. 104-115) 
My fourth problem from Sorensen comes from the consideration of premiseless paradoxes:
“Formally, premiseless paradoxes are surprises deduced from the empty set. If validly deduced, they are veridical paradoxes. The most celebrated example is Kurt Godel’s second incompleteness theorem: a consistent proof system that is strong enough to generate elementary number theory must be incomplete. There could be premiseless antinomies: two accepted inference rules might lead to opposite conclusions.”
One fell swoop, I promised: The reason why none of these last three objections sticks is very simple when we consider the question: For the traditional paradox pattern, to whom must the premises and the inference seem irresistible and the conclusion repugnant? The answer clearly is: To common sense!
Michael Clark (2002) comes very, very close to expressing what I would regard as the definitive view of the matter:
“But what, you might ask, counts as ‘acceptable’ and ‘unacceptable’? (Un)acceptable to whom? Isn’t Sainsbury’s account too vague? No, on the contrary the vagueness in that account is an advantage, since what counts as contrary to received opinion will vary with that opinion. What once seemed paradoxical may cease to seem so. Thus, although Quine treats Godel’s first incompleteness theorem as a paradox, it is not usually counted as one nowadays, since we have got used to distinguishing truth from proof.”
My reply to Sorensen builds on Clark’s main point above but draws a somewhat different moral. There was no point in time, I submit, when any of Godel’s theorems could sensibly be deemed paradoxical (in the philosophically fundamental sense of the term), since there was no point in time when the reasoning in his proof was available to common sense.  This, of course, applies to the last three problems from Sorensen: They all trade on a notoriously perverse consequence of the classical theory of validity. Common sense doesn’t reach that high. It takes a fair amount of theory to generate Sorensen’s objections. They all concern reasoning that would not be regarded as “valid” – or else be found deserving of any term of praise whatsoever – by the commonsensical audience, as any logic instructor is so painfully reminded every time the term “validity” is up for discussion in the classroom. 
So, I’ll subject WD to a minor revision and will call the resulting characterization “the canonical form” of paradoxes: A paradox is an argument in which, from a commonsensical point of view, an unacceptable conclusion is obtained by seemingly good inference from seemingly irresistible premises. 
The therapy will remain as previously prescribed by the traditionalists. The philosophically important paradoxes (as opposed to merely recreational fallacies with little or no instructional value) are not solved in the realm of common sense. That’s only where they are spotted.
I can now turn to the Paradox of Inference with a view to offering you a version of the problem in canonical form. 
2. “That familiar object of ridicule”
That’s how John Hawthorne (2004, 2003) describes Lewis Carroll’s Tortoise. Ridicule?! Try as I might, I’ve been unable to verify Hawthorne’s description.
In his Principles of Mathematics, Russell (1903) credits the Tortoise with having shown that “[w]e need…the notion of therefore, which is quite different from the notion of implies“. It takes quite of bit of exegetical interpretation to make Russell’s understanding of such a fundamental point transparent to the contemporary reader. But we have no need for such extreme measures here. The point is more forcefully stated in Principia (vol. 1, 1910), where he notes that “[t]he process of…inference cannot be reduced to symbols”. Here, he doesn’t mention Carroll, but the very same point from that passage in the Principles is made.
We know how the point has been received in contemporary philosophy. William and Martha Kneale claim that “[t]he distinction between rules and premisses was made clear by C. L. Dodgson (Lewis Carroll) in ‘What the Tortoise said to Achilles’”. Quine (1936, 1954) decisively used the point in his influential critique of conventionalism. So many have endorsed the claim that we owe such a lesson to Carroll’s Tortoise that we shouldn’t need another reminder. Here are some of the references I have at arm’s length: Susan Haack (1973: “What I have said in this paper…is foreshadowed in Carroll”), Anthony Quinton (1973: “a fact memorably demonstrated by Lewis Carroll’s story…”), Ivor Grattan-Guinness (1976: “Lewis Carroll showed that…”), Robert Nozick (1981: “The Lewis Carroll regress…”), Mark Sainsbury (1989: “As Lewis Carroll showed…”), Pascal Engel (1991: “Carroll’s paradox bears on the very status…”), P. N. Johnson-Laird and Ruth Byrne (1991: “Lewis Carroll’s (1895) classic paper…In this neat logical fable…”), John Pollock and Joe Cruz (1999: “This was apparently first noted by Lewis Carroll”), Paul Boghossian (2001: “a…far more powerful consideration – the argument outlined in that enigmatic note by Lewis Carroll…”), Roy Sorensen (2003: “Carroll’s puzzle does show that…”), Laurence BonJour (2004: “pointed out long ago by Lewis Carroll”), Michael Devitt (2004: “as Lewis Carroll made clear a century ago…”), Richard Fumerton (2004: “[My] view does at least remind one of Carroll’s famous dialogue…”), Alexander Oliver (2005: “[Carroll] hints at a deep problem about the epistemology of valid inference, demonstrating that…”).
This hardly reads like ridicule. Granted, the evidence that Carroll himself failed to draw the popular moral of his story seems unequivocal. Yet, even if Timothy Smiley (1995) is right when he remarks that “[a]ny attempt by Carroll to tackle the question of inference was bound to begin in confusion and end in constipation”, Carroll was certainly not alone in his day. And the lesson was taken anyway.
If there is any philosophy behind Hawthorne’s misleading description, it must be this: Where is the paradox?! Even if we can’t fairly object to Carroll’s describing his problem in a letter to a less-than-enthusiastic editor of Mind as “my paradox” (Dodgson 1977), where is the paradox for us?
I think we have just begun to understand that, if there is still anything worth discussing in Carroll’s tale, the discussion will be conducted where it always belonged: in epistemology. From Carroll to Quine, however, the conceptual resources were simply not there to see past the familiar admonition: “Mind the difference between premises and rules!” To this day, the best one-line summary of the popular version of Carroll’s lesson that I have come across is Russell’s in Principia: “the process of…inference cannot be reduced to symbols”. If that’s all that there is to the Tortoise, let’s all, please, stop calling the tale “a paradox”. That description is anachronistic.
Lately, however, we’ve learned where best to locate the Tortoise: It has become a major concern for advocates of inferential internalism, like Fumerton and BonJour. It turns out that the problem may concern the radically internalist intuitions that provide the underpinnings of inferential internalism. Maybe that’s what accounts for the persistent fascination that the Tortoise still holds for the unpolluted minds of our undergraduate students. In spite of the Tortoise’s clumsy presentation of her case for a regress, the non-philosophical audience still falls prey to…well, some important intuition that seems to lead to a regress of justification. Fumerton and BonJour have most definitely touched a nerve. Maybe Carroll is in their debt to some extent. 
But I think I can give you a Carrollian paradox that is not the exclusive concern of the inferential internalist. If I am right about how best to represent the crucial intuition in Carroll’s tale, it turns out that we all may have to worry about this epistemic Tortoise.
3. Another paradox regained?
My task is to offer you a paradox in canonical form. The argument below could be shortened if the legitimacy of each step to the naked eye of common sense were not one of my crucial concerns. Except in one case, every inferential step (indicated by “therefore”) is valid. The invalid step, I submit, must depend on some very simple and seemingly sound epistemic principle. I will make no effort to identify it here. All I will do in that regard is to provide a few comments (within square brackets) to facilitate the acceptance of some of the premises which may seem specialized or dubious, in order to establish that they actually are commonsensical.
Here’s the epistemic Tortoise:
(1) If you ask me to (rationally) accept the conclusion C of an argument just because I accept the premises of the argument (say, A and B), it is reasonable of me to ask you to furnish me with a good enough reason R for me to (rationally) accept the conclusion on the basis of the accepted premises. [Presumably, this is what proofs do. If you are not compelled to take an inferential step on your own, you ask for a proof. An inference rule is not a reason. It is not a proposition, not the content of a belief. Maybe it is a command (with “propositional content”, but not a truth bearer). It doesn’t matter what it is for our concerns here. What matters is that a proof is not a rule. Rules are only mentioned in proofs. A proof is a proposition. It is a reason for accepting a conclusion on the basis of given premises. Even if there is some sound objection to thinking that proofs invariably take a propositional form, there seems to be nothing wrong with thinking that a proof may be a proposition of the form “if you accept claims with such-and-such properties and you also accept such-and-such a rule, you may (rationally, validly, reasonably, etc.) draw a conclusion with such-and-such properties” (or “having accepted that such-and-such, rule X makes it rational for you to accept such-and-such”). A proof only “takes effect” once the rules that are mentioned in it are accepted. The proof simply “connects the dots”, so to speak. It’s a nudging in propositional form. It can be contested. It can be shown false. (“You claim that such-and-such follows from such-and-such by rule such-and-such. That’s false!”) It’s an ordinary statement, and ordinarily perceived as such. Caution: This is consistent with thinking that the terms “proof” and “argument” are interchangeable in most contexts – that is, that the term “proof” is usually applied to a set of at least two propositions, with a “therefore” (“so”, “consequently”, etc.) appropriately introducing one of the propositions. What matters here is the kind of claim that explicitly involves reference to a rule of inference (or, in any case, explicitly speaks of the epistemic appropriateness of an inferential step). That little report to the right of each sentence in your proof (when you do it textbook style) is what I am calling the proof. That little report is just a claim, not an argument. It’s the reason offered for the acceptance of the proposition to its left on the basis of some earlier acceptance. It’s only in this theoretical, “metalinguistic” sense that I use the term “proof” here. When you need the theoretical claim to take that inferential step, the theoretical claim is added to your reasons for the next acceptance.]
(2) If it’s reasonable of me to ask you to furnish me with a good enough reason R for me to accept the conclusion based on the premises, then it’s reasonable of me to refuse to accept C if I reject either A or B or R.
(3) [Assumption] I do think that the acceptance of R is a necessary condition of the acceptance of C on the basis of A and B, and R is a good enough reason to accept C on the basis of A and B. [“Logical compulsion” usually takes care of the matter, no doubt. Still, many of us will eventually find ourselves in a situation where we are in need of an explicit proof, a case in which we require that the speaker appeal to principles of reasoning of one sort or another. This is, of course, usual for philosophers. But we preach that the study of logic is not just an endeavor of intellectual interest. We preach that everybody should study logic and apply it to the exercise of rational persuasion. So, the fact that, to most people, claims that make explicit appeal to rules of inference are unintelligible is simply a consequence of the fact that the study of logic is not as widespread as we would like.]
(4) Therefore, it is reasonable of me to regard A, B and R as necessary for my acceptance of C.
(5) But, if I think that R is necessary for my acceptance of C on the basis of A and B, and R is a good enough reason for the acceptance of C on the basis of A and B, then I must (on pain of irrationality) accept that A, B and R are jointly sufficient for the (rational) acceptance of C. [Note that “good enough” is supposed to take care of defeasibility conditions too.]
(6) Therefore, I ought to accept that A, B and R are jointly sufficient for the acceptance of C.
(7) Therefore, A, B and R are jointly sufficient for the acceptance of C. [Having given myself an epistemic advice in 6, I take it here, performing the acceptance recommended by my reflection. Whether “therefore” is appropriately used here is a good point for discussion.]
(8) If I rationally reflect on what is sufficient for the acceptance of a given proposition P and conclude that Q is sufficient, then I ought to accept that Q is sufficient for the acceptance of P.
(9) Therefore, if I rationally reflect about what is sufficient for my acceptance of C, I ought to accept that A, B and R are jointly sufficient for the acceptance of C.
(10) Any proposition the acceptance of which is necessary for the rational acceptance of some other proposition must be included in the set of reasons for the rational acceptance of the latter proposition.
(11) Therefore, if I rationally reflect on what is sufficient for the acceptance of C, unless I accept 7 above, I may reasonably refuse to accept C.
(12) I rationally reflect on what is sufficient for my acceptance of C in 1-11.
(13) Therefore, 7 is necessary for my rational acceptance of C.
(14) However, according to 7, A, B and R are jointly sufficient for my acceptance of C, and, if a set of propositions is sufficient for the rational acceptance of some other proposition, nothing that’s outside the set can be necessary for that acceptance.
Therefore, 7 both is and is not necessary for my acceptance of C.
4. Concluding remarks
If I’m not mistaken, the above argument cannot be stopped by the old lesson. I don’t see how a distinction between premises and rules will have any bearing on this argument. Notice, also, that the argument does not rest on the kind of claim that characterizes inferential internalism. Supposedly, this Tortoise is every one of us in those reflective moments when we are moved by ostensive, consciously accepted epistemic advice.
1. The restriction to “sentences that are true if and only if false” is also ill-advised. The conclusions of paradoxical arguments can be either necessary or contingent falsehoods, either contingent or necessary truths – the whole gamut.
2. But the claim that this is “a traditional definition” is not backed up with references (although I think she is right, in that this is at least a very close approximation to what most philosophers have tacitly accepted since time immemorial).
3. Appearance to whom?! This will matter and I will get back to it shortly.
4. Olin: “There are…two principal options in providing a resolution for a…paradox: (i) we may dispel the illusion that the argument is air-tight…or (ii) we may explain away the appearance of falsity in the conclusion.
5. It is a “positive” instance, on his account, if the reasoner believes there are only truths among the premises and also believes the conclusion (which, in this case, is supposed to be a necessary truth). Otherwise, it is a “negative” instance, the object of the next objection.
6. It is noteworthy that Sorensen’s reasoner can’t have it both ways by his own lights: If he saw a paradox when he did accept the conclusion (taking it to be a necessary truth validly derived from a premise he also accepted), he will have to regard his jumble argument as a paradox. I don’t think Sorensen noticed this apparent inconsistency.
7. No, there is no harm in calling a specialized argument with a surprising conclusion a “paradox”. But there is no philosophical gain in doing so. We are after the meaning of the term in which the fundamental problems of philosophy may properly be called “paradoxes” (see Mates 1981).
8. The problem with the classical view of validity (as I have urged in my forthcoming paper on Moore’s Paradox), for the classically-minded among us, is just the view that any necessary falsehood or any necessary truth suffices to make an argument valid – even when no acceptable inference rule will take us from premises to conclusion.
9. A bit of my case against Olin’s view of paradoxes as valid arguments: Consider this valid argument offered by Lewis Carroll (my source is Quine 1974):
The only animals in this house are cats.
Every animal is suitable for a pet, that loves to gaze at the moon.
When I detest an animal, I avoid it.
No animals are carnivorous, unless they prowl at night.
No cat fails to kill mice.
No animals ever take to me, except what are in this house.
Kangaroos are not suitable for pets.
None but carnivora kill mice.
I detest animals that do not take to me.
Animals that prowl at night always love to gaze at the moon.
Therefore, I always avoid a kangaroo!
Now, suppose you accept the premises but reject the conclusion. Is the argument a paradox by your lights? Of course, not. For the average reasoner, the validity of the argument is not available to the naked eye, so to speak. But the persuasiveness of an argument to the average reasoner must be an essential element of the philosophical notion of a paradox.
10. If I’m not mistaken, it was Clark (2002) who coined the epithet “Paradox of Inference”. The label is not without its measure of inadequacy. “Paradox of Inference” is how W. E. Johnson (1922) calls J. S. Mill’s problem (or pseudoproblem) concerning how valid arguments can be informative. But we had better stick with Clark’s label anyhow. He is commended for giving the paradox a much-needed label.
11. It is interesting to keep in mind that the analysis of reasoning has long seemed to appeal to the kind of requirement championed by the inferential internalist. Here’s C. S. Peirce (1902) in his Baldwin Dictionary entry on reasoning:
“Reasoning is a process in which the reasoner is conscious that a judgment, the conclusion, is determined by other judgments, the premises, according to a general habit of thought, which he may not be able precisely to formulate, but which he approves as conducive to true knowledge…Without this logical approval, the process, although it may be closely analogous to reasoning in other respects, lacks the essence of reasoning.”
The devil, of course, is in the “approval” of a habit of thought.
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