In order to know, you've got not only to believe something, what you believe has got to be true, and you've got to have a good reason as for why you believe it is true.† But how do you know that's a good reason?† It itself has not only got to be true, but you've got to have another reason for why you hold it is.† And then for that reason, another.† Then another.† Then we're off to the races.
Virtually nobody, though, thinks we should run that very long race.† There are a variety of ways to shorten the distance.† Foundationalism is the thesis that we may stop giving reasons with a special set of beliefs called basics.† Coherentism is the thesis that at a certain level of reason-giving, we may stop when the beliefs in question have a special fit with the rest (or some special set) of our beliefs.† Contextualism is the thesis that we may stop with beliefs appropriate for the given circumstances.† Externalism is a rejection of the requirement that subjects need to give or be aware of further reasons when they obtain.†
Examples of philosophers and whole traditions abound for each of these theories of ways to leave off giving reasons, and we learn early in our philosophical education the criticisms these camps exchange.† But it is difficult to find real examples of the tradition that we might call infinitist, where reasons may iterate infinitely.† On the one hand, the old Pyrrhonean skeptical tradition has a version of accepting the requirement on knowledge, but rejecting the project of knowing.† This seems to be a plausible reading of the Agrippan modes and Sextus' use of them, but it is not a satisfying epistemological theory, since we are here looking for a theory of justification we can have.† The thesis that we should just give up on knowledge isn't enough.† Sometimes, in passing, Peirce is mentioned as a candidate for holding the theory, but little more is said.† Recently, Peter Klein (1999 and 2001) and XXXXXXXX have argued for the view and they have not only given accounts of the view's intelligibility but also its resistance to traditional arguments against it.
In light of these new developments in articulating and defending the view, I want to retrieve the question as to whether Peirce was an epistemic infinitist.† This is for two reasons.† First, it is important that we have an adequate textual backing for these classifications, even if they are tentative and little hangs on them for current epistemological theorizing.† We are scholars, and when we do scholarship, we should do it right.† Second, in pursuing another possible example of a philosophical view, we might deepen and broaden our conception of it.† Peirce's work is innovative in a number of areas, and it stands to reason that his views relevant to our current issue will be helpful.
I will argue for the following theses.† First, that infinitism may be understood weakly or strongly as a set of requirements for reason-giving.† Second, that weak infinitism is dialectically superior to strong.† Third, that Peirce is explicitly infinitist in his early work, namely the 1868 series of articles, "Questions Concerning Certain Faculties Claimed for Man," "Consequences of Four Incapacities," and "The Grounds of Validity."† Fourth, that Peirce's phenomenology of the relations between firsts, seconds, and thirds favors a weak infinitsm's mixed theory of justification.† The conclusion, then, is that were Peirce an infinitist, he would be a weak infinitist.† However, the prospects for this infinitism depend entirely on the prospects for Peirce's early semantics, which are not good.† Peirce himself revised the semantic theory later, and in so doing, it seems also his epistemic infinitism.
1. Two infinitisms
It is a common strategy for philosophers to use models to represent philosophical theses.† Sartre's empty box for the pour soi, Plato's divided line for the distinction between knowledge and opinion, possible worlds.† Epistemologists are no different.† A graphic representation of the infinitist (as well as a variety of other epistemological) thesis is with evidence- or justification-trees (J-trees).† Let some S believe that p on the basis of q and r, and q on the basis of s, and r on the basis of t.† The J-tree would look as follows:
J-trees have branches and nodes.† The nodes are the propositions S needs to justify S's belief that p.† The branches are the justifying relations between the propositions.† One important feature of J-trees is that, for our purposes, they must be pruned.† That is, they must represent only those nodes necessary for p's justification.† By pruning J-trees, we can represent the justificatory structure for a belief's dependence on other beliefs for their justification.† With pruning, we capture the thought behind the process of publicly justifying our commitments – we follow the rule of quantity when we give reasons for beliefs and decisions, and the rule is that you give only what's required to justify what's at issue, and that over-justification is overkill.†
Second, we should note that each node of the tree may confer justification to a higher node only if it itself is justified.† Without this requirement, all support for the proposition at the apex will only be conditional support.† So, a belief that p is justified by a belief q (or the set of beliefs q1-qn) only if q (or the set q1-qn) is itself justified.
It follows that J-trees should be considered iterated series of sets of propositions, so that the first set of propositions (p) is justified by the second set (q and r), and that set's justified by the third (s and t), and so on.† J-trees represent belief-structures, and what's required of the structures (in the context of justifying the belief in question), there are levels of justification – the first level is justified by the second by the third, and so on.† So, the schematic structure of J-tress is as follows:
††† †††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††/††††††††††† \
†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† /††††††††††††††††††† \†
††††††††††††††††††††††††††††††††††††††††††††††††††††††† Q1††††††††††††††††††††††††† Q2††† †††....
††††††††††††††††††††††††††††††††††††††††††††††††††††† /† /††† \††††††††††††††††††† /†† /†††† \
†††††††††††††††††††††††††††††††††††††††††††††††††† /†† /††††††† \†††††††††††††† /††† /†††††††† \††
††††††††††††††††††††††††††††††††††††††††††† R1† R2 ...†† Rn†††† Rn+1† Rn+2 ... Rm(m>n+2)† ....
Given this structural representation, a subject's justification for believing that p is the J-tree with p at the uppermost node for her.
A few terminological conveniences arise for epistemology in light of the model.† Foundationalism is the thesis that there must be terminating nodes for all branches of J-trees.† A basic beliefs needs of no J-tree itself for its justification.† Coherentism is the thesis that at a certain level on the J-tree, whole sets of beliefs play the role of terminating nodes.† Contextualism is the thesis that there may be terminating nodes on J-trees, but their status as such depends on some non-epistemic feature (e.g., to whom one is speaking, its usefulness, one's purposes, who is doing the evaluating...).
Infinitism is the thesis that reasons may go on infinitely. Given the J-tree model, we can formulate the thesis two ways.† A strong version of the thesis is that all branches of J-trees must be infinitely long.† There are no terminating nodes on J-trees.† A weak version of the thesis is that there may be some non-terminating branches of J-trees. So, there may be some terminating nodes on J-trees, but it is not a requirement that all terminate.† Graphically, the two theses can be captured as follows:
Strong Infinitist J-tree:†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Weak Infinitist J-tree:
††††††††††††† P††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† P
††††††††††† /†† \ ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††/††† \
††††††††† Q1†† Q2 ...††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Q1†† Q2
†††††††† /††††††† /††† \††††††††† †††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††/††† \
††††† R1††††† R2††† R3...††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† R1†††† R2
†††††† /††††††††† /††††††††† \†††††††††† ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††/†††† \
††† S1††††††† S2††††††††† S3 ...†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† S1†††† S2
†††† /††††††††† /††††††††††††††† \††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† \
††† ...††††††† ...†††††††††††††† ...††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ...
The thought of representing these trees as such is that on the strong infinitist tree, every branch of the tree requires a further reason iterated for each node on the tree.† On the strong infinitist requirement, no belief can be a terminating node on the J-tree, and as such, all J-trees are infinitely large in every direction for every branch.† On the weak infinitist model, there may be infinitely long branches of J-tress, but there may also be terminating nodes.† As such, weak infinitism is a mixed epistemological theory, as it allows that basic beliefs may play a role in the justification of a subject's belief.† However, it is not strictly a foundationalism, as the weak theory is not a requirement that all branches on J-trees have terminating nodes.
The structural difference between weak and strong infinitism is enough for the two versions of the infinitist commitment to be distinct, but there is a dialectical consideration that distinguishes the two.† One leading objection to infinitism is that infinite chains of reasons are, when considered as mere relations between beliefs, arbitrary.† The thought was captured early by Max Deutscher:
Could it be one vast delusion system?† Is a man reasonable in holding one belief merely because he holds another whose propositional content is suitably related to the first, even if he holds the second on account of a third which is suitably related to the second, and so on?† Might not a man just dream up a system and be ingenious enough to always extend his story in logical fashion? †How can the mere continuous extension of a belief system guarantee the rationality of the members of the system? (1973, 6)
We can make good on this worry by formulating the following way of extending reason-giving in a purely logical fashion.† Let the following beliefs stand in the support relation such that B1 is supported by B2, and B2 by B3, and so on.
B2: q & (q … p)
B3: r & (r … (q & (q … p)))
We can see clearly that if B2 is justified, it entails B1, so B1 is justified so long as B2 is.† And B2 is justified by B3, and B3 by B4, and so on.† So long as modus ponens is truth preservative and epistemic justification follows what are intuitively truth indicative inferences (these are at least plausible assumptions), p, on the infinitist thesis, is justified.† But now note that if we change p to not-p, we get the following set of beliefs:
B2': q & (q … ~p)
B3': r (r … (q & (q … ~p)
The problem is that on the infinitist theory, we have no rational way of telling the difference between one justifying set of beliefs that is conducive of truth and one that is not.† Infinitist justification, precisely because it is defined exclusively in terms of relations between beliefs, looses its moorings with truth.† Epistemic justification that does so loses the name.
But here the issue can be resolved by a question as to whether the sets of beliefs (B1...Bn) and (B1'...Bn'), if they are unadjudicable between their claims on the truth of p and ~p, are the only necessary components for a justificatory story.† If they cannot be rationally adjudicated by their own contents and inferential relations, then ex hypothesi, there must be some other feature that is necessary.† Insofar as any J-tree will be subject to the same structural issue (which is necessary, given the syntax of J-trees), only beliefs not in need of J-trees can play the role of rationally adjudicating the two set's claims on truth.† There, if there are infinitely iterating branches of reasons, must also be terminating nodes on those branches, too.† So, the difference between the two sets can be captured between the cognitive situation for some subject between B2 and B2':
B2: q & (q … p)
B2': q & (q … ~p)
Both simplify to:
B2a: q … p
B2'a: q … ~p
Given that J-trees are supposed to be snapshots of a subject's beliefs, and that a condition for belief is understanding, any subject caught in this dilemma must be capable of understanding the difference between these two conditionals.† One may ask our subject, "When q, not p?† Are you sure?"† A subject's understanding of the propositions may, it seems, provide immediate justification for accepting one conditional over another, say in cases of their relation being relations like the following:
∑ Analytic (or logical) entailment: If Sam's a bachelor, then he's male.† If something is blue, it's colored.† If something has properties F and G, it has property F.
If you drink a lot of that beer, you'll feel light headed (at least).
We can tell the difference semantically and epistemically between B2a and B2'a when they are formulated as such.† We may need further arguments to fix the consequent as justified, but we can tell the difference between and even assign justificatory status to one or the other depending on our understanding and experience relevant to these propositions.† If the foundationalist thesis is broadly right that understanding, moral sense, and experience provide at least a prima facie epistemic difference between propositions, we have a way of rationally adjudicating the two competing sets.† The conclusion, then, is that weak infinitism is dialectically preferable.† We have a means of deciding for ourselves which story is right.† And that means has nothing to do with beliefs that are inferentially justified, but those that are immediately so.
2. Peircian Infinitism
My argument is that a Peircian infinitism can be pieced together keys on the famous 1868 Journal of Speculative Philosophy paper "Questions Concerning Certain Faculties Claimed For Man."† The paper's main purpose is to show that all cognitions are inferential, not immediate or intuitional.
On the one hand, Peirce seems sympathetic to the issues motivating the foundationalist program – namely, the regress problem.† Question 7 is whether there is any cognition not determined by previous cognitions.† Are there intuitions?† Peirce notes that:
It would seem that there is or has been; for since we are in possession of cognitions, which are all determined by previous ones and these by cognitions earlier still, there must have been a first in this series or else our state of cognition at anytime is completely
determined according to logical laws, by our state at a previous time.
He then notes:
But there are many facts against this last supposition . . . . [It] is impossible to know intuitively that a given cognition is not determined by a previous one, the only way in which this can be known is by hypothetic inference from observed facts.
The argument yields for the conclusions (i) only beliefs can provide support for the other beliefs, and (ii) only inference can provide that support between beliefs.
The Peircian definition of an intuition on the semiotic theory is that intuitions are signs that refer immediately to their objects. (5.213).† So the question of the regress problem is whether there can be immediate reference.† Peirce's requirement for immediate reference is that if it is immediate, it must be immediate that it is immediate.† That is, we must intuitively know when we have intuitions.† Let us call this the meta-requirement.† So the difference between intuitive and non-intuitive knowledge itself must be intuitive.† Given that there has been (up to Peirce's time and up to now, too) debate about what is intuitive knowledge and not (the rationalism-empiricism debate, the current arguments against consciousness), it seems clear that such intuitions do not obtain.† The story given the meta-requirement, then, would be that for the second-order-intuition to be intuitive, we would need a third intuition, and a fourth for that.† Given that the second-order intuitions do not obtain and given that they are requirements for intuitions qua intuitions, the first order do not obtain either.†
Peirce's argument here depends crucially on the meta-requirement for intuitions.† We get an argument for the meta-requirement in what might be called the 1868 sequel for the "Questions" paper, "Some Consequences of Four Incapacities."† The thesis of the paper is that all mental events are results of the manipulation of signs.† Thoughts have a combinational component to them; they, to be thoughts at all, must be well-formed-formulae.† The components of these formulae are signs – feelings, images, emotions, representations.† For these thoughts to have content and for them to be of something, they must not only be well-formed, but they must be meant.† How thoughts have intentionality, in this account, is that signs have objects, because they are interpreted as being of something by other signs.† Without the interpretant, the original sign loses its meaning.† It is therefore essential that signs come as sets (not just subsets of well-formed formulae) but as sets that mean.† An analogy might make it clear:† our thoughts, as combinatorial entities, are like sentences, and sentences do not mean anything unless they are read and interpreted.† So it goes for thought.† (Or, more precisely, for logical comprehension of thought).† The rationale for the meta-requirement, then, is that the condition for a cognition to exist is for it to be known to.† As long as there is a problem with nailing down whether or not there are intuitions, they are not known to exist.† So they don't.† Pierce anticipates a familiar regress of analysis problem (an analogue of the epistemic regress) in light of this requirement.† If we cannot have thoughts without signs, then it seems for all interpretations, there must be interpretations of them, and then interpretations of them, and we're off to the races, but this time on a semantic level.† But Peirce accepts an infinitism on this level also, as he notes that the alternative requires that there be intuitions that play non-derived inferential roles.† To hold that there must be such notions "assumes the impossibility of an infinite series.† But Achilles, as a fact, will overtake the tortoise.† How this happens, is a question not necessary to be answered at present, as long as it certainly does happen."So Peirce is not only committed to there being a requirement of infinite analysis, he is committed to the thesis that these infinite analyses are actual.† ††Just as Achilles actually catches the tortoise, and in so doing traverses an infinity of spaces, we mean, and in so doing, involved in an infinity of interpretations.† The question, then, is how exactly this is brought to actuality.
The levels of sign-interpretation are triadic.† First, are emotional interpretants; second, energetic interpretants; and, third, are logical interpretants.† Most certainly thirdness is our primary object of interest for epistemology, and Peirce's argument against intuitionism is valid on the level of thirds.† It is only when thoughts are interpreted on this level that they may bear inferential relations with each other, and achieve the definiteness requisite to count as beliefs capable of either being justified or conferring justification.† However, this emphasis being exclusive to thirds is short sighted.†
Secondness has a place in the Peircian epistemic program, especially if we are considering the possible place of non-inferential justification for intuitions.† If an intuition is a thought that immediately refers to its object, instead of looking at the mediacy of reference on the theory of signs, the possibility of direct reference may be found in the ontology of the signs themselves.† The place of seconds in the semiotics is the requirement that for thought to have the determinant logical content it must have to be the kind of thought necessary for knowledge, something must exist, there must be something that can be knocked up against, something in consciousness that cannot be thought away.† Something that forces our determinate acknowledgement.† The place of secondness in our interpretive triads is that it is a condition not just for the being of the objects of our thought, it is a condition for the existence of the thoughts themselves:
For as long as things do not act upon one another, there is no sense or meaning saying that they have no being, unless it be that they are such in themselves that they may perhaps come into relation with others.
Guy Debrock (1997, 27) has termed this requirement 'the Peircian ontological principle' if there is no secondness, there is nothing at all.† Thirdness, as the nature or habits of thought requires that there be occurrences suitable to form those habits.
The consequence here is, then, that thoughts about the seconds that give rise to them are themselves referentially direct.† A person may say, "I am being appeared to thusly" or "He said that the car wouldn't look like that," or "Oh, this is how it feels to win the race." In making such references, the speakers do not rely on a context for their statements to have reference, and the speakers must be both aware of the objects of reference and capable of articulating what such references entail.† But the awareness of what they are experiencing is not provided by their awareness of the context or inferential and practical commitments or consequences.† We take aspirin because headaches hurt, not vice versa.† We say a car is brightly painted because it is canary yellow, not that it is canary yellow because it is bright.† These seconds, the pressure of the headache, the brightness of the yellow, have a force that we feel and articulate with assent.† Most certainly there is another interpretive element to bringing this force to full belief, but the interpretation depends on the material component of the sign for its impetus (CP 5.292).
So there is a sense in which, on the Peircian theory, there can be non-doxastic support for beliefs.† Seconds impel us to formulate commitments.† These formulations themselves, on the semiotic, require further inferences, but the impetus itself is not inferential.† It is the condition for inference.
Peirce's argument from "Questions" depends on a robust Cartesian requirement for intuition – namely that the intuition provides certainty.† From this thought, the meta-requirement for intuition arises.† If we are to defeat skepticism, then we must not only show that we know, but show that we know when we know.† The mistake was to take the requirement unqualifiedly to entail that certainty, infallibility, incorrigibility and the like could provide the goods for such a demonstration.† But intuitions may come in all forms, and Perice's own semiotic is posited on requiring that some non-inferential support function at the heart of the interpretive enterprise.† The conception of the progress of inquiry itself requires some feature of non-inferential, defeasible support:
Besides positive science can only rest on experience; and experience can never rest on absolute certainty, exactitude, necessity, or universality (CP 1.55).
And in the 1903 Harvard lectures, Peirce is clearly committed to the thesis that "perceptual judgments are the first premises of all our reasonings" (5.116), and that "all our knowledge rests on perceptual judgments" (5.142).† Peirce was (at least by 1903) committed to there being asymmetric and non-doxastic support, but he denied that the criteria for such a role was to be the restricted categories arising from the project of refuting skepticism.† It does require that if there is knowledge, there is non-inferential support.
However, it seems clear that the argument from the meta-requirement in "Questions" and "Consequences" is still in play.† Beliefs bearing this indexical relation to the seconds from which they arise must still have further beliefs mediating their inferential relations with other beliefs. Sensations so far as they represent something are determinate according to a according to a logical law by previous cognitions (5.291).† That we have such and such an event may be provided by the index, but what the sensation is of must be provided by the event's relation to other beliefs.
Peirce's early theory of justification, then, is committed to two seemingly inconsistent theses.† On the one hand, there must be an inferential or interpretive feature to all epistemic support. On the other hand, there can, and it seems must, be a non-doxastic support for premises for them to even exist and be justified at all.† Requiring that all justificatory support be inferential, but also allowing seconds to justify seems contradictory.† But remember that the weak infinitist story is that it is intelligible that doxastic and non-doxastic support can appear on J-trees.† Terminating nodes may occur on J-trees on the requirement that they do so in combination with support from a non-terminating node.† Further, the terminating nodes themselves are revisable and testable given experience – they are observational judgments justified by the contents of the experience.† But they do not play a singular role in the cognitive life, as their relation to other beliefs is mediated by other beliefs.† For a subject's belief that she is being appeared to like this to be relevant to her other beliefs, the appearance must be interpreted as a representation of something, or that it is a symptom of being in some situation that gives rise to certain representations.† Given that J-trees represent the necessary components of a belief's justification (the trimming requirement from section 1), the theory runs that no terminating belief is sufficient for the justification of another belief, but that they may always be accompanied by a further non-terminating node.† So, if there is a terminating node supporting a belief, there must also be a non-terminating node, too.
Peirce explicitly rejects the notion that his argument shows that there must be a first cognition.† His arguments are precisely designed to show that there are not first cognitions.† Rather, what the arguments have shown is that inference is necessary for any cognitive element of life to have semantic or epistemic value.† This is not to deny that our cognitive states never begin rationally.† Instead, Perice's argument shows that cognition arises by a process of unfolding (CP 5.263) where non-inferential material features of signs are the motivating features for cognitions.† He notes in the third paper of the 1868 series:
[I]t does not follow that because there has been no first in a series (of premises), therefore that series had no beginning... for the series may be continuous, and may have begun gradually... (5.327).
It is clear that the Peircian Perspective in the 1868 articles ties the question of cognition's content closely to its epistemic status.† The double role of interpretation is the locus of this connection.† For a thought to mean, it must be interpreted.† Some interpretations are better than others by their responsiveness to the signs and their syntax, and as such some are justified or not.† Interpretations have both semantic and pragmatic roles to play.
One question is which role is conceptually prior.† Could Peirce, say, be a semantic infinitist independently of his epistemic infinitism, or vice versa?† One thought regularly expressed among contemporary epistemologists is that epistemological theories cannot be articulated independently of a semantics.† If we are interested in inferential justification, then some notion of the semantics of the variables between the logical function must be implicated.† Inference is a semantic notion.† A justified inference, then, is a conceptually dependent notion – we've go to get our semantics nailed down to do epistemology right.
Peirce's project is structurally similar – it is Peirce's semantic notion that thought must be understood as inferential that yields (a) the meta-requirement for intuitions, and (b) the intelligibility of a regress of interpretations, and (c) the demonstrative relation between thoughts and the seconds that give rise to them.† Peircian infinitist epistemology rides piggyback to Peircian infinitist semantics.
But Peirce's 1868 semantics were flawed on two fronts.† First, because Peirce's theory of indexicals was inadequate to the task.† All the demonstrative indexes are limited to seconds and thirds in consciousness. This leaves Peirce's semantics incapable of analyzing reference to objects in the world.† Peirce's case for demonstratives in "Consequences" is that they bear a "real physical connection of a sign with its object, either immediately or by its connection with another sign" (CP 5.287).† The question, then, is how these connections are established.† On Peirce's semantics, it could only occur as a static system of relations between judgments.† That is, if intuitions have the meta-requirement, then demonstratives do too.† As a consequence, on the Peircian model, there is no escape from the circle of judgments.† Peirce must have realized this problem, as "Fixation" in 1878 has it such that "external objects affect our senses according to regular laws, and that we may, on the basis of this knowledge, come to know them." (CP 5.384).† The mediacy of judgments and interpretations of knowledge of things other than judgments had been dropped.
Second, it is unclear how Peirce's semantics of interpretation in the 1868 essays has anything to do with truth.† We know that signs, in order to mean, must be interpreted by other signs.† But we have no criteria for what a correct or incorrect interpretation is.† Peirce must †be committed to this difference, as it is crucial to save his theory that all thought is modeled on valid inference.† He explains cases of fallacious reasoning as cases where one misinterprets or confuses a rule of inference, and thereby uses the wrong one. (CP 5.282).† But what is this notion?† Are the correct inferences ones established by other inferences?† How, then, could they be distinct from the wrong ones, since ex hypothesi they are both logically determinate enough to be cognitively significant?† If it is a brute fact of the matter which is right, then there are components of consciousness that have their cognitive contents independently of interpretation.
If the theory were extended to epistemic evaluation, we would get the manifest absurdity on the following argument from analogy:
Surely this would be unacceptable, as we most certainly have epistemically unjustified beliefs.
A closing question is whether another Peircian epistemic infinitism could be abstracted from the later (1901) model for inquiry and truth:
Truth is that accordance of the abstract statement with the ideal limit towards which endless investigation would tend to bring scientific beliefs. CP 5.565.
The problem for such a model for our purposes is that it underdetermines the theory of justification necessary for the task. †Endless investigation does not mean infinite brances of J-tress.† Take a J-tree with a finite number of nodes with p at its apex.† Endless investigation of p's truth, at any time, may change which nodes are on the tree, change their order, or add new nodes.† Or it may reveal that the tree does not provide the requisite support for p by defeating some branch necessary for p's justification.† Or it may add new nodes to the J-tree infinitely, so that the tree may stretch in all brances (as the strong model would go) or may only have one (as allowed by the weak model) to infinity.† But there is nothing to the notion of infinitely extended inquiry that means that the J-trees it yields are infinitist (strong or weak) or finitist.† Endless inquiry, I believe, is indicative of the fallibilist component of Peirce's epistemology and its requirement of revisability than it does with a model for infinitism.† We may be justified now, but that does not make our beliefs immune to correction when new information comes in.† We may be open to infinite iterations of correction and revision, but such a structure itself does not require that the supporting reasons themselves be infinite.† In this respect, Peirce's later theory of inquiry may be called epistemically infinitist, because his epistemology does not rule out the possibility of infinitely extended branches on J-trees, but such an infinitism (as opposed to his views in 1868) is hardly explicit enough to merit little more than a qualified attribution of the view.† The later view is, on what's given, consistent with infinitism, but hardly an endorsement.
Peirce was an infinitist in his early work, namely in the 1868 series of articles in The Journal of Speculative Philosophy.† I've argued that the infinitism may be understood on what I've termed weak infinitism.† However, the infinitism was held on the basis of a flawed semantic theory.† Peirce himself later recognized these defects, and in correcting them, it seems he also took leave of the epistemic view.† In any case, he no longer felt a compulsion to explicitly endorse the view.† (5433 words)
Augustine. Contra Academicos.
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 See Sosa (1974, 113), Cornman (1977), McGrew (1995 and 1999), Cortens (2002, 25), and BonJour (2003, 10).
 See Clark (1988, 347) for the requirement formalized: (p) [ Jp → ($q) (Jq & qRp)].† The requirement, then, is that each node of S's J-tree must be justified for S.† See also Sosa (1974, 115), Cornman (1977, 291), Chisholm (1977, 17), and BonJour (2003, 10).
 Klein (1999 and 2000) may be reasonably interpreted as committed to a version of the strong thesis, as his argument for infinitism is posited on a denial that there can be any basic beliefs (or terminating nodes).† XXXXXXX may be interpreted as a weak infinitist, as his analysis of the regress problem allows that some beliefs may be immediately justified, but that immediate justification may not the only necessary component of a node's justification.
 Versions of this argument and a simpler form that I call 'the simplification reductio' (forthcoming a) have been made by: Deutscher (1973, 6), Pollock (1974, 28-9), Oakley (1976, 227-8), Cornman (1977, 290), Foley (1978, 313), Post (1980, 32-5; and 1987, 88-91), Moser (1985, 67), and Cling (2004, 110).
 See Davis (1972, 8) for an account of the meta-requirement.† Note, also, that this requirement is analogous to the Pyrrhonean requirement for regress-ending hypotheses (PH 164-177) and also has found other forms in BonJour's meta-justifications argument (1985, 30-3), and have figured widely in response to foundationalism.† See Oakley (1976, 222-3), Possin and Timmons (1989, 206) and Klein (1999, 277-9).
 Compare with CP (6.338)
 CP (5.262)
 CP (5.250)† (Peirce's emphasis)
 CP (1, 25)
 See Peirce's discussion of pure demonstrative applications of thought-signs.† (CP5.296)
 Cf. Delaney (1993, 89 and 111-8) who takes this point as indicative of a weak foundationalism in Peirce's early work.
 This strategy is noted by Floridi (1997, 54), who extends Perice's response to skeptics as an argument for convergent realism.
 See CP 5.318 for Perice's explicit endorsement of this thesis.
 See Thompson (1978, 79) and de Waal (1996, 436) for versions of this problem for Peirce's early semantics.
 Hookway (2002, 18 and 29-30) notes this difference and attributes Peirce's change of mind to Royce's influence.† See also Short (2004) for an account of the development of Peirce's semiotics and theory of indexes.
 Author's acknowledgements will appear here.