Sunday, March 28, 2010

Different Senses of the Quantifers?

Carnapians want to say that different things can be truely said to exist when speaking in different language-frameworks. So the existential quantifier "Ex" will mean different things in these different frameworks. But can there really be multiple different meanings for these different uses of Ex, which would qualify as different kinds of e.g. existential quantification?

An argument that you can't is: The meaning of Ex is determined by its introduction and elimination rules. So any putative kind of existential quantifier would need to obey them. Hence different senses E1 and E2 from different frameworks would both have to obey the standard introduction and elimination rules for Ex. But if E1 and E2 obey these rules, then you can prove E1x from E2x and vice versa. Hence there is no room for ambiguity.

This argument can't be right though, if restricted quantification ('There is nothing in the fridge'. 'All the beers are in the fridge') - something that even the most ardent anti-Carnapians accept- counts as `a kind of' quantification. And intuitively it is. Hence in order to seem like a kind of quantification, a connective need not obey the full introduction rules. It suffices if there's a more limited range of instances of the introduction schema
P(x) --> Ex P(x) that speakers accept, together with all corresponding instances of the elimination schema Ex P(x). (A^B^C..^P(z) > F) ---> F (in cases where z does not occur free in A,B, C... or F). This is what we have for beers in the fridge.

Why can't the Carnapian claim that the same thing goes on with different linguistic frameworks? The different choices for when P(x) --> E2x P(x) is acceptable will each correspond to a different meaning for the existential quantifier. We can even formally represent these different possible senses for existential quantification formally, by saying a kind of existential quantification E_i corresponds to each subset S_i of the set of predicate-expressions (i.e. to each choice of what predicate-expressions the introduction and elimination schema are supposed to hold for).

You are probably worrying that this turns the Carnapian into a kind of maximalist (all the objects in question really exist, different frameworks just correspond to different framework restrictions) but I can't actually see any argument for that. So speak up if you can!

Thursday, March 18, 2010

Seeing Sets

I used to laugh about (early) Pen Maddy claiming that we could see sets. But now I think that's almost right- though not in the way that Maddy intended it.

I can see that my program doesn't infinitely loop, or that the 1000th prime is 7919 by pressing enter, waiting a few seconds and then looking above the command line prompt on my computer. These are all claims about mathematical objects, yet (given suitable equipment and background knowledge, we would ordinarily say that I can see these things to be true).

This seems just as literally true as the claim that I can see that the electricity is on, when I look at the lit windows of the house next door.

In both cases I immediately form the belief, probably am justified, am depending on a lot of contingent assumptions about electronic wiring etc.

But maybe we should distinguish seeing Xs from seeing that some fact about Xs obtains? Maybe there's something especially problematic about believing in objects which you can't see?

-If seeing x = seeing that x exists, then I can see that there is a 1000th prime in the above example (suppose I wrote the program but had never seen the proof that there are infinitely many primes)

-If we take a more intuitive approach to seeing xs (i.e. is it awkward to say I am now looking an X) then:
a) certainly it is awkward to say `I am now looking at a number'...hmm though we might say `I am now seeing the line of the program that causes the crash (and lines in programs are abstract objects, just like lines in poems),'.
b) it's also pretty awkward to say `I am now seeing a drought', or `I am now seeing North America' or 'I am now seeing a proton'.

If you can see a drought when you look at a color map of precipitation, why can't you see a pair of twin primes by looking at a chart?

Overall conclusion:

Seeing that P really means little more than having some visual experience which causes you to immediately believe that P, which you might cite as part of your justification for believing that p. So if you can know things (e.g. all the background mathematical beliefs involved in the program case) about numbers, then it's not too hard to arrange to see things about them.

Of course, the anti-platonist won't think that you can know things about numbers either - well that's where my thesis comes in. But if we can know some things about the numbers, its not hard to arrange things so that we can see further things about them ie rig up reliable methods for forming beliefs about them whose last step involves visual experience.

Tuesday, March 16, 2010

"Ideal" vs. "Ideal"

Scientific explanations, which explain the behavior of an actual object by relating it to the behavior of an ideal object, don't usually involve a normative element. It's not as if we think that inclined planes should be frictionless, or planets should be perfectly spherical. These ideal models aren't somehow better then the actual objects in question, they are just easier to think about.

I wonder if psychological explanations of actual human behavior by relating it to rational human behavior ("the price rises because if everyone was a homo economicus with this set of beliefs and desires they would..." "actually, getting a beer is what a fully rational person with Jim's beliefs and desires would do right now..") are just instances of this. If they are, then the normativity makes no difference to the explanation. The idea that one ought to be rational (assuming there is such a fact) plays no more role in the success of the explanation than the claim that inclined planes ought to be frictionless plays in the success of the ordinary physical explanation.

Potter and the Loch Ness Monster

M. Potter asks why some philosophers intuitively require so much less evidence for introducing abstracta than for concrete objects. How come the requirement not to "multiply entities beyond necessity" doesn't apply to these? Without an answer taking this relaxed attitude towards positing yuppie cliques and category theoretic arrows, while being very skeptical about the Loch Ness Monster looks a bit unprincipled. Well here's a sketch of an answer.

Start with a reliability based notion of justification: we evaluate a creature's justification by thinking of it as having certain faculties i.e. mechanisms that reliably produce true beliefs (e.g. infra-red vision, smell, first order logic). We say a belief is justified when it is the result of one of these reliable mechanisms 'working as intended'. Now in order for mechanisms that produce contingent beliefs to be reliable, they will typically have to be causally sensitive to facts about the outside world - so that e.g. they tend to only produce the belief "there's a llama" in situations where there is actually a llama. In contrast, you can build a faculty that reliably produces the right results about necessary aspects of the world, without using any such external input. And if there are necessary truths such as: whenever there are yuppies behaving in such and such a way there's a clique of yuppies, you can build in a reliable mechanism that makes this transition immediately, without requiring any further input from the environment. So it's not surprising that the reliable belief forming mechanisms we humans have should require less justification for introducing necessary abstracta, or ordinary objects whose existence is necessitated by already known facts about other objects vs. for introducing concrete objects (like the Loch Ness Monster) which lack either of these properties.

Now obviously, what I just said won't convince anyone who has some *other* other reason for rejecting abstract objects, and ordinary objects, to believe in them. But it does provide a unifying explanation, and hence (I think) a way for those who a) have the intuition that introducing abstract objects needs less justification and b) are inclined to take this intuition at face value to defeat Potter's challenge that their intuitions about justification are unprincipled. Quite to the contrary, this distinction falls out of a reliable-mechanisms theory of justification almost immediately!


If (all) propositions intrinsically have a logical structure, then does an english speaker's utterance of "I will go to the store unless you already bought milk" typically express a proposition with the structure ~P>Q, or one with the structure PvQ?

Does it depend on the situation? Who bought milk last time? :)

It seems better to say that propositions expressed by natural language sentences only have a logical structures only relative to a choice of logic, and a method of translation.

Sunday, March 14, 2010

Carnap Disenchantment

[Sigh, I can never make up my mind about Carnap. I guess I'm feeling anti today]

I understand what it is to say that it's "merely a pragmatic choice" whether to use first order logic with the usual connectives vs. with the sheffer stroke. In both cases you will be expressing truths when you derive things in accordance with the logical laws, so the only harm you can do with choosing the sheffer stroke is make your proofs take longer.

But the "choice" of accepting a weaker vs. stronger and (say) inconsistent logical system, does not have this feature. In one case, you will be deriving truths. In the other case, you will now be deriving some falsehoods/crash your whole language so that none of your sentences are meaningful at all.

So I don't see what Carnap can mean by saying that adopting a system is merely pragmatic choice. Adopting a consistent system is a hard epistemic task! The only pragmatic choice is choosing which system - of a menu of systems of reasoning which are coherent enough to give their terms meaning and count as truth-preserving - to use.

Saturday, March 13, 2010

Paradox of Analysis

The paradox of analysis is roughly this: If a conceptual analysis of a term like justice was successful, then the two sides of the analysis should mean the same thing, so it should also be trivial.

The notions of cognitive triviality (analyticity?) and sameness of meaning are infamously hard to spell out, but I think we can get much of the intuitive puzzlement of the paradox of analysis by rephrasing it as follows:

If you know already what 'justice' means, how can it be useful to you to have a conceptual analysis that says an act is just if and only if it is ____?

If you accept this restatement of the problem, I propose the answer is this:

Your "knowledge of what `justice' means" consists in something like a disposition to accept some collection of methods of inference, which - under favorable conditions- tend lead to your beliefs about what's just correctly tracking the facts about what's just. Call the particular algorithm for making and revising judgements about what's just α. So your understanding of the word justice consists in the fact that your brain implements α.

The potential usefulness of conceptual analysis comes from the fact that your brain can implement α without:

a) your knowing what algorithm α is (e.g. some processes in your brain recognize grammatical english sentences, but you don't know what these processes are).
b) your knowing that the descriptions of actions which algorithm α ultimately gives a positive verdict on are exactly those which have property B. (this is useful when your usual methods of checking for B-hood are faster/easier to deploy than your usual methods of checking for justice)
c) your knowing that property C applies to most of the things which A would ultimately give a positive verdict on, but C is easier to apply, and all the purposes normally served by considering which actions are just would be served even better by thinking about which actions have property C. (the classic definitions of computability and limit are examples of this kind)

Thursday, March 11, 2010

"Coherence" and Mathematical Existence

When I say that "the more practically benign a system of proto-mathematics is, the more likely it is to count as expressing largely true claims about some domain of objects", I realize that this sounds horribly woolly. People naturally ask me: but how practically benign does a practice have to be to guarantee that it succeeds in talking about some object? (not to mention: how do you measure `largely true'?)

Why Be Woolly

Here's a classic example of a claim that isn't woolly:

H: Any logically consistent system of mathematical beliefs counts as expressing truths about some suitable domain of objects.

We can see H is false because it implies that if I believe ZFC+{X} and you believe ZFC+{~X} where X is some statement about number theory independent of ZFC, since we both have logically consistent systems of belief, we will both be right - just talking about different objects.

But what goes wrong?

Note the problem isn't that there aren't enough mathematical objects (if we just have sets every first order consistent theory has a model). Rather (I claim) it's because actual people will use words in the mathematical theory like 'finite' or 'smallest' or 'number' which have meaning that goes beyond their role in this first order logical stipulation.

When we both say that by the "numbers" we mean (among other things) the smallest collection containing 0 and closed under successor, smallest (intuitively) means the same thing for both of us, so it is NOT correct to then interpret each of us as talking about whatever larger non-standard model makes our claim true.

Hence our informal use of the words like "smallest" or "all possible collections" imposes constraints on interpreting us which go beyond the first order logical content of our mathematical statements.

If you buy this, here's why you should be wooly. In general there will be some vagueness with regard to how wrong you can be about Madagascar, Christmas or to use Quine's famous example, atoms, and still count as talking about these things. Once a theory is sufficiently wrong it can be a tossup whether to say that the objects in question are real, and the person is wrong about them, or that there are no such objects. But this is exactly what we face with regard to mathematical objects as well! We have an amorphous informal practice, and a norm that people count as referring to whatever the most natural object is that best satisfies their methods of reasoning about these putative objects, provided there is one that matches suitably well.

There's no bright line about how wrong your various formal and informal beliefs about some putative object can be while you still still count as referring - for the same reason in math as in physics or history.

Hence, I don't try to draw one, and that's why I'm woolly on this issue, and why you should be to!

All we can say generally is: Mathematicians can posit new objects, and the more logically consistent their reasoning about these objects is, and the less their intuitions about consequences of reasoning about these objects lead to false conclusions about other things, the more likely it is that they will count as expressing largely true claims about some suitable piece of the mathematical universe.

The other non-woolly alternative is to give a list: a mathematician counts as referring if they have x beliefs (which are true of the integers), y beliefs (true of the reals), z which are true of imaginary numbers, w for quaterinians, v for sets, k for arrows ... and thats all the mathematical objects that it is metaphysically possible to think about! But surely this is insane.

trolly problems and literature examples

I've heard it suggested that moral philosophers should consider examples from literature rather than simplified cases as in trolley problems. Here's a theory of why examples from literature might be particularly bad for moral philosophy purposes.

Kant says (as I understand him) that the experience of beauty happens when observing an object provokes the "free play" of the conceptual faculties, producing a harmonious volley between the intellect and the imagination. This works most naturally for novels and poems, where reading a line can set off a chain of thoughts which aren't logical deductions, but are still somehow naturally suggested by the line.

In contrast, in much moral philosophy you are looking for (relatively) general principles [it's an interesting question why this is], that different people might agree to and be guided by even when particular interest leads them in different directions. So you want something like "all actions of X kind are impermissable", For these purposes, you want to show that your general principle is acceptable even in, as it were, the worst case scenario, even in the most perverse instances. You also want to avoid features that would be distracting, from the question of whether the given action is permissible, and also unclarity about what the descriptive scenario is supposed to be.

Now, if we buy the kantian idea about beauty we get a quick explanation for why literature examples will tend to be bad for the purposes of moral philosophy. Beautiful cases will be ones that promote the free play of the intellect, considering all kinds of different aspects of what's being described, and reaching out into all sorts of other questions. Hence they are particularly likely to involve a) simulatious application of multiple apparent moral reasons for and against b) interesting factual questions about what the situation really is (do we really know that soldier is unpersuadable?) c) other different but related moral/philosophical issues - that one might easily confused with the issue of unpersuadability.

So literature examples may be good a suggesting questions, but there's some reason to think they are - so to speak- actively engineered to be distracting when consider as examples in debate about some particular moral principle.