Suppose we want a theory that tells us for all sentences in our language in what possible contexts their utterance is true. Call those functions from contexts to truth values "A-intensions". A
systematic theory should tell us how the A-intension of complex
sentences depend on their constituents. Here are some theories which
are not very satisfactory in this respect.
Theory 1. Each sentence consists of a sentence radical and a fullstop. (The sentence-radical is the entire sentence without the
fullstop.) All sentence radicals have the same semantic value:
God. The semantic value of the fullstop maps this semantic value to a
truth-value. But whether it maps God to true or false
depends on the context of utterance. For instance, in a context in
which it doesn't rain and the utterance of "." is preceeded
by an utterance of "it rains", the value of "." maps God to
false; in a context where "." is preceeded by "2+2=4", it maps
God to true; and so on.
Warning: another pointless exercise in conceptual geography.
Can intrinsic properties have their causal/nomic role essentially? It seems not. Suppose something x is P. If P essentially occupies a
certain causal role, say being such that all its instances attract one
another, we can infer from x's being P that either there are no other
P-things in x's surrounding or x and the other things will (ceteris
paribus) move towards one another. But if we can infer from x's being
P what happens in x's surrounding, P cannot be intrinsic. Being
intrinsic means belonging to things independently of what goes on
in their neighbourhood.
Two unrelated notes that will not interest any of my readers.
First, my LaTeX Paper CD Case Generator can now be run on the server. The Redhat LaTeX packages seem to be rather old, so new stuff doesn't work well. Maybe I'll manually install a newer tetex version sometime.
Second, I've installed the Hoary Hedgehog on the little Powerbook. The only problem, as usual, was my keyboard layout. (My keyboard is German.) Here is the .Xmodmap file I wrote to fix this, just in case anyone -- my future self, in particular -- runs into the same problem.
Suppose we find a proof, in ZFC, that ZFC is inconsistent. Does
it follow that ZFC is inconsistent?
On the one hand, if we could infer from ZFC
![$m[1]](/texer/images/?format=gif&fontsize=12pt&tex=%5Cmodels)
~Con(ZFC) that ZFC is inconsistent, we could
contrapositively infer the consistency of
ZFC & Con(ZFC) from Con(ZFC); and since ZFC & Con(ZFC) obviously entails Con(ZFC), ZFC & Con(ZFC) would thereby entail its own consistency. Which it only can if it is inconsistent (Gödel's second incompleteness theorem). So it seems that we can only infer that ZFC is inconsistent from the observation that ZFC entails its own inconsistency if we presupposes that ZFC &
Con(ZFC) is inconsistent.
Fundamental (or 'perfectly natural') properties are properties on whose distribution in a world all qualitative truths about that world supervene. That is, whenever two worlds are not perfect qualitative duplicates, they differ in the distribution of fundamental properties.
This is not the only job discription for fundamental properties. If it were, far too many classes of properties could play that role. For instance, all qualtiative truths trivially supervene on the distribution of all properties, or on the distribution of all intrinisic properties, or (for what it's worth) on the distribution of all extrinsic properties. (That's because no two things, whether duplicates or not, ever agree in all extrinsic properties.)
A structural property is a property that belongs to things in virtue of their constituents' properties and interrelations. For instance, the property being a methane molecule necessarily belongs to all and only things consisting of suitably connected carbon and hydrogen atoms.
There is two-way dependence: Necessarily, if something instantiates a structural property, then it has proper parts that instantiate certain other properties; conversely, if the proper parts of a thing instantiate those other properties then, necessarily, the thing itself instantiates the structural property.
Some people intuit that
- the subject in a Gettier case has knowledge;
- Saul Kripke has his parents essentially;
- "Necessarily, P and Q" entails "Necessarily, P";
- whenever all Fs are Gs and all Gs are Fs, the set of Fs equals the
set of Gs;
- the liar sentence is both true and not true;
- the conditional probability P(A|B) is the probability of the
conditional "if B then A";
- it is rational to open only one box in Newcomb's problem;
- switching the door makes no difference in the
Monty Hall problem;
- propositions are not classes;
- people are not swarms of little particles;
- a closed box containing a duck weighs less when the duck inside
the box flies;
- spacetime is Euclidean;
- there is a God constantly interfering with our world.
They are wrong. All that is false.
Imagine a world with nothing but infinitely many duplicate dragons, aligned in one long row. Consider the second dragon in the row. Call it "Fred".
Fred could have failed to exist. There are many worlds where he does not exist. (The actual world is probably one of them). Some of these worlds where Fred does not exist contain no dragons at all, others contain some of the other dragons in Fred's row. In particular, there are worlds where all the other dragons exist, but not Fred. The dragons are after all distinct existences, and there are no necessary connections between those.
Robbie Williams pointed out that in my recent musings on worms and stages, I ignore the following straightforward characterizations:
Worm Theory: the semantic value of predicates like "rabbit" is a set of 4D worms.
Stage Theory: the semantic value of predicates like "rabbit" is a set of 3D stages.
He's right. I believe that these theories both cannot work, so I don't want to define stage and worm theory that way.
I've set up a new server, so that now this blog should support LaTeX commands. As a test, here is what I take to be the most general version of the fixed Principal Principle:
![$m[1]](/texer/images/?format=gif&fontsize=12pt&tex=%24%5Cmathrm%7BC%7D%28A%5C%3B%7C%5C%3BH%29+%5C%3B+%3D+%5C%3B+%5Csum_%7Bx%7D%5C%2C%0D%0A%5Cmathrm%7BC%7D%28%5Cmathrm%7BP%7D%28A%29%5C%21%3D%5C%21x+%5C%3B%7C%5C%3B+H%29+%5Ccdot+%5Cmathrm%7BC%7D%28A%0D%0A%5C%3B%7C%5C%3B+H%29%24)
Hm. The 'x' is a bit blurry.