Sensitivity and Closure

Kelly Becker, in his book Epistemology Modalized, gives a nice modal account of knowledge:

S knows that p iff:

1. p is true
2. S believes that p
3. S’s belief that p is formed by a belief-forming process or methodw that produces a high ratio of true beliefs in the actual world and throughout close possible worlds (reliability condition).
4. If p were false, S would not believe that p via the methodn S actually uses in forming the belief that p (sensitivity condition). (Epistemology Modalized, p.88)

Methodsw are individuated very narrowly, but not so narrowly as to include specific belief contents.  Specific belief contents are however included in methodsn.  Becker individuates a methodw as the narrowest specific-content-neutral method or process that is causally operative in belief formation.  An example of a methodw might be forming beliefs about which people are in the vicinity based on quick looks in at least dim lighting.  A methodn on the other hand might be something like If what I am looking at now has short legs and floppy ears (and such and so other features) then it’s a dachshund.

Becker also makes a serious and interesting case against closure under known entailment, and he takes it, as epistemologists generally do, that sensitivity is incompatible with closure:

The sensitivity component of our theory somehow predicts this result – I do not know not-[sceptical hypothesis] because, if it were false, I would believe it anyway. (Epistemology Modalized, p.120)

But in fact, it isn’t clear that his account of sensitivity is incompatible with closure.  Take a standard BIV case.  I believe I am not a BIV, yet 4 holds: if I was a BIV I would not believe that I was not a BIV via the methodn I actually use in forming the belief that I am not a BIV.  My method, after all, involves coming to know ordinary propositions about the world around me by interacting with it, and inferring from these ordinary propositions that I am not a BIV.  Since brains in vats cannot employ the same kinds of methods that embodied humans do, my belief that I am not a BIV is sensitive according to Becker’s analysis.

How to Eschew Metaphysics

Here’s how Blackburn describes pragmatism:

You will be a pragmatist about an area of discourse if you pose a Carnapian external question: how does it come about that we go in for this kind of discourse and thought?  What is the explanation of this bit of our language game?  And then you offer an account of what we are up to in going in for this discourse, and the account eschews any use of the referring expressions of the discourse; any appeal to anything that a Quinian would identify as the values of the bound variables if the discourse is regimented; or any semantic or ontological attempt to ‘interpret’ the discourse in a domain, to find referents for its terms, or truth-makers for its sentences … Instead the explanation proceeds by talking in different terms of what is done by so talking.  It offers a revelatory genealogy or anthropology or even a just-so story about how this mode of talking and thinking and practising might come about, given in terms of the functions it serves.  Notice that it does not offer a classical reduction, finding truth-makers in other terms.  It finds whatever plurality of functions it can lay its hands upon. [Simon Blackburn, Expressivism, Pragmatism and Representationalism: 75]

I'm interested in the claim often made by pragmatists, such as Simon Blackburn or Huw Price, that they are eschewing metaphysics, in contrast to platonists, fictionalists, error theorists and the like. Pragmatic accounts of a discourse provide a genealogy, or some consanguineous account, of why it is we go in for this way of talking and, as it may happen, this account may be metaphysically deflationary.  So it may be that the motivation for talking about, say, mathematical objects, does not involve representing how things stand with a domain of mathematical objects.  If there is some such story—if we can account for the uses of mathematical talk, without invoking mathematical objects—then we have an ontologically deflationary pragmatic account of mathematical discourse.

But so far, what’s been said about pragmatic accounts of mathematical discourse is open for the fictionalist to adopt.  The difference between the fictionalist (who is apparently engaged in metaphysics) and the pragmatist (who apparently eschews metaphysics) is that the fictionalist claims that mathematical talk is, strictly speaking, false, whereas the pragmatist does not.

Fictionalists and pragmatists then agree in methodology: provide an account of the usefulness of mathematical (or moral, or possible worlds) discourse that makes the existence of mathematical (or moral, or modal) objects orthogonal to the practice.  Their point of divergence is not methodological or ontological, but semantic: whether one opts for fictionalism or pragmatism depends on what one takes the meaning of existential quantification to be.  Here, the pragmatist reads the pragmatic purpose of quantification over mathematical objects back into the semantics of quantification over mathematical objects, and the fictionalist does not.  A truism: people can engage in ontological disputes.  There is something at stake between someone who claims that the Higgs boson exists and someone who claims that it does not, or between someone who claims that God exists and someone who claims that he does not.  The interlocutors in these debates are in disagreement over what the world is like.  So, sometimes at least, quantificational talk is used to express disagreements about what the world is like.  Ultimately then, the difference between the fictionalist and the pragmatist lies in what they take the meaning of existential quantification to be.  Fictionalists take existential quantification to be univocal: it always expresses claims about what the world is like.  Pragmatists (are committed to) taking existential quantification to be multivocal: within discourses whose purpose is to describe the world existential quantification expresses claims about what the world is like; within discourses whose purpose is not to describe the world, existential quantification does not express claims about what the world is like. (Note that the point of divergence is not, or need not be, over semantic minimalism. The person engaged in metaphysics need not couch what he is doing in terms of finding truth-makers or referents to be relata in substantive relations of truth or reference to given sentences; he can simply couch what she is doing in terms of whether such and such objects exist. Hartry Field is a case in point.)  

The take-away claim: whether one gets to eschew metaphysics depends on whether existential quantification is univocal or multivocal.

Kant and the Necessary 3-Dimensionality of Space

Here’s a thought.  Kant took it to be necessary that space was 3-dimensional.  Bracketing the possibility that space is transcendentally ideal for the moment—of course, you might think I’m bracketing the most interesting thing here—most people who are paid to think about these things now reject the necessary 3-dimensionality of space on the grounds that spaces with more dimensions are possible, and the standard argument for this is the following.  A 3-dimensional space can be modelled as $\mathbb{R}^3$ = $\mathbb{R} \times \mathbb{R} \times \mathbb{R} $ with each n-tuple representing a point in 3-dimensional space.  Methods of this sort allow for higher-dimensional spaces to be represented, since extending or generalizing the model to represent higher-dimensional spaces is quite straightforward. 4 dimensional space is represented as $\mathbb{R}^4$, 5-dimensional space as $\mathbb{R}^5$, and so on.

But why think a thing like this constitutes grounds for taking higher-dimensional spaces to be metaphysically possible?  Why think that because a model of 3-dimensional space can be extended in this sort of way (and remain coherent), higher-dimensional spaces themselves are possible, or even coherent?  Why think that because (i) there is a space which can be represented using $\mathbb{R}^3$, and (ii) there is nothing in consistent about $\mathbb{R}^4$, that (iii) there could be a space that is represented by $\mathbb{R}^4$?  Now, there may be other good reasons to think that (iii) is true, but the standard argument looks to be enthymematic at best.

[Cross-posted at Kant and Laws]