In Defense of Infinitism

Preface

This essay will attempt to the problem of regressive epistemic justification. My response was developed to respond to the following question:

Explain the regress argument for foundationalism. Must there be such things as ‘basic beliefs’? If so, then what exactly are the ‘basic beliefs’? If not, then how can we respond to the regress argument?

Introduction

The regress problem explores the structure of justification; epistemologists aim to solve the regress problem in various ways. The foundationalist, for instance, advocates basic beliefs that are non-inferential in the sense that their justification does not depend on any other belief. Coherentism, as a rival theory to foundationalism, is usually considered in adequate depth by the foundationalist. In contrast, this essay argues that foundationalists seldom evaluate infinitism thoroughly: usually, infinitism is too hastily dismissed as blatantly erroneous at the mention of John Williams’ finite mind objection (Williams, 1981, pp.85-88). I argue that this objection alone is insufficient – a further objection is required to debunk infinitism: the conjunction objection. This essay therefore aims to strengthen foundationalism with a more thoughtful rebuttal of infinitism. I will begin by outlining the regress argument for foundationalism, followed by the infinitist position. Then, I will demonstrate how a combination of the finite mind objection and the conjunction objection undermine infinitism.

Regress Argument for Foundationalism

The regress argument for foundationalism is best understood through a problem-solution structure – the regress problem is presented; foundationalists ponder the trilemma between foundationalism, coherentism and infinitism before concluding that foundationalism is the only position that does not suffer from “insurmountable difficulties” (Turri, Klein, n.d.).

The regress problem arises due to the principle of justification: knowledge is adequately justified true belief.[1] Traditionally, every belief must be supported by another justificatory belief. Therefore, belief B1 is justified by B2; B2 is justified by B3 et cetera – epistemic chains begin to form. The question is this: which type of epistemic chain or ‘evidential ancestry’ (Klein, 1999, p.298) best warrants our beliefs and avoids undesirable scepticism?

The foundationalist, at this stage, tends to swiftly reject infinitism without sufficient consideration; the infinitist history is certainly one of neglect and rejection (Turri, Klein, n.d.). A more thorough evaluation of infinitism and its critiques features later in this essay.

Next, coherentism is rejected. I do not intend to go into any depth on this discussion, but I briefly outline that foundationalists usually reject coherentism due to some argument based on a circularity claim (Watson, n.d.). The coherentist allegedly justifies B1 with B2, B2 with B3, and B3 with B1 – this circularity is fallacious because the coherentist uses B1 as both their conclusion and as their supporting reasoning.

The foundationalist finally presents their solution. In short, basic beliefs are somehow self-evident or self-supporting. These anchor the epistemic chain and act as the bedrock source of justification. Foundationalism therefore revises the justification principle; not all beliefs require another belief to justify them. These basic beliefs take various forms. This essay need not consider competing versions of foundationalism, since all versions can unanimously benefit from a more careful consideration of why infinitism fails, the focus of this essay.

Infinitism

Infinitism proposes that knowledge may be justified by an infinite non-repeating chain of justifications. Note that infinitists agree with foundationalists in that they reject the allegedly circular reasoning of coherentism (Klein, 1999, p.298). Infinitism is also similar to traditional coherentism in that it maintains that only reasons can justify a belief (Klein, ibid.). Klein is the most notable contemporary proponent of infinitism; his view is centred around two principles (Klein, 1999, pp.298-299):

i. “Principle of Avoiding Circularity (PAC)

For all x, if a person, S, has a justification for x, then for all y, if y is in the evidential ancestry of x for S, then x is not in the evidential ancestry of y for S.

ii. Principle of Avoiding Arbitrariness (PAA)

For all x, if a person, S, has a justification for x, then there is some reason, r1, available to S for x; and there is some reason, r2, available to S for r1; etc.”

The PAC simply expresses infinitist opposition to circularity – Klein takes PAC to be uncontroversial because circular reasoning is a simple logical fallacy; he also states that PAC undermines most forms of coherentism.[2]

The Principle of Avoiding Arbitrariness requires more explanation. The PAA emphasises Klein’s rejection of inappropriate or thoughtless justificatory reasoning. Firstly, the PAA states that a person requires epistemically justified reason to believe each statement (as opposed to the causal reason or context for their belief) (Klein, 1999, p.299); this flexibly captures the potential demands of the justification principle. Secondly, the reason(s)/beliefs must be available. The notion of available belief is contrasted against the notion of occurrent belief: an available belief is one that is somehow related to the persons’ pre-existing beliefs but is in a state of doxastic potentiality. Available beliefs may express themselves through a disposition (direct or ‘second order’) to form a certain belief (Klein, 2014b, p.279). Klein exemplifies through a simple demonstration: “the proposition “352 + 226 = 578” is available even though it might never be consciously entertained” (Klein, ibid.). This proposition is available because of the persons’ mental capacity to make calculations of this sort. I wish to further clarify this notion of ‘available reasons’ through my own analogy.

Suppose a mathematician develops a theory Q. In excellent scientific spirit, she has a high degree of belief in theory Q but is not certain. Theory Q contains numerical value Ɀ. Ɀ has infinitely many decimal digits; the mathematician can meticulously infer each digit individually from the previous digit. Each digit is mathematically interdependent with its adjacent digits. She has not calculated every single digit of Ɀ, since it is infinite. However, each calculation of a further digit of Ɀ increases her degree of belief in theory Q, since her equations remain balanced and sound. The 10,000th digit of Ɀ has doxastic potential because she has the capacity to calculate it – it is an available reason for her belief in theory Q.

The Finite Mind Objection

A common objection to infinitism is John Williams’ finite mind objection. I argue that foundationalism ought to be strengthened by bolstering the finite mind objection, which alone does not sufficiently undermine infinitism. Simply put, he argues that it is psychologically, if not logically impossible for a finite mind to utilise an infinite regress of justificatory beliefs (Williams, 1981, p.85). Robert Audi is another proponent of the finite mind objection. Even when we consider, for instance, an infinite set of simple arithmetical beliefs (e.g. a set of beliefs that may result from counting 0,1,2,3… ad infinitum), he emphasises the absurdity of a human mind actually grasping an infinite set of justifications (Audi, 2010, p.211 and Audi, 1993, pp.127-128).

However, I believe Klein’s infinitism possesses within it a solution to the finite mind objection. The crucial inclusion of ‘available reasons’ as part of the infinitists’ epistemic chain counters the apparent problem. As emphasised through my analogy, the mathematician does not occurrently possess the infinite number of possible beliefs related to Ɀ – they are only potential beliefs. As potential beliefs, they may form links in the epistemic chain without ‘filling up’ the finite mind. The mathematician does not have an infinite mind, but could reliably actualise and utilise the infinite potential justifications within Ɀ. Hence, the finite mind objection alone does not undermine infinitism; foundationalism must find another way to reject infinitism.

Note an important feature of Klein’s infinitism concerning the nature of justification. Klein implies that beliefs and justification must be understood in gradual degrees. In other words, “a belief-state is reason-enhanced” as the person supports their belief through additional justifications (Klein, 2014a, p.105). Reconsider the mathematician example: she is never certain, but her degree of belief in theory Q gradually increases. This aspect of my analogy highlights Klein’s commitment to the fact that “warrant, and with it rational credibility, increases as the series lengthens; but the matter is never completely settled” (Klein, 2014b, p.281).

However, foundationalists would disapprove of this ‘reason-enhancement’ theory of justification. Watson captures contrasting positions: “Inferential justification is said to transmit justification, not create it; therefore, an infinite chain of justifying beliefs would have no source of support to transmit” (Watson, 2018). Watson’s point is that Klein’s proposed method of generating justification through a vast number of reasons holds very limited epistemological merit, because these reasons are ultimately ungrounded. Contrary to this, foundationalism understands justificatory steps as “promissory notes” (Ross, 2018) that are handed down the epistemic chain: suppose B1 is dependent on B2, and B2 on B3. The collective force of B2 and B3 provides no meaningful epistemic support for B1 unless B3 is a foundational belief. Without an anchoring foundation, these promissory notes are never satisfied.

Thus, we have two ways to understand the nature of justification: either, qua foundationalism, it is something that is transmitted in steps or, qua infinitism, it is gradually enhanced. I argue that even if we grant that justification is gradually reason-enhanced, Klein’s infinitism nonetheless falls victim to a second critique: the conjunction objection.

The Conjunction Objection

The conjunction objection is based on probabilism. Probabilism, akin to Klein’s infinitism, understands belief in degrees. The laws of probability dictate that even if a person’s beliefs are all highly likely, if there is a vast number of beliefs, the likelihood of at least one belief being false is very high (New, 1978, p.342). This is because the conjunction of two probabilities <1 will always produce a probability lower than either of the two individual probabilities. For instance, the probability of not landing a 36 on a roulette wheel is high (≈97.3%); however, with an increasing number of spins aka conjunctions, the chances of not landing a 36 become increasingly low. If one had an infinite number of spins, landing a 36 would be certain.

I propose that the infinitist’s chain of beliefs can be understood through the laws of probability. Recall Klein is a proponent of degrees of belief: he states that a belief’s rational credibility may increase but is never certain. Therefore, each justificatory belief or ‘available reason’ is not certain, but if given a charitable interpretation may be highly likely. Each ‘available reason’ may be highly probable individually, but across the entire infinite chain, the likelihood of at least one fallacious justificatory belief is, in fact, certain. Note that the presence of a fallacious belief is detrimental to the infinitist: since the infinitists’ beliefs are part of an interdependent chain, the entire epistemic chain collapses if one is shown to be false. An infinite chain of non-certain beliefs produces certain error; this presents a grave problem for infinitism.

Klein may respond by stating that, in practice, an infinite number of justificatory reasons is not pragmatically required to provide warranted knowledge or credible belief (Klein, 2014b, p.281). If, as Klein suggests, the highest possible degree of warrant is not required, then Klein is no longer committed to infinite justificatory reasons aka conjunctions. Recall it was the infinite number of conjunctions that produced certain error for Klein’s infinitism. However, this response would be inadequate and inconsistent. Klein must be rigorous and accept the consequences of an infinite chain, rather than resort to an ambiguous middle-ground that features an infinite chain in theory, but only ever a finite chain in practice.

Conclusion

In summary, foundationalists too hastily reject infinitism. This essay demonstrated how one may reject infinitism with more thorough consideration and improve the regress argument for foundationalism. I present two objections to infinitism. Although Klein’s ‘reason-enhanced’ justification theory aptly responds to the finite mind objection, I argue that an infinite chain of beliefs in degrees falls victim to the conjunction objection. Klein’s commitment to degrees of belief forces him to accept that the conjunction of an infinite chain of beliefs produces certain error, undermining the entire ‘evidential ancestry’ of infinitist reasons.

Word Count: 1997

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[1] BonJour clarifies that the proper solution to precisely what this adequate justification entails does not affect discussion surrounding the regress problem (BonJour, 1978, p.2).

[2] Note that some coherentists endorse PAC and maintain that their reasoning is non-circular. They argue that a “web of belief” (Quine, 1970) is more plausible because it simply has greater mutual support. Klein maintains that a “larger circle” merely makes the logical fallacy of circularity harder to detect; it remains a logical fallacy (Klein, 1999, pp.300-301).