Supplement to 2.5: Schwarz on naturalness and induction.

This is one of a series of posts setting out my work on the Nature of Representation. You can view the whole series by following this link

Wolfgang Schwarz’s paper “Against Magnetism” gives a rather different perspective on the Lewisian treatment of induction than the one I have been developing. Wo notes, and I acknowledge, that Lewis himself did not buy into the architectural assumptions I have been throwing in throughout this subseries of posts. In particular, he denied that we should assume sentence-like vehicles of individual of belief. Lewis liked an account where holistic belief-states were attributed to agents. And just to be clear: my version of radical interpretation per se is not committed to the sentence-like structure. Just like Lewis, I see it as a hypothesis about the contingent cognitive architecture that we may or may not have. I just don’t see why we shouldn’t be interested also in what the theory predicts under those hypotheses. It would be interesting, though, if Lewis himself had found a way to connect up naturalness to inductive reasoning and radical interpretation without the aid of such hypotheses. Wo sketches one way in which this arise:

Here is the key quote from Wo’s paper:

“…rational agents should assign high prior probability to the assumption that nature is uniform. But uniform in what respect: should one believe that emeralds are uniform with respect to green or with respect to grue? Here we can appeal to natural properties: one should assign high probability to worlds that are uniform with respect to patterns in the distribution of fundamental properties. At least in worlds like ours, attributes like green (unlike grue) supervene on intrinsic physical features of their instances: perfect duplicates never differ in colour. Hence if unobserved emeralds are similar to observed ones in their fundamental physical properties, it is plausible that they will also be green and not blue. It does not matter, for this proposal, whether green, or being in a world where all emeralds are green, are themselves particularly natural.”

The general idea is that if one has high prior probability in uniformity among patterns of fundamental properties, one needn’t know in detail exactly how other, macroscopic properties relate to the fundamental properties, to know that they too will be uniform.

The question is how to leverage that insight. Schwarz emphasizes the following feature of green/emerald: that they locally supervene on intrinsic physical features of their instances. Now, that’s compatible with there being a large set of different physical descriptions, any one of which would suffice for being an emerald, with no-non-disjunctive single description being necessary and sufficient. The same goes for being green. That observation fits with Schwarz’s remark that he is not assuming that the properties themselves are particularly natural. Unfortunately the same feature means that there’s no real reason to think that the confidence in the uniformity of fundamental patterns will generate confidence in emerald/green uniformity. Suppose that being P or Q or R is necessary and sufficient for being an emerald, and each LHS disjunct is an intrinsic physical description. And suppose A or B or C is necessary and sufficient for being green (again, with the LHS disjuncts intrinsic physical descriptions). Now, it might be that all observed emeralds are P-emeralds and all observed emeralds are A-green, and so all observed Ps are A. With high probability, we could project this pattern in the fundamentals: all Ps are As. But that alone tells us nothing about the colour of Q-emeralds or R-emeralds. They could be A, or B, or C (i.e. green), or none of the above (i.e. not-green), without there being any lack of uniformity in patterns in the fundamentals.

But there is another element in the quote from Schwartz. He at one point adds the assumption “unobserved emeralds are similar to observed ones in their physical properties”. This does not follow from the stated supervenience assumption: things which are P and which are R can be quite unlike, physically.  So this should be seen as an additional assumption concerning intra-world (lack of) variation: though emeralds (/green things) may have all sorts of different physical features in other possible worlds, within the actual world, all emeralds (/green things) share the same (non-disjunctive) physical property, E (/G).

Now, this still allows emeralds/green things to be “unnatural” by many Lewisian measures–any necessarily equivalent description might be infinitely disjunctive. But if it’s going to do work for us, we do need I think to assume there is some extensional characterization of them that is not disjunctive, or at least, is the sort of description that we can reasonably take to specify a “pattern” in the physical fundamentals, in the sense relevant to the uniformity constraint on the priors.

Let’s introduce the label “E1” for the (unknown-to-the-agent) property in fact possessed by all and only the things that are emeralds. Let “G1” plays the same role for green things. And now, the crucial thing is that we have high confidence that all E1s are G1, conditional on all observed E1s being G1, as part of the general fundamental uniformity assumption. The same goes for many other analogous instances: all Eis being Gj, for arbitrary i and j.

Now, I can imagine the story running as follows. First, some information about the agent’s prior credences:

  1. For every i,j: C(all Ei are Gj|all observed Ei are Gj)=1.
  2. C([(x)(x is an emerald iff x is E1) v (x)(x is an emerald iff x is E2)…..])=1
  3. C([(x)(x is green iff x is G1) v (x)(x is green iff x is G2)…..])=1

(1) is the uniformity constraint on priors. (2) and (3) are the assumption the agent is certain that actual emeralds (/green things) are physically similar to each other in some respect or other. We now argue that by (2) and (3) the agent’s credal space divides into cells, according to which property necessarily covaries with being an emerald, and being green. Within the emerald=Ei/green=Gj cell, we can cite the appropriate instance of (1), to get that C(all emeralds are green|all observed emeralds are green&Ei=E&Gj=G) is high. When we glue these cells back together, we get that C(all emeralds are green|all observed emeralds are green) is high.

Okay, this reasoning needs studying. But if it’s what Schwarz intended, it provide different kind of perspective on the way that inductive generalizations could relate to considerations of naturalness.

A crucial question is what grounds the agent’s confidence in (2) and (3) comes from. In virtue of what are we so confident that actual emeralds/green things are physically unified? Is this based on evidence, or are we somehow default-justified in assuming such uniformity? Notice that these assumptions play no role in the story outlined in the last two posts. And they would be false for many things on which we are prepared to induct (as Jackson notes being observed by Sam can be a perfectly good “projectable” property, so long as Sam isn’t the one doing the projecting). Wrongness is not an intrinsic property of acts, but we inductively generalize upon it. So we really do have a different account here (though one compatible with the same overarching theory of radical interpretation, but embedded within very different architectural and normative assumptions).

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