A number of people have noticed that the problem about probability I mentioned last week is not really a problem about infinite probability spaces, but rather about possible distributions over such spaces. For instance, a Gauss distribution over the reals
will yield well-defined probabilities at every interval. But in the case of
the arbitrary real number, the distribution would have to be a line parallel to the x-axis, and how could the segments of the area under this line possibly add up to 1?
I'm still not sure if talk about
probability really does not make sense in such cases or if it does, but
we (or at least I) lack an adequate mathematical treatment. For example, are the following
three conditions logically inconsistent?
1. Atom A will decay at some time in the future.
2. The probability of decay at any day is the same as at the preceeding day.
3. Future is infinite.
Just in case anyone reading this blog is also interested in Lewis exegesis, Dave Chalmers usually knows the answers to my questions: The answer to the question raised at the end of this posting, whether Lewis himself believes the deducibility
claim he attributes to Jackson, is 'yes'. And about mental properties, apparently Lewis still prefered the realizer view in 1998.
Supervaluationsism and structuralism ('eliminative structuralism', not
the kind of structuralism that postulates structures) almost coincide.
Structuralism about something t says that any sentence 'F(t)' is to be
interpreted as 'for all x (if x is a candidate for t then F(x))'. For
example, arithmetical structuralism says that '2+2=4' is to be interpreted
as 'for all [N,0,',+,.] (if [N,0,',+,.] satisfies the peano
axioms then 0''+0''=0''''). If we translate 'x is a candidate for t' as 'x
is the referent of a precisification of 't'', we get: F(t) is (super-)true
iff it is true on all precisifications of 't'.
One of the many drawbacks of studying philosophy in Germany is that nobody kicks you when you spend years on your M.A. thesis. So hereby I publicly kick myself. As a consequence, this blog might become even more occupied with David Lewis than it already is.
To start this trend, here is a question about Lewis' functionalism. When a property P is multiply realizable, we cannot identify it with its realizers because then we would identify the realizers with each other, too. All we can do is locally identify P-in-k with its realizer in k, where k is a world or species or individual. Now what is P itself? In Lewis' papers on mind, he usually says that 'P' is systematically ambiguous or indeterminate, denoting the contextually salient realizer. (At least this is what I take him to say. He is not particularly explicit here.) The alternative would be to identify P with the diagonal property of being a P-realizer. The main reason why Lewis rejects this option (e.g. in 'Reduction of Mind') is that it is difficult to see how this diagonal property can occupy the causal role associated with 'P'. Difficult, but not impossible: In 'Finkish Dispositions' he proposes a solution, and consequently prefers identifying fragility not with the contextually salient realizer but rather with the diagonal property. Since then he apparently hasn't written anything more on the issue, so I would like to know if he has changed his mind on mental states (and heat, etc.) as well. (There may still be problems if the theoretical role is not entirely causal, so that even if the diagonal property can do the causal work, it might not be able to do the rest.) Or has he completely endorsed the third alternative – to simply leave the question unanswered: 'The folk well might have left this subtle ambiguity unresolved' ('Void and Object'). Can anyone help?
In part II of Meaning and Necessity Carnap defines 'L-determinate designators' for rather specific languages (coordinate languages). I think that a more general definition is possible that pretty much meets Carnaps ideas. This more general definition simply identifies L-determinacy with what we nowadays call rigidity.
Does anyone know a good resource on probability theory with infinite
probabilty spaces (if there is such a thing)? For example, I would
like to know if the probability that an arbitrary real number lies
between 0 and 1 is defined, and if so, how the obvious awkwardness of
any answer can be explained away.
If you recently received an email from somebody called 'Wolfgang Schwarz'
mentioning wolfgang@umsu.de and containing a strange attachment, please
don't open it. It is the worm W32.Bugbear@mm. If you have opened the
attachment, this page tells you how to remove it. Also
please don't reply to the sender who is not me and completely innocent,
since the mess really spread from my old windows machine. I'm very sorry
about this.
It is often said, correctly I think, that there are contingent but a priori
sentences, e.g. "water is the dominant liquid on earth". Are these
sentences analytic or synthetic? That is, what puts you in a position to
know these sentences? Does understanding suffice, or do you have to invoke
some other a priori means, like Gödelian insight? To me this seems
wildly and unnecessarily mysterious. Of course understanding suffices, at
least in ordinary cases. So there are contingent but analytic sentences. I
wonder why this is hardly ever said. Does anyone really believe that those
statements are synthetic a priori?
Since narrow content is not determined by external factors, it depends
much more on other propositional states than wide content. For example, if
you believe that Aristotle was human whereas I believe he was a poached
egg, the narrow content of all our beliefs about Aristotle will differ.
When I believe that Aristotle was Alexander's teacher, you can't have a
belief with exactly the same narrow content unless you also come to believe
that Aristotle was a poached egg. Likewise for imaginings: When we both
imagine Aristotle teaching Alexander, our imaginings cannot have the same
narrow content.
Similarly, I think, if Ted believes that for any atoms there is a
fusion, whereas Cian disbelieves this, they cannot share any imagining
about atoms.
Dave Chalmers kindly explained his views on deducibility to me. He thinks that anything one could reasonably call non-deferential understanding of the fundamental truths would suffice for being able in principle to deduce macrophysical facts, provided that these fundamental truths, unlike my P, contain phenomenal facts and laws of nature. He also notes that I shouldn't have called these restrictions (to non-deferential understanding and the rich content of fundamental truths) assumptions, since they are really just restrictions. I'm still not sure if any kind of non-deferential understanding would suffice, but with the restrictions in place it's not as easy to come up with counterexamples as I thought.