Problems for Infinitism

Infinitism in epistemic justification is the thesis that the structure of justification consists in infinite, nonrepeating series. Although superficially an implausible position, it is capable of presenting strong arguments in its favour, and has been growing in popularity. After briefly introducing the concept and the motivations for it, I will present a common objection (the finite minds problem) as well as a powerful reply which couches Infinitism in dispositional terms. I will then attempt to undermine this counterobjection by drawing parallels between it and the problems raised against semantic dispositionalism by Kripke’s exegesis of Wittgenstein’s private language argument.


I
One of the most obvious responses to infinitism is the finite minds objection.The objection itself if extremely simple, but its ramifications are rather complex.Given the assumption that we are in fact finite creatures (with finite minds), and given that propositional justification consists in infinite non-repeating chains, it follows that we can never have doxastic justification for any proposition whatsoever.Thus, infinitism so conceived renders us impotent against the sceptic who claims that we can never be justified in our beliefs, regardless of whether or not the proposition itself is justified.
The response of the infinitist is to claim that doxastic justification does not consist in the agent "containing" the entire infinite chain in her head, but instead in the agent having an appropriate second-order disposition such that, when prompted, she will justify her belief with the proposition which is in fact the next in the chain.Thus, although she does not contain in her head the entire infinite list of reasons, and the correct order, she has something similar to a schema the mastering of which gives her the ability to make any of an infinite number of steps, despite it being only finite itself.This is analogous to mathematical functions, or meta-linguistic rules of inference which can patently be defined over an infinite domain, yet fully grasped by an infinite agent.
2155-4838 | commons.pacificu.edu/rescogitansII I will now sketch what I believe to be an insurmountable problem for this dispositional response, with reference to Kripke's interpretation of the private language argument, or "Kripkenstein".The argument attempts (I feel successfully) to demonstrate that the notion of assigning some fixed, internal "meaning" to our terms is an incoherent one.This is shown by the infamous addition/quaddition dilemma.Define quaddition as the function Q such that for some z For all numbers x and y less than z, quaddition behaves exactly like addition Q(x,y) = 5 otherwise The challenge is to give a reason for supposing that we "meant" addition rather than quaddition with our use of "plus" (when z is sufficiently large).Both are equally compatible with the evidence.The standard response to this is semantic dispositionalism, which relies on a dispositional account of the meaning of our use of "plus".Thus, although quaddition and addition are both compatible with the evidence so far presented, had we been asked "What is z plus z?" we would have given the value of z+z rather than 5.The problem with this is simply that it misses the point.If there is a matter of fact about which function we mean by "plus", this results in normativity such that the agent is only justified in her answer to questions containing "plus" if she is doing it right and sticking to the semantic rule.In other words, we are not asking for an account of which response the agent would have given, but an account of what justifies that response.Merely giving a descriptive account of what an agent happened to be disposed to do is insufficient.After all, we are disposed to get it wrong sometimes, and given our finite lives we will never be able to answer certain "plus" questions.Simply giving an account of the speaker's disposition towards the use of the term cannot give us the meaning unless we can weed out these wrong uses.The fact that we are disposed to use the term incorrectly demonstrates that meaning cannot be defined as our dispositions.We are disposed to use "plus" in all sorts of ways in various circumstances, the important issue is to find out which pattern of usage is the right one.The crucial point is that there is no non-circular means by which we can pick out the circumstances in which the agent is using the word correctly.To see this, we only have to ask ourselves why we would not count certain "slip-ups" as also constituting our meaning of "plus".The only answer is that they do not count because they were instances in which we were mis-using the term.But this is patently circular.Similarly, the only way the semantic dispositionalist can line up our dispositions and the addition for immense numbers (which we could never actually calculate in our lives) is by stipulating certain properties (such as immortality, larger brain capacity etc.).But there are infinitely many ways of tweaking the agent such that she has these properties, and many result in different dispositions (addition-like, quaddition-like etc.) the only way the semantic dispositionalist can make use of these idealising thought experiments is to specify which tweaking is the right one.But again this is wholly circular, since 2155-4838 | commons.pacificu.edu/rescogitanschoosing an addition-like tweaking begs the question.This is the essence of what is dubbed the Problem of Multiplicity It is crucial to observe that the Kripkenstein argument does not undermine there being well-defined mathematical functions such as addition and quaddition.What is undermined is the coherence of designating one of these as the "meaning" of our term "plus".
We are now ready to construct our objection against the more sophisticated form of infinitism.As we saw above, assigning semantic properties on the basis of a disposition over an infinite domain led to severe problems.What about epistemic properties?For a proposition p, there are infinitely many infinite non-repeating chains which purport to justify p.Some work and some don't (they will have faulty steps in at least one place).Now, just as the sceptic above questioned the coherence of claiming that one function in particular could be extracted from our disposition as its "meaning", we can question the coherence of the infinitist's claim that one particular chain of propositional justification can be extracted from our disposition as the actual justificatory basis of the agent.The upshot of this would be this: since there is no matter of fact about which chain the agent is following, and doxastic justification depends on which grounds the agent believes the proposition, it follows that there is no fact of the matter as to whether or not the agent is doxastically justified.After all, we will occasionally slip up when it comes to justifying our belief when prompted.And as the chain becomes more and more abstract we will take longer and longer to provide the next step in the chain, such that conceivably somewhere along the chain the justificatory step will take longer than we live.Both of these observations are obviously parallel to the semantic argument, and we can see that in order for the infinitist to respond, he must be able to give a reason as to why certain justificatory steps we are disposed to make aren't constitutive of which chain of propositional justification which forms the grounds of our belief, and must give a reason as to why one idealised agent which follows chain A should be thought to represent the actual agent rather than another idealised agent which follows chain B. However, it should be obvious that this cannot be done.Just as the semantic dispositionalist cannot give a distinction between mistakes and the following of a rule which does not appeal circularly to the rule, so too the dispositional infinitist cannot explain why certain justificatory inferences which the speaker is disposed to make does not constitute his justificatory grounding without explicitly presupposing some inferences are mistakes.

III
Where do we go from here?The infinitist has a seemingly fatal problem on his hands: his theory entails that, not only can we never show whether or not an agent is doxastically justified in a belief, there is not even a fact of the matter about doxastic justification.The infinitist perhaps would look for important areas of dissimilarity between the epistemic and semantic arguments, and seek to show that these areas are crucial to the private language argument.Most prominently, he might point to the 2155-4838 | commons.pacificu.edu/rescogitansimportant notion of idealised conditions.The semantic dispositionalist ultimately failed due to his inability to give a non-circular account of those conditions which separated correct uses of the word from the incorrect ones i.e. he failed to account for the normativity which is essential to rule-following and thus private language.Can we assume that there is equally no such non-circular account of ideal conditions for justificatory inferences?It seems not.While it is obvious that in the case of language, misuses of "plus" can only be said to be so simply because it wasn't what the agent really "means", we seem to have a better grasp when it comes to epistemically ideal conditions.For one thing, they crop up a lot more in philosophical literature.Chalmers relies heavily on the notion of an ideal reasoner for his account of two-dimensional semantics.Putnam, Cripsin Wright and others use the notion of epistemically ideal conditions in one way or another to define truth.But this is not as promising as it might first seem.Chalmers speaks little or nothing about what ideal conditions consist of.Instead, he simply defines an ideal modal reasoner as someone whose epistemic space perfectly corresponds to metaphysical space, giving no reductive account of how this might come about.As for the accounts of truth, definitions of epistemically idealised conditions range from insufficiently spelled out to inadequate.One common approach is to define ideal conditions as having all relevant evidence readily available.This clearly fails as an adequate account since, as well as the relevant information, coming to a true belief requires executing the appropriate inferences following from that information.Our aim here is to demarcate when our justificatory disposition is being properly followed and when it is not, yet clearly, the idealised conditions mentioned above do not guarantee this in any way.One important point to note is that, even if we could find a reductive account of conditions which would result in infallible inferences, this would be missing the point.For what we are looking for is a reductive account of conditions when an agent follows their actual disposition accurately, and since people can ultimately be unjustified and thus following a faulty disposition, these conditions must make those agents make faulty inferences in the right places.Thus, this project in the end is less promising than it originally seemed.However, a more successful response might be to exercise a companions-in-guilt argument and claim that if the conclusion can be swallowed for something even more central i.e. meaning, perhaps the same can be done for justification.Indeed, Kripkenstein offers a sceptical solution which accepts the private language argument and provides a deflated account of meaning.Roughly, mastery of a word no longer consists in the internalisation of a specific rule, which has been shown to be an incoherent notion, but instead consists in the approval of the linguistic community as a whole.Thus, using "plus" correctly is not determined by whether or you are following a semantic rule properly, but instead by whether or not your behaviour and performance is approved by those around you.Perhaps the infinitist could say that ascribing justification is similarly a pragmatic process which occurs when the agent meets the 2155-4838 | commons.pacificu.edu/rescogitansstandards of his community.This ties in with what Klein suggests elsewhere for other reasons.Klein suggests that the ascription of doxastic justification is acceptable for pragmatic purposes even when the agent has only shown herself to be following the correct chain to the extent which the epistemic context requires (i.e. in everyday talk we do not have to check many steps at all before we can acceptable say that our conversational partner is justified).Of course, this differs greatly from the sceptical solution in that, when proposing this, Klein still believes that there is an over-arching fact of the matter as to whether or not the agent is justified proper.However, once this pragmatic idea is in place, the jump to a sceptical infinitist solution is not too far.
But the prospects of the infinitist appropriating this response appear dim when we note some particular consequences of the sceptical solution.One (obviously) is that the concept of private language is abandoned.There is no talk of what "meaning" an agent has in his head attached to each term and no talk of whether someone is using language properly without respect to a community.What you "mean" by a term (and thus what it takes for you to use the term "correctly") therefore consists merely in the approval of the community.Thus, a Robinson Crusoe from birth cannot speak a language and cannot be said to mean anything.This is an acceptable result (and even perhaps attractive one for someone won over by Quine), but the analogous consequence for the infinitist seems much worse.Are we to accept that Robinson Crusoe could never be justified?Does justification merely consist in your inferences satisfying the standards of your evaluators?Perhaps this is an acceptable conclusion to some.Maybe the reason why the conclusion that Robinson Crusoe would never be justified seems wrong is because we are imagining him and finding his inferences acceptable or unacceptable.The matter therefore requires us to observe the distinction between context and circumstance of evaluation.Here the analogy with language comes apart somewhat, since for us to claim his behaviour has meaning since the appropriate behaviour would only come about due to interaction with others.Those who are fond of deflationary accounts which, say, reduce meaning to communal approval or truth to agreement may be happy to define justification as being able to satisfy others in the community.
These deflationary accounts are of course highly contentious.Having to swallow this outlook in order to maintain their position in epistemology will undoubtedly be an unwelcome result to many infinitists.However, to those who are already sympathetic to them, this result may in fact be a positive find, a step towards a coherent view which spans the philosophical spectrum.