Carroll’s Regress and the Epistemology of Logic

PATRICE PHILIE

CARROLL’S REGRESS AND THE EPISTEMOLOGYOF LOGIC1

ABSTRACT. On an internalist account of logical inference, we are war-ranted in drawing conclusions from accepted premises on the basis of ourknowledge of logical laws. Lewis Carroll’s regress challenges internalism bypurporting to show that this kind of warrant cannot ground the move frompremises to conclusion. Carroll’s regress vindicates a repudiation of inter-nalism and leads to the espousal of a standpoint that regards our inferentialpractice as not being grounded on our knowledge of logical laws. Such astandpoint can take two forms. One can adopt either a broadly externalistmodel of inference or a sceptical stance. I will attempt, in what follows, todefend a version of internalism which is not affected by the regress. Themain strategy will be to show that externalism and scepticism are not sat-isfying standpoints to adopt with regard to our inferential practice, and thento suggest an internalist alternative.

KEY WORDS: epistemology, externalism, inference, internalism, logic,scepticism

1. INTRODUCTION

This paper addresses one of the most fundamental aspect ofthe epistemology of our inferential practice. It does so bystudying a well-known problem in the foundations of logic,that of Lewis Carroll’s puzzle about inference as exposed inhis 1895 Mind paper ‘‘What the Tortoise said to Achilles’’.The problem elicited by the regress can be seen like this. Onan internalist account of logical inference, we are warrantedin drawing conclusions from accepted premises on the basisof our knowledge of logical laws. Carroll’s regress challengesthe internalist assumption by seeking to show that knowledgeof logical laws cannot ground the move from premises to

Philosophical Studies (2007) 134:183–210 � Springer 2006DOI 10.1007/s11098-005-7269-z

conclusion. Thus, Carroll’s regress vindicates a repudiation ofinternalism and leads to the espousal of a standpoint that re-gards our inferential practice as not being grounded on ourknowledge of logical laws.

Such a standpoint can take two forms. One can adopt abroadly externalist model of inference, according to which nointernalist warrant is necessary to move from warranted pre-mises to a conclusion. The externalist model construes war-rant in terms other than knowledge of logical laws. Anexternalist might cash out his notion of warrant by appealingto the idea that it is enough, in order to be warranted todraw an inference, to be a reliable agent, to arrive at the con-clusion via appropriate methods, to be in the right circum-stances, and so forth. However, someone who accepts theconclusion of the regress but wishes to resist the notion of anexternalist warrant will embrace scepticism – namely, theview that there are no norms of inferential practice.

This issue has been discussed recently by Paul Boghossianand Crispin Wright in a stimulating exchange in which theysettle the terms of an important debate concerning the natureof rationality and logical knowledge.2 Armed with a criticalperspective, I will follow their lead and finally attempt to de-fend an internalist solution to the problem. The main strategywill be to show that externalism and scepticism are not satis-fying standpoints to adopt with regard to our inferentialpractice, and then to put forward a modified internalismwhich is not threatened by the regress.

2. THE REGRESS

Let us consider this simple inference:

(1) If Socrates is a man, then Socrates is mortal.(2) Socrates is a man.(C) Therefore, Socrates is mortal.

On a broadly internalist account of inferential knowledge,you are warranted in drawing the conclusion (C) on the basisthat the premises are warranted and also on the basis of your

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knowledge of the logical law of modus ponens. Indeed, forthe internalist, mere warrant for the premises is not enoughto draw the conclusion: you must also know that the premisespossess a valid logical form.

How should this knowledge be represented? Well, you mustknow that modus ponens is a valid logical form, that is, youmust know:

(W) If an argument displays the logical form of modusponens, then it is valid.

W, then, is to be understood as the warrant enabling you to‘move’ from the premises to the conclusion in our originalargument. It is a piece of knowledge that you must possess.The question now is: how does knowledge of W warrant youin drawing the conclusion (C)? It seems that you need to knowtwo things. One, you need to know W, and second, you needto know that the argument above is an instance of modus pon-ens. So what you need to know can be represented like this:

(a) W: If an argument displays the logical form of modusponens, then it is valid.

(b) The argument above ((1), (2), (C)) is an instance ofmodus ponens.

(c) Therefore, the conclusion (C) – that Socrates is mortal– can be drawn.

The problem is immediate: the step from (a) and (b) to (c)is itself inferential and possesses the modus ponens form. Soin order to be warranted in drawing the conclusion (c), youneed to possess an additional piece of knowledge, namely:

(i) W: If an argument displays the logical form of modusponens, then it is valid.

(ii) The argument above ((a), (b), (c)) is an instance ofmodus ponens.

(iii) Therefore, the conclusion (c) – that the conclusion (C)can be drawn – can be drawn.

It seems that this pattern could be repeated indefinitely; ifso, it means that claiming knowledge of W when drawing

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inferences leads to a regress of warrants. This in turn wouldmean that knowledge of the validity of the modus ponensschema cannot act as a warrant for the conclusion (C).

Carroll’s regress as understood here seems to establish thatknowledge of logic (that is, in this case, modus ponens) is un-able to warrant simple inferences – but this is, arguably, pre-cisely the role of logic. Invoking one of the most fundamentallaws of logic is of no help to warrant the logical move fromthe premises to the conclusion. So according to Carroll’sstory so understood,3 it is impossible to be internally war-ranted in drawing inferences. There is an infinite regress ofnecessary warrants. That is a point deserving emphasis: whatCarroll’s regress suggests is not, as it is often assumed, thatthe mind cannot move – the mind does move, we have noproblem to effectively draw simple inferences. It’s ratherabout the ground of the move, about whether (or how) logiccan make the mind move. We usually assume – in this kindof case – that our move from the premises to the conclusionis warranted by appealing to logical laws. But Carroll’s storyindicates that this a misunderstanding of our inferential prac-tice. The move, if we accept the regress and its conclusion, isnot warranted the way that the internalist model requires.

3. INTERNALISM, EXTERNALISM, SCEPTICISM

Let us think carefully about the conditions under which thereis a regress. In order for the regress to go through, it must bea requirement that the inferrer possesses an internal warrantfor drawing the conclusion. In other words, the inferrer needsto know that modus ponens is a valid rule of inference. Thisadditional piece of knowledge is illustrated by W.4 Thisrequirement amounts to what Wright labels ‘simple internal-ism’. It is to be defined thus:5

Simple internalism: In order to acquire knowledge by infer-ence, the thinker must know that the rule used is valid.

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Defined in this way, the simple internalist position is sub-jected to the regress. In contrast, a ‘simple externalist’ stand-point about inference does not face the threat of the regress.According to this externalist position, an inferrer does notneed to know that modus ponens is a valid rule of inferencein order to draw the conclusion (C): all that is required isthat the rule is in fact valid – knowledge is not a necessarycondition. Simple externalism is characterised thus:

Simple externalism: In order to acquire knowledge by infer-ence, the rule used must be valid. Knowledge of this rule isnot necessary.

It is clear that one who espouses this view will not be affectedby the regress, since it is not necessary to be capable of pro-ducing a warrant such as W in order to draw the inference.All that is necessary is that the rule is indeed valid. In thissense, the challenge posed by Carroll’s regress is one that willaffect only a broadly internalist conception of inferentialknowledge.

There is another stance about our inferential practice whichis not subjected to the regress – namely, scepticism about war-rant. According to scepticism, when one makes an inference,there is no warrant in play. That is, on this view, inferring theway we do is not governed by any norm. Talk of a rule being‘valid’ is illusory according to the sceptic. Let us define it:

Scepticism: Inference is not guided by any kind of warrant(internalist or externalist) and there are no norms of correctreasoning.

This is a sceptical view about our normal conception of infer-ential practice in the sense that it suggests the superfluousnessof there being a norm that we follow when we draw infer-ences. Indeed, talk of warrant easily transfers into talk aboutnorms. The warrant – say, W – allows me to draw the infer-ence, and that warrant expresses a norm (the validity ofmodus ponens) that I (must) follow. So the sceptical predica-ment is that inferential practice is not normative, if we under-stand norms in the way just suggested. Let us now turn to acritical examination of scepticism and externalism.


4. LOGICAL NORMS AND SCEPTICISM

Dispensing with norms is to embrace the sceptical stance,namely that when we infer, there is no warrant in play – wedo not, and cannot, appeal to logical laws when inferring. Toespouse scepticism is to repudiate the apparent fact that thereare norms that we follow in our inferential practice. Or, toput it in other words, it amounts to repudiating that there arevalid logical rules. On this view, logic (by that I mean logicas exposed in logic books, logical systems, and so forth) is atbest a mere representation, a ‘rational reconstruction’ ofinputs and outputs. This reconstruction, under the aegis ofscepticism, has absolutely no normative force: we can’t usethe reconstruction as a warrant, otherwise we wouldn’t beembracing scepticism anymore. Indeed, scepticism does seemto be, on the face of it, committed to repudiating both normsas such6 and the notion of validity.

Here are two legitimate questions to ask with regard to thesceptical stance: What is the place of logic in such a picture?and: Is there even a place for it? Here we face a dilemma.The first horn has it that logic is a mere representation, arational reconstruction of the way most of us ‘infer’. But thesceptic embracing this horn has to claim that this representa-tion has no normative force. However, this strikes me asimplausible. It’s comparable to claiming that maps representa country (say) while holding that the map has no normativestatus. Of course not: the whole point is to use the map as anormative (if conceivably flawed) tool. Similarly, the point ofdevising a representation of how we infer is to guide ourinferential practice. If that much is conceded, the scepticalstandpoint is abandoned: when inferring, we appeal to normssuch as W.

There is a possible reply by the sceptic. She could claimthat we use the representation as a guide, but we don’t haveto. It is not necessary for correct inferential practice to appealto norms. When we do refer to logic books, the scepticcontinues, we are merely taking a shortcut instead of goingthrough the process ourselves. What makes an inference

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correct is not its compliance with a norm, but something notnormative (whatever that may be). To come back to the mapanalogy, it is not the map that makes true which road totake, but rather the actual road system, out there in theworld. But it is precisely at this point that the map analogythreatens to break down: what corresponds to the ‘right’road, out there in the world, when we talk about inference? Itwould have to be something akin to a ‘logical fact’, and ifone believes that there are such facts, then one is not a scep-tic anymore.

The other horn of the dilemma is to repudiate the logic-as-representation view and hold that logic books and logical sys-tems are unnecessary. The problem here is immediate: whathave we been doing all these centuries? Were we just wrong indevising logical systems? Were we wasting our time? Obvi-ously not. What are students learning in logic classes, if notnorms of correct reasoning? It seems that if we are to be con-sistent sceptics about the normativity of logic, we should fol-low a paraphrase of Hume’s advice and ‘commit logic booksto the flames, for they contain nothing but sophistry and illu-sion’. This point of view entails nihilism about logical norms– anything goes as far as logic is concerned. If furthermoreBoghossian (2001, 15–19) is right (and I think he is) in stress-ing the relation between the existence of logical norms andthe possibility of objective epistemic reasons, then scepticism(or nihilism) about logical norms entails relativism about epi-stemic reasons. This amounts to much more than a merelocal scepticism about logical knowledge – it amounts to afull-blown relativism.

The dilemma, then, is this. If the sceptic endorses the logic-as-representation view, then it’s hard to avoid the recognitionof the normativity of logic. If, on the other hand, the scepticdecides to do without logic altogether, the view leads to afull-blown relativism. In a nutshell, the sceptical conclusionjust seems to be inconsistent with the way we infer, since wedo, as a matter of fact, reason according to rules of inferenceand we often justify our inferences on the basis of them. It iscommitted to the view that such talk is illusory, and this is,


I believe, intolerable. I acknowledge that the sceptic, however,may just be unmoved by such arguments – the considerationsjust advanced give us reasons to resist the sceptical conclu-sion, but they do not refute it. The sceptical point of viewshould be seen as a challenge, as a position to be supplantedby a standpoint saving objectivity and answering, rather thanrefuting, the sceptical considerations. As Boghossian says,‘‘[t]he correct project in epistemology is to show how knowl-edge is possible. It is not the refutation of arbitrarily extremeskeptics’’ (2001, 37).

5. THE EXTERNALIST PROPOSAL

Let us now turn to externalism. According to externalism, noknowledge of validity is necessary for correctly inferring.Faced with appropriate premises whose truth we recogniseforces the acceptance of the conclusion without any processof self-reflection involved in the way the regress demands –without, that is, any ‘reflectively appreciable warrant’7 beingnecessary. Indeed, the regress goes through only if one has torecognise that the premises fulfil a certain condition (display-ing a valid logical form), and then, on one’s recognition thatthey do fulfil these conditions, one is licensed to infer theconclusion. But if it is claimed that inferring the way we do issimply a matter of recognising the truth of the premises andthat the conclusion will ‘force’ itself on us because of exter-nalist considerations (for instance, reliability), such an appealto the required condition (the warrant) doesn’t play any role– it becomes superfluous. Thus, upon’s one recognition thatall men are mortal and that Socrates is a man, the conclusionthat Socrates is mortal forces itself on us – in terms of inputand output, given the input of the premises, the natural out-put is the conclusion. To put it in reliabilist terms, the infer-rer is reliable – he doesn’t know the rule, but he follows itwith near-perfect regularity.

However, externalism about inference suffers from impor-tant problems. For instance, Boghossian, in his aforemen-

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tioned paper (27–28), objects to externalism on the groundthat externalist considerations such as reliability fall short ofbeing justifications. Boghossian uses the following example toshow that being a ‘reliable’ inferrer does not amount to pos-sess a justification:

1. x, y, z, and n are numbers and n is greater than 2.2. Therefore, xn + yn „ zn.

Here, Boghossian argues that someone may reliably drawthe inference without being justified in the usual sense – with-out, that is, knowing anything about the proof of Fermat’stheorem.8

Boghossian’s objection reveals a deep problem concerningexternalism in general that could be formulated in this way:can externalism make sense of the idea of justified belief?However, the objection to externalism I would like to drawattention here appears when we try to make sense of the met-aphor in play when talking about the conclusion ‘forcing’ it-self on us. To make matters clearer, think of the parallel herewith a famous case, that of the recognition of chicken sexes.Just as it can be said that recognisers of chicken sexes areable to do so without being able to justify their capacity byappealing to a criterion, one could hold that we infer cor-rectly without appealing to a rule of inference – all that mat-ters is that the rule is indeed, as a matter of fact, valid.Exposure to chickens over a period of time will somehow in-stil the capacity to distinguish reliably between their sexes;similarly, exposure (so to speak) to premises displaying a defacto valid logical form will, for any competent speaker,prompt the capacity to infer correctly – and that’s it, nothingmore is required, strictly speaking.

The immediate problem is that there is a dissimilaritybetween the two cases. It is uncontroversial that there is a factof the matter about chicken sexes – the controversial external-ist thought is the offered explanation of the capacity to dis-criminate between the sexes (no need for internalised warrant),not whether there is a fact of the matter or not about the sexof chickens. But if we push the parallel further, it appears that


in order to be able to offer an externalist explanation of ourcapacity to infer correctly, there has to be a fact of the matterabout logical inference. The externalist offers an explanationof the route between an external fact and the capacity of acognitive agent to recognise that fact. In the case of inferentialpractice, a proper externalist explanation will have to proceedalong these lines – namely, that the inferring agent is in con-tact with something over and above the recognition of thetruth of the premises, and that this something is what enablesthe agent to draw the inference. This ‘something’ would haveto be a valid rule of inference. But this seems to involve anontological commitment which is puzzling. What is it that werecognise when we infer correctly? What are we responding to,and how? Are we somehow in contact with logical facts – thesame way that the chicken sexes discriminator is in contactwith the relevant features?

The externalist thought about chicken sexes has some cred-ibility since the ontological issue is not in dispute. No onedisputes the existence of chicken sexes. But what is externalabout a rule of inference? What does it mean to say that weare in contact with logical facts and that we, as it were, trackthese facts without possessing a reflectively appreciable war-rant – without knowing how we’re doing it? These ontologi-cal questions about the status of logical norms and validityare extremely difficult ones. The externalist is committed, ifthe foregoing is right, to a realism about validity and logicalfacts. Such a view is difficult to sustain in the face of scepticalobjections – the sceptic would ask for an explanation of theselogical facts. The internalist would also want answers to thesequestions. An account which does not embody such strongontological commitments is preferable from the present per-spective (answering sceptical doubts).

In the last two sections, I have reviewed both a generalclaim that has to be endorsed by the proponent of scepticism(the denial of norms) and an externalist account of our infer-ential practice. It is notoriously difficult to counter-argue in adecisive manner against the externalist and the sceptic. Thechallenge here is perhaps not to be understood in terms of

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refuting scepticism or externalism, but rather to save our in-ternalist intuitions by putting forward a plausible account ofthe epistemology of our inferential practice. If it is possible toshow: (1) the plausibility of an account of our inferring prac-tice doing justice to the idea that W must be, in a sense to bedetermined, intelligibly internalised in order to draw infer-ences, and (2) that this proposed account does not fall intothe regress, then, I believe, the non-internalist standpointswill prove to be much less attractive since there would be aninternalist view doing better justice to our intuitions. Let usnow turn to a proposed account, that of Boghossian,attempting to steer between the Charybdis of externalism andthe Scylla of simple internalism.

6. EPISTEMIC RESPONSIBILITY

Simple internalism is characterised in terms of knowledge oflogical rules. Indeed, on this view, in order to perform a justi-fied inference, the inferrer must know that the logical ruleused is valid. Seen from this perspective, the cleavage betweenexternalists and internalists is construed around the notion ofknowledge: the internalist claims that knowledge of validity isnecessary, whereas the externalist holds that it is not. Accord-ing to Boghossian (2001, 29–31), however, this is a misrepre-sentation of the contrast. The debate is better characterisedaround the notion of epistemic responsibility rather than thatof knowledge. In order to see that, let us reflect on Boghos-sian’s objection to externalism mentioned earlier, where a reli-able but ignorant inferrer draws a conclusion whose warrantconsists in the proof of Fermat’s theorem. What it shows isthat an inferrer can perform a correct inference yet be episte-mically irresponsible. The inferrer, in that case, has no justifi-cation for making the inference; he is correctly drawing aconclusion without having any epistemic ground for doing so.This, on Boghossian’s interpretation of the debate, is the cruxof the internalism/externalism debate: epistemic responsibility– and it need not be construed in terms of knowledge. The


minimal condition for being (internally) justified, according toBoghossian, is not to know (it is too strong a condition andleads to Carroll’s regress), but rather to be epistemicallyresponsible. This is a ‘weaker’ condition in the sense that it ispossible to be epistemically responsible without having a‘reflectively appreciable warrant’ in the mind’s eye, as it were.

In the present context, the notion of epistemic responsibil-ity can be used fruitfully with regard to rules of inference. Ifit can convincingly be shown that I do not need to know thatmodus ponens is valid in order to draw the inference but thatI am being epistemically responsible in performing the step inthe Socrates example, then knowledge of W is not necessary.And since I am being epistemically responsible, then I aminternally warranted. Thus, even if knowledge of W can behad, it does not constitute a precondition under this kind ofinternalist explanation. Therefore, this type of account would– if it can be made to work – remain internalist in spirit andavoid the Carroll-type objections.

The main difficulty is to cash out the idea of epistemicresponsibility in a satisfactory manner. Under which condi-tions can we be said to have an epistemically responsible be-lief without having a reflectively appreciable warrant for thatbelief? Boghossian suggests that ‘meaning-constituting’ rulesare good candidates for that purpose. According to this lineof thought, since, say, the rule of modus ponens constitutes –at least partially – the meaning of ‘ fi ’, it follows that usingsuch a rule in an epistemically responsible manner need notimply the possession of a reflectively appreciable warrant. Iam epistemically responsible in such a case, according toBoghossian, because the rule determines a given meaning. Thisis what grounds the responsibility, if I understand Boghos-sian’s suggestion.

Boghossian’s idea receives support from Prior’s well-knownconnector ‘tonk’.9 The rules defining ‘tonk’ are unsound –thus they do not determine a meaning. For that very reason,on Boghossian’s account, I would not be epistemicallyresponsible if I used them. A correct, responsible use of a setof rules must be grounded, minimally, on their soundness.

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With ‘tonk’, this is not the case. Conversely, if a set of rulesdoes determine a meaning, then I am using them responsibly.Such is the case with the rules defining the conditional: theyare sound, they determine a meaning, therefore I am episte-mically responsible when using them. Responsibility, on thisview, is a matter of reasoning according to valid rules.

Such an account of epistemic responsibility, if it could bemade to work, would solve two problems: it would, as justmentioned, solve the problem of ‘tonk’,10 and it would solveCarroll’s regress – indeed, since I do not need to know W

reflectively, as it were, I am still entitled to reason accordingto it in a fashion congenial to internalism. Since what causesthe regress in the simple internalist view is the necessity toknow that modus ponens is valid and that such knowledge isnot necessary under Boghossian’s suggestion, then there is noregress, and we save the spirit of internalism.

Boghossian’s hope, in a nutshell, is that:1. meaning-constituting rules guarantee epistemic responsi-

bility2. epistemic responsibility suffices for warrant.

Wright, in his commentary (56–59), disputes both theseclaims. He concocts counter-examples devised to show thatthere are meaning-constituting rules that do not guaranteeepistemic responsibility and that there are epistemicallyresponsible patterns of inference that are not warranted. Iwill now review Wright’s objections. Their study will bring tolight constraints that should be respected in the elaborationof an internalist epistemology of our inferential practice.

With regard to his criticism of (1), Wright avails himself ofHume’s Principle and transforms it into a set of introductionand elimination rules:11

N-I :ð9RÞðF1� 1rGÞ

Nx : Fx ¼ Nx : Gx

N-E :Nx : Fx ¼ Nx : Gx

ð9RÞðF1� 1rGÞ


Such rules, like the usual formulation of Hume’s Principle,are arguably meaning-constituting of the cardinality operatorN, ‘the number of’. However, Wright argues that the merefact that these rules determine a meaning for the cardinalityoperator is not, by itself, enough to regard them as sufficientfor epistemic responsibility. Indeed, opponents to Neo-Freg-eans are quick to point out that despite the meaning-consti-tuting character of such rules, it remains possible to disputetheir correctness on various grounds – such as their structuralsimilarity with Basic Law V, for instance (this law, as is well-known, is inconsistent). This shows, according to Wright,that meaning-constitution as such cannot guarantee epistemicresponsibility, since if Hume’s Principle turns out to be toosimilar to Basic Law V, then using the N-rules would be epi-stemically irresponsible – one would not be justified in usingthem.

Under this train of thought, the meaning-constituting char-acter of a rule does not, as such, imply the possibility ofclaiming a (non-reflective) warrant because, if I understandWright correctly, there are meaning-constituting rules that dosucceed in determining meaning but that may be used in anepistemically irresponsible manner. However, we can ask, inBoghossian’s behalf: what would it mean to use the N-rulesirresponsibly? Presumably, it would mean that these rules areunsound. However, it is not clear how a rule can determine ameaning and yet be unsound. Take the connector ‘tonk’. Therules for this connector do not determine a meaning preciselybecause they are unsound. The point of the ‘tonk’ example isto show that unsound rules do not determine a meaning.Applied to the N-rules, it would show, if critics are rightabout Hume’s Principle and its structural similarity with Ba-sic Law V, that they are unsound – thus, that they fail todetermine a meaning for the cardinality operator. TheN-rules, under the hypothesis that those who object toHume’s Principle are right, might have a meaning-constitut-ing character, but they do not constitute a meaning. If this iscorrect, it follows that Wright’s point, namely that meaning-

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constituting rules can be used irresponsibly, is not establishedby his example. His point should rather be taken as a warn-ing to the effect that even though some rules appear to deter-mine a meaning, this mere fact does not entitle us to usethem. In other words, arguments are needed to show that agiven rule is, in fact, meaning-constituting. Otherwise, wemight end up using unsound rules.

Wright also objects to the second claim made by Boghos-sian, the one according to which epistemic responsibility suf-fices for warrant. Let us formulate, he suggests, Basic Law Vof the Grundgesetze as a pair of introduction and eliminationrules for the course-of-values operator:

Course-of-values-I:ðxÞðFx$ GxÞfxgFx ¼ fxgGx

Course-of-values-E:fxgFx ¼ fxgGxðxÞðFx$ GxÞ

As we now know thanks to the discovery of Russell’s para-dox, these rules are unsound. However, as Wright says, itmight have taken decades before the discovery was made, andit is conceivable that generations of students could havelearned these rules and used them routinely. These studentsof Frege’s system could hardly have been accused of episte-mic irresponsibility. Indeed, in the context just defined, it isreasonable to hold that using these rules is blameless. To beguilty of epistemic irresponsibility seems to require that theuser did something wrong, something he shouldn’t have donegiven his present state of knowledge. But in this case, it seemsunreasonable to draw a charge of irresponsibility. However,since justification and warrant in the usual sense requiresvalidity in the case of rules of inference, it follows that ab-sence of epistemic irresponsibility does not amount to war-rant. Thus, if we identify absence of irresponsibility withresponsibility, it follows that epistemic responsibility does notalways suffice for warrant, contrary to what Boghossianholds.


Let us put the matter in this way. Boghossian suggests thatmeaning-constitution suffices for epistemic responsibility, andthat the latter suffices for warrant. Thus, he hopes to make acase that an inference made in accordance with a meaning-constituting rule suffices for warrant. However, Wright arguesthat the link between epistemic responsibility and warrantmight not be as strong as Boghossian seems to suppose. Hedoes so by setting up a situation in which an unsound rule isused in a blameless manner. This means that absence of irre-sponsibility cannot amount to the possession of a warrant. Ifwe take the further step of identifying absence of irresponsibil-ity with responsibility, we have a counter-example to Boghos-sian’s claim that epistemic responsibility suffices for warrant.

However, it is possible to draw a different conclusion fromWright’s observation. Instead of concluding that epistemicresponsibility is not sufficient for warrant, we can insteadresist the identification between absence of irresponsibilityand responsibility. Wright’s example is constructed aroundthe notion of blamelessness – strictly speaking, then, heshows that blamelessness does not need to correspond to war-rant, not that responsibility is not sufficient for warrant. Thisleaves the door open to the following interpretation ofWright’s counter-example: Boghossian is right in stressingthat meaning-constitution suffices for lack of irresponsibility –I am not being epistemically culpable in using the course-of-values rules in the example. But what is needed is not an ac-count of blameless beliefs. It’s rather an account of responsi-ble beliefs. We need to know what responsibility amounts to,not merely lack of irresponsibility. Wright’s example showsthat lack of irresponsibility need not always correspond toresponsibility. What is required, then, is an account ofresponsibility.

To sum up, the questions prompted by the discussion ofBoghossian’s suggestion and Wright’s criticism of it are thus:How should we distinguish responsible beliefs from irrespon-sible ones? What are the criteria for holding an epistemicallyresponsible belief? I shall come back to these questions in thenext sections.

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Wright points out that his criticisms do not address the gen-eral intent of Boghossian’s idea. His idea – if we put aside thereference to the meaning-constituting character of basic rulesof inference – is to make room for warrants that are not knownin the usual sense, but that are nevertheless held responsibly.What we need, then, is a condition allowing us to consider thatsome beliefs concerning rules of inference are held in an episte-mically responsible manner, and a convincing explanation ofresponsibility in the case of inference. Boghossian toyed withthe condition of meaning-constitution, but Wright showedimportant limitations to this idea. In sum, Wright’s objectionsto Boghossian’s proposal have taught us two things:

• the meaning-constituting character of a rule is not, by it-self, enough to ensure epistemic responsibility

• in order to be able to save the insight according to whichit is possible to follow a rule of inference in a responsiblemanner, we need an account of responsibility – we needto show why, in a given case, an inferrer is being epistemi-cally responsible.

Thus, what is needed is a condition which, unlike meaning-constitution, can be backed up by an appropriate conceptionof epistemic responsibility. In the following sections, I will de-fend such a condition.

7. CONSTITUTIVE NORMS

There is another line of thought available as a response toCarroll’s regress which is not vulnerable to Wright’s objec-tions. It draws on Davidsonian ideas about the nature ofpractical reasoning. In order to see how these ideas can beexploited in the present context, it is necessary to distinguishbetween rules of inference that are constitutive of our practiceof inferring and those that are not constitutive. The focus hasso far been on modus ponens and not on other logical laws.I will argue in what follows that if a satisfying internalist pro-


posal is to be possible, then the constitutive role of modusponens has to be recognised.

Let us assume, for the sake of the argument, that logicallaws have a normative status in the sense that they governour inferential practice – that is to say, acceptance of a logi-cal law implies a commitment to recognise as valid argumentsmade with it, and also implies that we can make use of it inour inferring practice. It seems that acceptance of a logicallaw carries with it the idea of ‘obeying’ to it. In this sense,logical laws are norms. Modus ponens is such a norm ofinferential practice, but it is also a constitutive norm in thesense that the very practice of inferring presupposes modusponens. Carroll’s regress reveals this interesting fact aboutinference.

To see why, let us take this simple inference, an inferencenot using modus ponens:

(20) Socrates is mortal and Aristotle is Greek.(C*) Aristotle is Greek.

On an internalist account of our inferential practice, thisinference is subject to the regress. Indeed, to be able to drawthe conclusion (C*), you must be credited with the knowledgethat the argument above displays a valid argument-form, thatis, you must know that the rule for &-elimination is valid.How does this knowledge warrant you to draw the conclu-sion (C*)? Well, this warrant must be represented like this:

(a*) (W*): If an argument displays the logical form of the&-elimination rule, then it is valid.

(b*) The argument above ((20), (C*)) is an instance of the&-elimination rule.

(c*) Therefore, the conclusion (C*) – Aristotle is Greek –can be drawn.

But this is another inference, an inference justifiable only ifone is warranted to accept modus ponens. So in order to bewarranted in drawing the conclusion (c*), you must be cred-ited with the knowledge that:

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(i*) W: If an argument displays the logical form of modusponens, then it is valid.

(ii*) The argument above ((a*), (b*), (c*)) is an instance ofmodus ponens.

(iii*) Therefore, the conclusion (c*) – that the conclusion(C*) can be drawn – can be drawn.

And so on. Whatever rule of inference is used in the origi-nal argument, an ultimate inference made with modus ponensmust be made if we are to remain faithful to the internalistmodel. This is because no matter which rule is appealed to inthe first place, there will be a corresponding argument havingthe form (a), (b), (c) or its starred equivalent. Any such argu-ment will be warranted, ultimately, on the basis of modusponens since for any instance of any logical law, to knowthat the conclusion follows from the premises requires grasp-ing an argument which has a modus ponens form – namely,an inferential pattern having this form:

(30) If an argument displays a valid logical form, then it isvalid.

(31) Argument x displays a valid logical form.(32) Therefore, argument x is valid.

It is readily seen that the correctness of any inference ofthat form is contingent upon the validity of modus ponens.

Before we proceed any further, we need to address twoobjections to the claim that modus ponens is constitutive ofour inferential practice. The first objection draws our atten-tion to the fact that modus ponens is not necessarily neededsince other rules could do the same job – namely, the disjunc-tive syllogism and the reductio ad absurdum. Let us take thedisjunctive syllogism as our case-study. It could be said thatinstead of using modus ponens in the inference needed to jus-tify a move made on the basis of a logical law such as the &-elimination rule, the inferrer can use instead the disjunctivesyllogism. Indeed, why not say that the crucial pattern ofinference is this one instead of (30)–(32)?:


(40) Either an argument does not display a valid logicalform, or it is valid.

(41) Argument x displays a valid logical form.(42) Therefore, argument x is valid.

The disjunctive syllogism, then, would do the job as well asmodus ponens as a candidate for constitutiveness. This will beparticularly the case for someone who works with a logicwhere ‘p fi q’ and ‘�p v q’ are equivalent. Indeed, if modusponens is equivalent to another rule of inference, then thisother rule of inference can be used in all places where modusponens is used, including where I claim that modus ponens isconstitutive. So I have to concede, on these grounds, that it ismodus ponens or one equivalent to it (disjunctive syllogism andthe reductio ad absurdum) which needs to be constitutive ofour inferential practice. This does not threaten the central idea,which is that there has to be, on an internalist model of infer-ence, a logical rule warranting the inference from the recogni-tion of the validity of a given argument-form ((31) and (41), inour examples) to the claim that the argument under study isvalid. This rule will be modus ponens or one equivalent to it.

What if, on the other hand, someone works with a logic inwhich modus ponens, disjunctive syllogism, and reductio arenot equivalent? In this case, the situation remains basicallythe same, since in any logic there are inferential moves madesuch that their justification will depend on an argument usingeither one of these three rules (or another equivalent rule).The crucial point still holds: a rule, either modus ponens orthe disjunctive syllogism or the reductio, is presupposed inany inferential justification. This is enough for our purposes.

The second worry is the following. Some paraconsistent log-ics, such as Graham Priest’s Logic of Paradox (LP), do not in-clude any of the three rules we just mentioned, namely modusponens, the reductio, and the disjunctive syllogism.l2 In thesesystems theorems are proved, results are achieved, etc, withouttaking modus ponens and the others as unrestricted rules ofinference. Prima facie, then, it seems that LP is a counterex-ample to my claim. But matters are not that straightforward:

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as we shall see, the absence of these laws in a logical systemdoes not mean that the inferrer does not make an implicit useof one of them when justifying his inferences. The reason isthis. LP, as a logic, includes rules of inferences. The paracon-sistent logician using this system will, under an internalist ac-count of inferential practice, be warranted in drawinginferences only if these two conditions are satisfied: the infer-ences drawn are made according to the rules of the systemand the inferences thus drawn are justified in the appropriatemanner. Thus, let us imagine that our paraconsistent logicianwishes to infer a conclusion q from a given set of premisesaccording to a rule R of his system. This logician will justifyhis inference by a piece of reasoning looking like this: ‘‘Mylogical system LP tells me that when an argument displays thelogical form exemplified by R, it is a valid argument. Thisargument has the required logical form. Therefore, I am justi-fied in drawing the conclusion q.’’ But this piece of reasoningis only valid if modus ponens is. So the justification for it willappeal to modus ponens – or one equivalent to it.13 Hence,even the inferrer using a logic which does not include modusponens must rely on something like it in order to be internallywarranted in his logical reasoning.

The foregoing considerations show why, once appeal to W

is required, the regress is unavoidable. It is because all rulesof inference presuppose, for their warrant, a further argumenthaving the modus ponens form that Carroll’s regress will gothrough for them. More precisely, it is because any explicitappeal to any rule of inference when justifying a particularmove from premises to conclusion is itself dependent onanother rule of inference – modus ponens – that there is a re-gress. Here we have the precise diagnosis of the regress: thesecond-order justification made with modus-ponens. Accord-ingly, the remedy to stop it will consist in removing the needto appeal to it in the way the regress demands.

To sum up, we have seen that under a broadly internalistmodel, modus ponens is constitutive of our practice of infer-ring in the sense that inferring requires a warrant involvingthis logical law. In other words, no internally warranted


inference is possible for us without using modus ponens. Incontrast, the law of excluded middle does not seem to be con-stitutive, in the sense just mentioned, of our practice of infer-ring. Indeed, it is not required when inferring using the lawof conjunction or the law of modus ponens. It has to beacknowledged that nothing in this argument excludes thatthere might be other constitutive logical laws. It is enough forour purposes that there is one such law.

8. AN INTERNALIST PROPOSAL

Now how does the distinction between constitutive andnon-constitutive logical laws help us in solving Carroll’s re-gress? Taking my lead from Davidson’s ideas about akrasia, Iwill suggest that we can draw a parallel between elements ofhis proposed solution of this problem of practical reason andthe present difficulty.

Akrasia occurs when an agent’s action does not conformto what she recognises as the right action to take, everythingconsidered. For instance, someone is offered a cigarette. Hereis how we could picture the agent’s practical reasoning. Onthe one hand, she considers that staying in good health isimportant and refusing the offer would be the right thing todo given that reason. Thus, according to this line of thoughtshe should refuse the offer. On the other hand, she pondersthat pleasurable things must be pursued and smoking wouldbe pleasant. Thus, according to this other line of thought sheshould accept the offer. Weighing her reasons, she arrives atthe following conclusion of her practical reasoning: all thingsconsidered, she should refuse the offer. Nevertheless, she ac-cepts the offer and lights the cigarette. She acts contrary towhat her own ‘all things considered’ (ATC) judgement pre-scribes her to do – she is an akratic agent. How can we ex-plain the phenomenon of akrasia? It seems irrational toaccept the offered cigarette, the agent recognises that it isirrational, but still she accepts it. Akrasia is a challenge foranyone interested in clarifying the nature of practical reason-ing, just as Carroll’s regress is a challenge for those inter-

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ested to get a clear view of the nature of our inferential prac-tice.

According to Davidson (1970), the explanation of akrasiais the following. The akratic agent is simply acting irratio-nally. The irrationality of the act resides in the fact that theagent ignored a central maxim of practical reasoning, namelythe principle of continence:14

Principle of continence: One ought to act on what one regardsas the best option, all things considered.

This principle prescribes acting in accordance with the (ATC)judgements formed as the outcome of practical reasoning. Inthe example above, this means refusing the cigarette – this iswhat she regards as the best option, everything considered.But she does not act on it. She is acting irrationally. David-son’s analysis contains many twists and subtleties that neednot concern us here, but let us focus on the status of theprinciple. In practical reasoning, one cannot choose to acceptor to reject the principle of continence since any instance ofpractical reasoning presupposes use of the principle. The veryidea of rejecting the principle would involve use of the princi-ple itself. Therefore, one cannot rationally reject the principle– it cannot be rational to abandon the principle since it isimpossible to give a reason for the rejection. This is the cruxof Davidson’s analysis of akrasia: the akratic agent is irratio-nal since she acts contrary to the principle of continence.Moreover, and crucially, according to Davidson the principleof continence ‘‘is not just another premise to be accepted orrejected. It is constitutive of the concept of a creature withpropositional attitudes capable of practical reasoning’’(Davidson, 1999, p. 404, my italics). So the principle is con-stitutive of practical reasoning and is no ordinary premise tobe added freely when justifying a particular judgement. Itdoes not and should not figure as a premise in practical rea-soning.

Similarly, we can say that since modus ponens, on an inter-nalist account of logical knowledge, is constitutive of ourpractice of inferring, it is impossible to reject it on rational


grounds.15 Its validity is grounded on the very practice ofinferring. The fact that it is impossible to rationally reject itreminds us of its special status: in contrast, it is possible toreject the law of excluded middle without being irrational –the validity of excluded middle is not grounded on inferentialpractice as such.

It is here that we can make the link with epistemic respon-sibility. In order to be a responsible inferring agent, you haveto obey modus ponens – the very practice of inferring (on aninternalist understanding of it) presupposes the validity ofthis law (that is to say, it presupposes W). You are clearlynot responsible if you do not obey the rule. Indeed, not obey-ing it would mean that you are not engaging in inferentialpractice, i.e., that you do not reason in such a way so thatyour thoughts follow a logical pattern. It’s irrational to en-gage in inferential practice without it.

In contrast, the validity of a non-constitutive rule such asexcluded middle can be discussed on rational grounds in away that is not possible for modus ponens. Indeed, one canreason without using excluded middle as a logical law – in-tuitionists do it all the time. Thus, it is at least theoreticallypossible to be a responsible inferrer and yet not reasonaccording to the law of excluded middle. However, no onesharing our inferring practice – including intuitionists andothers – can infer without acknowledging, via the very prac-tice of inferring, the validity of modus ponens. We have seenin the previous section that this is true even of paraconsistentlogicians: assuming an internalist model of our inferentialpractice, inferrers using a logic not containing modus ponensas a rule of inference must ultimately appeal to it whenthey justify a move from a non-empty set of premises to aconclusion in their system.

Obeying the constraints of rationality is certainly a mark ofepistemic responsibility, if there is any. It seems correct to saythat a thinker who obeys minimal constraints of rationality isnot merely showing his lack of irresponsibility – he is notmerely blameless. He is, naturally, responsible. We cannot, itis true, blame him. But it seems reasonable to acknowledge

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more in this case and praise him for satisfying minimal con-straints of rationality. He is not merely not doing anythingwrong – he is doing something right in virtue of his naturalrationality. To obey fundamental norms of rationality is aninternal matter. Indeed, what pertains to rationality lies with-in the thinking agent, not outside of him. Obeying modusponens is an internal matter, especially if we reflect that beingrational is an ‘internal’ notion.

Thus, the idea suggested remains internalist and yet, itavoids Carroll’s regress. Indeed, it is in virtue of our naturalrationality that we obey the rule of modus ponens, not in vir-tue of recognising, as a reflectively appreciable warrant, thatmodus ponens is a valid rule of inference. In order to justifi-ably draw inferences possessing the modus ponens form, it isenough to be an agent engaging in inferential practice –awareness of an explicit warrant such as W is unnecessary.

If this is right, then Carroll’s regress is blocked: W is notto be added as a further premise fulfilling a justifying role.Remember that in the exposition of the regress, it startedwhen knowledge of W had to be added as a further premise,namely the premise (a). But if modus ponens, because of itsconstitutive status, does not figure as an extra premise, thenthere is no regress. What happens when we are faced with anargument displaying the logical form of modus ponens is this:we recognise that the argument is of a certain form, and wedraw the conclusion – we are internally justified in doing soin virtue of our natural rationality.

9. CONCLUSION

Although some aspects of the proposal suggested need furtherclarification, it remains our best hope for a satisfying episte-mology of our inferential practice given the problems withexternalism and scepticism raised in sections 4 and 5. Themajor obstacle for an internalist account is Carroll’s regress,and the proposed solution deals with it in a manner avoidingthe pitfalls (illustrated by Wright) of using the condition of


meaning-constitution. Indeed, recognising the constitutiverole of modus ponens allows for a satisfactory grounding ofepistemic responsibility.

NOTES

1 Patrick Greenough, Gerry Hough, Jesper Kallestrup, Duncan Prit-chard and Stewart Shapiro provided me with very helpful comments onnumerous occasions. I am grateful to all of them. I would also like toexpress my gratitude to the anonymous referee for directing my attentionto crucial aspects of the argument. Special thanks to Crispin Wright,whose forceful and constant criticisms prompted me to revise and refinemany of the ideas contained in this paper.2 See Boghossian’s ‘‘How are Objective Epistemic Reasons Possible?’’(2001) and Wright’s commentary ‘‘On Basic Logical Knowledge’’ (2001).3 There are other interpretations of the regress. Quine offers one in his(1936, 103–104), and Wright mentions another one in his exchange withBoghossian, (see Wright, (2001, 76; and 84, footnote 19)). Discussion ofthese various regresses and their impact on the issue at hand would in it-self require another paper.4 Strictly speaking, a further requirement is needed. It must be the casethat claiming knowledge of W is always necessary – that is, it must be areasonable demand to always require, on the part of the inferrer, knowl-edge of the validity of modus ponens. This requirement seems plausiblegiven that one espouses simple internalism. Indeed, if the inferrer mustpossess an appropriate warrant to acquire knowledge by inference in onecase, then there seems to be no reason why we should not require such awarrant in the other cases. That is, if knowledge of the validity of modusponens is required to draw the inference (C), then it is also required todraw the inferences (c) and (iii). There does not seem to be any room foran internalist to stop the regress by saying that the warrant is only neededin the first instance but not in the others.5 The formulations of simple internalism and simple externalism areborrowed from Wright (2001, 51).6 The status of the norms thereby followed does not need to concern ushere. The sceptical stance simply does not allow norms, whether they areunderstood abstractly, naturalistically, or non-cognitively.7 The expression is Wright’s (2001, 52).8 I only mention this objection to externalism in passing, since I will re-fer to it in a later section and since discussing it in details would lead ustoo far from the goal at hand.9 See Prior (1966). To remind the reader, here are the introduction andelimination rules for ‘tonk’:

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Tonk-I :A

A tonk B

Tonk-E :A tonk B

B

By using these rules, it is possible to infer any conclusion from any givenpremise.10 Wright and Boghossian refer to this problem as ‘bad company’. SeeBoghossian (2001, 33–34) for his discussion of bad company in relation tomeaning-constitution.11 The introduction and elimination rules offered below are equivalent toHume’s Principle, which says that the number of F’s is the same as thenumber of G’s if and only if there is a one-to-one correspondence betweenthem, in symbols:

ðHPÞðNx : Fx ¼ Nx : GxÞ � ð9RÞðF1� 1rGÞ:12 See Priest (1979) for an exposition of LP.13 I will omit the qualification henceforward.14 See Davidson (1970, 41–42) on the principle of continence and its sig-nificance for akrasia. For further discussion, see for instance Lazar (1999)and Davidson’s reply (1999, 402–405).15 Rejecting it would require its use, since it would have to be the out-come of an inference. Thus, the mere attempt to reject it depends on itsvalidity.

REFERENCES

Boghossian (2001): �How are Objective Epistemic Reasons Possible?�,Philosophical Studies 106, 1–40.

Carroll (1895): �What the Tortoise said to Achilles�, Mind 4, 278–280.Davidson (1970): ‘How is Weakness of the Will Possible?’, in Essays onActions and Events (pp. 21–42), Clarendon Press.

Davidson (1999): ‘Reply to Lazar’, in Hahn and Lewis Edwin (eds.), ThePhilosophy of Donald Davidson (pp. 402–405), Open Court.

Lazar (1999): ‘Akrasia and the Principle of Continence or What the TortoiseWould Say to Achilles’, in Hahn and Lewis Edwin (ed.), The Philosophy ofDonald Davidson (pp. 381–401), Open Court.

Priest (1979): �The Logic of Paradox�, Journal of Philosophical Logic 8, 219–241.

Prior (1966): �The Runabout Inference Ticket�, Analysis 21, 38–39.Quine (1936): ‘Truth by Convention’, in The Ways of Paradox (pp. 70–99),Harvard University Press.


Wright (2001): �On Basic Logical Knowledge: Reflections on Paul Boghos-sian’s ‘‘How are Objective Epistemic Reasons Possible?’’�, PhilosophicalStudies 106, 41–85.

Department of PhilosophyUniversity of OttawaArt Building70 Laurier EastOttawa, OntarioCanada KIN 6N5E-mail: [email protected]

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Carroll’s Regress and the Epistemology of Logic

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