Friday, June 1, 2012

A school paper:

I wrote what follows last fall as my final paper for 'Epistemology'. 

Some Restrictions on “Knowledge”1
Epistemology is concerned with the nature of knowledge: what we know, and how we
know it. In practice, philosophical epistemology2 is not concerned with cataloguing particular instances of knowledge, but rather with giving a clear definition of “knowledge” and then using it to decide if some arbitrary statement or belief should qualify as knowledge. I wish to put two explicit restrictions on the sorts of definitions we can give to the word “knowledge” (but not to put restrictions on what-may-be-known), and to examine the consequences of these restrictions with respect to some major historical schools of epistemology.

To be acceptable, any definition of “knowledge” must satisfy some basic restrictions; at
minimum, its extension must have at least some overlap with the extension of the incomplete intuitive conception of knowledge (a rigorous epistemology that says nothing about propositional beliefs and everything about good growing conditions for carrots would not be satisfying). This restriction I will call the 'extension requirement'.

Not all the necessary restrictions on “knowledge” are as obvious as the extension
requirement. I contend the extension requirement is already implicitly agreed to by all those willing to try epistemology (all those willing to admit that people really are talking about something when they use the common word “knowledge”) and as such it should not be controversial to anyone wanting to read this paper. Thus I assume the extension requirement.

The extension requirement contradicts the skeptic and asserts the existence of knowledge. I will analyze a typical skeptical argument in light of this requirement to see how the argument fails, as I argue it must. Using this analysis I will derive a second restriction on “knowledge”: that of necessary multivalued-ness.

The derivation of necessary multivalued-ness undermines mainstream epistemology
since, historically, the three main branches of epistemology (foundationalism, coherentism and infinitism) have grown from the incompatible implicit assumption of binary-valued knowledge. When taken together with skepticism the three main branches of epistemology have been thought to be exhaustive, and therefore a claim to undermine all four would seem to leave us empty-handed. But the multivalued-ness requirement, by specific virtue of not being binary, opens a fifth category of epistemology, which bears a passing resemblance to foundationalism and infinitism, but differs from both in that it only requires finite, not infinite, regression, while not claiming the existence of special foundational beliefs. Thus beginning with a relatively benign assumed restriction, I derive a second restriction which undermines the motivation for four broad classes of traditional epistemology while introducing a whole new class of epistemologies.

The skeptic argues for the null epistemology that there can be no knowledge.
Contrariwise it is commonly, though not universally, accepted that there are things that can be (or even that are) known. Further, it is nearly universally desired that a defence of knowledge, a non-empty epistemology, can be given. On face value these claims are irreconcilable. I contend that until a satisfactory direct reply to the skeptic is formulated, we should accept the validity of the skeptical argument as a working hypothesis. If we cannot refute it, we have no intellectually honest grounds for dismissing it. I have already claimed that the extension of “knowledge” must overlap the extension of our intuitive grasp of knowledge, in particular it must have non-empty extension in the 'real-world'. A platonic conception of “knowledge” that does not allow us to claim that anything is known or even can be known may be self-consistent, but, like a dissertation on the medicinal properties of dragon's blood, it is fruitless and uninteresting. If we wish to avoid contradiction while accepting these two propositions: that there is knowledge, and that the skeptical argument rules out 'knowledge', the kind of knowledge we claim to have has to be a weaker conception of “knowledge” than that used by the skeptics in their argument. We need to dissect the shared form of the various skeptical arguments, see what proposed properties of “knowledge” they rely on, and accept only those formulations of “knowledge” which are not
susceptible to the skeptical arguments.

Prime among the skeptical arguments is the problem of the criterion, or how it is that we can get knowledge without question begging:
1. A proposition P is claimed as knowledge.
2. The skeptic demands justification for the belief that P, since all knowledge requires  justification.
3. A justifying proposition j(P) is given.
4. The skeptic demands justification for the belief that j(P)
5. ... (continuing similarly for jn(P))
Thus it is claimed that unless some special instance of self-justifying knowledge can be given
(foundationalism), or a defence of circular justification be raised (coherentism), or an infinite regress is allowed (infinitism), knowledge is impossible. It is not clear that a foundation can be given; any defence of circular justification is itself as a (conjunction of) proposition(s)
susceptible to problem of the criterion type arguments; and infinitism plainly fails to give an
account of knowledge that satisfies the extension requirement, since human minds are finite in power and scope. Much ink has been spent, largely fruitlessly, defending (and attacking) these three -isms in hopes of salvaging a “knowledge” with non-empty extension, a knowledge safe from the skeptic. But less has been said about the form of the argument. It is usually taken to be obviously valid. And if it is valid then all epistemology is limited to four avenues of enquiry: either we concede to the skeptic and accept the trivial epistemology of no knowledge whatsoever, or we argue for one of the three -isms of foundations, coherence, and infinite regress. Fortunately for epistimologists a structurally identical argument has already been dealt with by borrowing from another discipline: biology.

The (refuted) argument goes by the name the Prime Mammal fallacy (Dennett, 127),
though as originally given it was unnamed (Sanford, 512).
1. Claim: There exists a mammal M.
2. Claim: All mammals are born of mammals.
3. Thus there exists a mammalian parent of M: p(M).
4. ... (continuing similarly for pn(M))
I hope the analogy to the skeptic's argument is clear to the reader: mammals correspond to items of knowledge, parents to justifications, and grandparents to justifications of justifications, etcetera. We might well re-rename the argument “the mammal skeptic” or “the problem of the first ancestor”.

Thus either there was a Prime Mammal (analogously foundationalism), or3 an infinite
ancestry of mammals. But there does not seem to be any non-arbitrary choice of Prime Mammal, and the number of mammals that have ever lived is clearly finite (though large). Therefore mammals are impossible. Clearly this conclusion is false, there are mammals. It follows there must be some flaw in the argument.

Implicit in the second claim is the premise that mammal-hood is binary and fully
heritable. All things are either full (essential) mammals born of full (essential) mammals, or they are essentially non-mammal. Therein lies the problem. It is a kind of sorites argument that asks us to partition animals into mammals and non-mammals, like partitioning collections of sand into heaps and non-heaps. The only apparent escape from this argument4, and the one seemingly given by historical reality, is to accept “mammal” as a vague or multivalued predicate. Doing so allows us to reformulate the second claim as: “All things of a particular degree of mammal-hood are born of other things with a similar (ε-close5) degree of mammal-hood”. Substituting this reformed claim into the Prime Mammal argument we find that analogical foundationalism and coherentism still fail, since with regard to foundations it is now not clear that “Prime Mammal” has any meaning at all (Prime mammal-of-what-degree?), and the grandparent paradox of analogical coherentism is untouched. On the other hand infinitism taken through the analogy is entirely transformed into a finite claim. No longer does accepting the argument while rejecting foundations and coherence lead to infinite mammals but only to at minimum, the possibly large but finite number: ceiling(ε^-1) mammals. Unlike the conclusion of “infinite mammals”, the claim of “a large number of mammals” does not lead to a contradiction. Therefore the Prime Mammal argument with the implicit premise of binarity removed, no longer asserts the impossibility of mammals.

Having solved the argument in the analogous Prime Mammal form, I will now attempt to carry back the solution to the argument of interest. Just the same as we did with the Prime Mammal we must modify the heredital claim: that knowledge must be justified by (is begotten of) other knowledge. The modification too is similar. We must accept “knowledge” as a multivalued object and reformulate the heredital claim as: “All propositions that are to some degree knowledge must be justified by further propositions that are to a similar (ε-close) degree knowledge”. With this modified premise the skeptical argument can be answered. A finite regress of sets of propositions each being to a degree knowledge at most ε less than the degree of knowledge of the statement being justified suffices to give ultimate justification for some proposition. Thus the skeptical argument with its implicit premise of binarity removed, no longer asserts the impossibility of knowledge.

Thus the weakened skeptical argument allows a fourth approach to grounding knowledge aside from the three traditional -isms, and, counting the skeptical, a fifth class of epistemologies. The approach of a finite branching regression of bits of knowledge of decreasing rank, bottoming out on propositions which need not be knowledge at all.

But this approach requires us to accept that a proposition can be more certain, can be to a greater degree 
knowledge, than any of the propositions that compel us to accept it. At first sight this seems unacceptable and in outright contradiction to our common conception of knowledge. But this appearance stems from a naive analysis: I claim our intuition is correct in that a statement justified by a single further proposition cannot be more certain than its justification (if only because I cannot see past my own intuition and cannot see a way for that sort of justificatory relationship to hold), but it does not follow that a multiply justified statement cannot be more certain than each of its justifications are individually.

For example, let us presume to characterize degrees of knowledge as probabilities with all the standard rules associated. Suppose I am contemplating an object Q, and suppose I claim to know with probabilities ½ that Q is blue (p(B)=½), and Q is red (p(R)=½). From Q is red (resp. blue) I can infer (justify) that ¬(Q is green) with a probabilistic degree of knowledge p(¬G) = 1-½ = ½; from either colour claim alone I can do no better than p(¬G) = ½. But from both claims together--since R and B are mutually exclusive--I find that:

p(¬G) = 1-p(G) = 1-[1-((p(R)+p(B))] = 1-[1-(½+½)] = 1

which is strictly greater than the strength of my knowledge that either R or B alone. Not only does this show that there exist models of multivalued knowledge on which an ascending level of justification can be mathematically defended, but I feel it regains the assent of intuition which gives me confidence that I'm still writing about agreeable definitions of “knowledge” overlapping the intuitive, and that I have not wandered off into carrot horticulture.

Assuming only that we agree to search for a non-empty, non-trivial epistemology: we find that to answer the skeptic's sorites we must have multi-valued knowledge. Having thus derived multivalued-ness of “knowledge”, a fifth response to the skeptic opens up. It is a finitistic response, without the hopeful question begging of foundationalism and without embracing circularity. Working through a gradual building up of knowledge from things which are not knowledge, just as evolution builds up mammals from things which are not mammals.

1: Through the course of the work I have attempted to properly respect the use-mention distinction and to make very clear when I am writing about the common vernacular “knowledge” (here mentioned in quotes, when used it appears without quotes) and the epistemological word whose definition I am writing on, which will typically be rendered with quotes since I am discussing the word and concept itself and not its referent.
2: That is to say the study of knowledge as a term and concept, as opposed to practical epistemology, the use of philosophical epistemology to form belief.
3: The Prime Mammal analog of coherentism would be the suggestion that some mammal is it's own (time-travelling?) great-great-...-grandparent. I reject this position out of hand since I do not need to carry the refutations of the individual analogous Prime Mammal -isms back through the analogy to the skeptical argument.
4: ... without calling into question classical logic and classical metalogic. I for one want to hold on to the Law of Bivalence.
5: Where “ε-close” is defined as: belonging to the ball of radius ε centered at the point in question, in some arbitrary given “degree-of-knowledge” metric space.

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