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Infinitism again

I'm thinking about my old infinite sequence of justification again:

Oliver Black (1996, 2003) has been arguing that the common tendency to dismiss any appeal to infinite sequences of justification is at least too steady. He points out that the existence of infinite sequence of justifications can be made plausible by the following argument (A) from 1,2 and 3 to 4 (Black 2003):
1. A belief is justified only if a justified belief is a reason for it.
2. There are justified beliefs.
3. The proper ancestral of the reason-relation is irreflexive.
4. There is an infinite sequence of justified beliefs each of which is a reason for its predecessor (he calls such sequence a J-sequence).
This argument, however, is often put aside because 4 is taken to be implausible. Black takes 4 to be dependent on two other issues:
Q1. Does there exist an infinite sequence of beliefs held by the same person?
Q2. If so, can such a sequence comprise only justified beliefs?
Of course, Black is supposing, here, that in order for a J-sequence to exist, it has to be such that all beliefs in the sequence are held by the same person. He then takes Q1 and Q2 to be relevant in a decision concerning whether or not we should accept that there are J-sequences. It is commonly considered that Q1 and Q2 should be answered negatively. Black then shows that we have reasons to answer yes to both Q1 and Q2. Here is an example of a sequence of beliefs that would satisfy what is required in Q1 and Q2: we have the (occuring or dispositional) beliefs that for any natural number n, n+1>n. This is a sequence of infinite beliefs that is clearly held by one person; each belief is justified by a standard appeal to number theory. So, we answer yes to Q2 and therefore to Q1. (Details in Black 2003).
That is of course not enough to establish whether there are J-sequences. In fact, our sequence of beliefs is not one where each belief justifies the following one no matter how we arrange the beliefs in a sequence. Black takes that the claim that 4 is implausible requires to be supported somehow other than insisting on a negative answer for Q1 and Q2. In fact, much has rested on the implausibility of 4; it is often part of an argument for some form of foundationalism and it is used to support the idea that justification cannot be merely a matter of reason-relations (therefore, negations of 1 and 3 above respectively). Without J-sequences it might seem that justification is either proceeding from a starting point or going around in circles. But Black offers reasons to suspect that the dismissal of J-sequences can be to hasty. This dismissal has to be grounded in something other than an appeal to the implausibility of sequences of justified beliefs. Black then points out that if an example of J-sequence is given, then 4 should no longer be held as implausible.
Now, here is an example of what I take to be a J-sequence:
S justifiedly beliefs that 'x is red' because S believes she is justified in believing that she knows what is red, and that she is justified in believing that she knows what it is to know what is red, and that she is justified in believing that she knows what it is to know that she knows what is red etc.
Each belief in the sequence is justified by the next one and yet every belief has to be present if S is to justifiedly belief that 'x is red'. The justiifcation of one belief requires the justification of all the beliefs in the sequence. It seems to me that this is a J-sequence. So, for example, Sellars (1956) argues that in order for one to know by observation that 'x is red' one needs to know that she is a reliable reporter of red things; this, however, is possible only if she knows she is a reliable reporter of a reliable reporter of red things etc. A J-sequence needs to be invoked to justify the observational report. If this is so, we make use of J-sequences all the time and most cases of justification seem to invoke an infinite regress. J-sequences seem to be not only possible but abundant.
My example of J-sequence, however, depends on the acceptance of an internalist view of knowledge that has that one needs to know that one knows in order to know. According to internalism, justified beliefs are such that the believer can provide justifications for what is believed. Epistemological internalism has become less popular in the last years; causal theories of knowledge, different forms of reliabilism, naturalist takes on knowledge and other sorts of externalism have criticized the need for one to be able to provide a justification in order to be justified. Externalism breaks the need for an infinite sequence of justifications by claimimg that one can be justified without possessing the justification. The justification doesn't need to be in the head (or, at least, doesn't need to be in the head of the believer). The manoeuvre to cut finite a J-sequence can be considered in comparison to what is often said about the famous infinite regress requirement for Modus Ponens attributed to Lewis Carroll:
1. If p then q
2. p
3. If 1 and 2 then q
4. If 1, 2 and 3 then q
We claim that a conclusion can be drawn from 1 and 2 only because we take the meaning of the words––the connectives––to be established somewhere else and not in the argument. The meaning of these words constitute what makes the rule of inference an effective constraint on what we think. This authority cannot be given solely by any number of rules as Wittgenstein's (1953: 185-201) remarks on rule-following make clear. Something external to the rules has to play a role. Similarly, something external to a J-sequence has to play a role if effective justification is to happen. Externalism takes justification to happen somewhere outside a J-sequence (and the head of the believer) for the sequence itself will always fail to provide any justification.
A J-sequence can be taken as an infinite deferral of justification, nothing is ever justified but justification is postponed for good. We can counter this by saying that if the sequence is to be understood as actual, Cantorian infinite then justification is no more than suitable infinite deferral. This would maybe count as an outline of a defensible (internalist) theory of justification: if we can place a belief in the end of a (suitable) J-sequence, then it is (sufficiently or tentatively) justified. I cannot give a list of sufficient and necessary conditions for a J-sequence to be suitable but a plausible necessary condition would be that it is expressible in a recursive manner. A suitable J-sequence is one that can be expressed in a finite number of recursive clauses.
Such sketched theory of justification is not compulsory, not even for internalists, because we can somehow find the first argument (A) above not cogent. One can be a foundationalist (deny 1), or a coherentist (deny 3) or even a skeptic (deny 2). The externalist, on the other hand, can insist that a justification is in the world (or in our practices) and our beliefs are not isolated from the world that justify them.1 The externalist can deny both 1 and 3 in A.
Further support for an infinite regress (internalist) account of justification can be provided if we consider a similar infinite regress account of what it takes to establish that a belief is true. Consider the following thesis.
(T) In order for us to establish that a belief b is true we have to establish that the belief 'b is true' is true and also that ' 'b is true' is true' is true and etc.
At first sight, competing accounts of what it takes to establish that a belief is true would be of three kinds: 1) an appeal to a starting point, something that has always been established as true, like a foundation for all other judgments of truth; 2) a refusal to take any belief as true and 3) a claim that at some point ' 'b is true' ... is true' can be established as true only by establishing that b is true. These three alternatives resemble the steps in argument A. If one is persuaded that none of these alternatives are encouraging, T seems reasonable. Now, one could think of a further alternative to T that would have an externalist spirit to it:
(ET) One does not need to establish the truth of 'b is true' in order to establish the truth of b.
This externalism about truth seems far less reasonable than externalism about justification. It seems like ET harms Tarski's material adequacy condition for truth: the locution 'is true' can always be added to a sentence if its translation is held. If we find reasons to hold that b is true, we have already found reasons to hold that 'b is true' is true.
If it is reasonable that we have to establish an infinite number of truths in order to determine a single truth, it is also plausible to extend the idea to justifications and accept J-sequences as a starting point for an internalist infinite regress epistemology. While it is not compulsory, it is a competing theory of justification: infinite regress justification cannot be dismissed easily by an internalist epistemology. Maybe, indeed, what is needed to make internalism plausible is precisely to bite the infinite regress of justification bullet. J-sequences, the starting point for an infinite regress epistemology, seem to be everywhere.


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