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Back to the question of deducibility.
According to the deducibility thesis, the fundamental truths (plus
indexicals, plus a 'that's all' statement) a priori entail every truth.
More precisely, when P is a complete description of the fundamental
truths and M any other truth, then, according to the deducibility thesis,
the material conditional 'P
M' is a priori.
Dave Chalmers agrees that any concept can be explicitly analyzed by an infinite conjunction of application-conditionals. But he wants to restrict 'explicit analysis' to finite analyses. That certainly makes sense, but I doubt that there are any concepts for which the application-conditionals cannot be determined by finite means. For example, I think it will usually suffice to partition the epistemic possibilities into, say, 50 zillion cases and specify the extension in each of these cases. Admittedly, I can't prove that, but the fact that concepts can be learned and our cognitive capacities are limited seem suggestive.
Dave Chalmers told me to read some of his papers. I have, and I'll probably say more on the deducibility problem soon. Here is just a little thought on conceptual analysis.
Chalmers suggests that we don't need explicit necessary and sufficient conditions to analyse a concept. Rather, we can analyze it just by considering its extension in hypothetical scenarios. What is it to consider a hypothetical scenario? The result seems to depends on how the scenario is presented. For example, 'the actual scenario' denotes the same scenario as 'the closest scenario to the actual one in which water is H2O'. But the difference in description could make a difference for judgements about extensions. Chalmers avoids such problems by explaining (§3.2, §3.5) that to consider a scenario is to pretend that a certain canonical description is true. Hence to analyze a concept, we evaluate material conditionals of the form 'if D then the extension of C is E', where D is a canonical description. (Are there only denumerably many epistemic possibilities or can D be infinite?) Now fix on a particular concept C and let K be the (possibly infinite) conjunction of all those 'application conditionals' (§3) that get evaluated as true. Replace every occurrence of 'C' in K by a variable x. Then 'something x is C iff K' is an explicit analysis giving necessary and sufficient conditions for being C.
There may not always be a simple, obvious, or finite explicit analysis, but at least there always is some explicit analysis. If moreover satisficing is allowed, it is very likely that we can settle with something much less than infinite.
When I tried to spell out the 'modus tollens' I mentioned on monday, I came across something that may be interesting.
Frank Jackson argues that facts about water are a priori deducible from facts about H2O:
1. H2O covers most of the earth.
2. H2O is the watery stuff.
3. The watery stuff (if it exists) is water.
C. Therefore, water covers most of the earth.
1 and 2 are a posteriori physical truths, 3 is an a priori conceptual truth.
Here are, very quickly, some more thoughts on the matters I talked about here and there, inspired by another discussion with Christian.
You don't have to know much about plutonium to be a competent member of our linguistic community. One thing you have to know is that plutonium is the stuff called 'plutonium' in our community. Maybe that alone suffices. Of course, if noone knew more about plutonium than this, the meaning of 'plutonium' would be quite undetermined. To fix the meaning, it would suffice if a few persons, the 'plutonium experts', knew in addition that this element (where each of the experts points at some heap of plutonium) is plutonium.
Are all truths a priori entailed by the fundamental truths upon which everything else supervenes? If 'entailed' means 'strictly implied', this is trivially true. The more interesting question is: Are all truths deducible from the fundamental truths (deducible, say, in first-order logic) with the help of a priori principles?
If yes, then it seems that Lewis' 'primitive modality' argument against linguistic ersatzism (On the Plurality of Worlds, pp.150-157) fails. Recall: Lewis argues that if you take a very impoverished worldmaking language then even though it will be feasible to specify (syntactically) what it is for a set of sentences to be maximally consistent, it will be infeasible to specify exactly when such a set represents that, e.g., there are talking donkeys. Now if all truths are a priori deducible from fundamental truths, and -- as seems plausible -- fundamental truths are specifiable in a very impoverished language, then we can simply say that a maximal set of such sentences represents that p iff p is a priori deducible from it.
Unfortunately, I find the 'primitive modality' argument quite compelling. So, by modus tollens, I have to conclude that not all truths can be a priori deducible from fundamental truths. Does anyone know whether Lewis himself believes the deducibility claim he attributes to Jackson in 'Tharp's Third Theorem' (Analysis 62/2, 2002)?
This is a continuation of my last post and also partly a reply to concerns raised by my tutor Brian Weatherson.
Imagine a small community consisting of three elm experts A, B, and C.
First case: Each of A, B, and C knows enough to determine the reference of 'elm', but their reference-fixing knowledge differs. However, they belief that their different notions of 'elm' necessarily corefer. This is the case Lewis discusses in 'Naming the Colours'.
Some days ago, Christian and I had an interesting discussion about two-dimensionalism. While I don't agree with many of his criticisms (forthcoming in Synthese), I do agree that two-dimensionalism works best if both dimensions belong to an expression's public meaning. I think that Christian thinks that this holds only for context-dependent expressions. I think it holds almost universally. But this may be a matter of terminology: For me it is part of the meaning of 'the liquid that actually flows in rivers' that this would not denote H2O if it would turn out that XYZ flows in rivers, whereas for Christian this is a metasemantic fact. Anyway, problems for two-dimensionalism come when the first dimension doesn't belong to public meaning.
Don't miss Brian Weatherson's very insightful answer to my posting on rigidity (from which I've just stripped some irrelevant formalities). I happily agree with everything he says, so I'll just add a footnote here.
Many advantages of the counterpart theory derive from its denial of the equivalence between 'a=b', 'possibly a=b', and 'necessarily a=b'. For example, this allows for a statue to be identical to a lump of gold even though it might not have been. Since, as Weatherson argues, the rejected equivalence is built into the customary ('strong') concept of rigidity, that concept must be weakened to be useful for counterpart-theorists.
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