Tuesday, February 7, 2012

Kripke's Paderewski and Frege's Common Coin

Maybe this was already obvious to everyone but ...

Kripke's Paderewski example can be modified to refute both the attractive principle Frege's Common Coin, below, and contemporary weakenings of that principle which exempt indexical and demonstrative sentences, or require that all speakers be normally linguistically component.

Frege's Common Coin:
a) When two sincere speakers "disagree over" a sentence*, there is a single proposition expressed by this sentence in this context which one believes and the other does not believe,(and indeed believes the negation of).
b) When two sincere speakers "agree about" a sentence, there is a single proposition expressed by this sentence in this context which they both believe.

*[I realize this is an awkward locution, but I just mean the intuitive kind of disagreement which occurs when I say "snow is red" and you say "snow isn't red" but doesn't occur when I say "I'm tired" and you say "I'm not tired". It sounds far more natural to say `disagree over a proposition' but we will see that it is actually not clear whether these scenarios involve disagreement over a proposition.]

Consider the following drama involving Pierre, a man in a thought experiment of Kripkie's who knows the musical statesman Paderewski in two different ways, and doesn't realize that Paderewski the pianist is Paderewski the anarchist. Suppose all the following utterances are sincere,
Act 1 (musical evening)
pierre:"Paderewski is tall" p1
alice:"yes, Paderewski is tall" p2
Act 2 (on the street)
alice:"Paderewski is tall" p3
bob:"yes" p4
Act 3 (political rally)
bob:"Paderewski is tall" p5
pierre:"no, he's not!" let p6 be the proposition that Pierre *denies*

Frege's Common Coin tells us that there are propositions p1…pn which speakers express attitudes towards in all the different phases of our play, that p1=p2, p3=p4, p5=p6 and that Pierre believes p1 and does not believe p6.

But this is a very bad thing to say: By the fact that a person like Alice or Bob who has a single grip on Paderewski presumably says the same thing by asserting this sentence on the street vs. at a political rally or a musical evening p2=p3 and p4=p5.
By transitivity of identity p1=p6.
Thus Pierre believes p1 and does not believe p1. Contradiction.

[If you are worried about the fact that Pierre is unusually ignorant for his society, and hence may not count as "normally linguistically compent" substitute in the name "John". Now cases like the one above will turn out to be so ubiquitous that denying that Pierre, Alice and Bob know enough to have linguistic competency would imply that no one ever has linguistic competency for names.]

Possible Moral: If we want to take propositions to be the objects of belief then, contra Frege's Common Coin, each sentence must be associated with (something like) a class of different propositions which someone could sincerely assert that sentence in virtue of believing.

Friday, December 9, 2011

The Sheffer stroke, and the blandest antirealism ever

In college my metaphysics prof said realism was the doctrine that there is a complete true description of the world, though it might take infinitely many sentences to express this description. Do the following incredibly bland considerations about different truthfunctional connectives commit me to "antirealism"?

(1) If you want thinking about propositions to play a role in psychology, then you need to individuate propositions narrowly enough that truthfunctionally equivalent sentences express different propositions.

(2) If you individuate propositions as narrowly as required by (1) then tautologies involving the sheffer stroke will be different propositions than any corresponding sentences which use more standard truthfunctional connectives.

(3) The same argument goes for all the other infinitely many n place truthfunctional connectives.

Conclusion: no language with finitely many basic connectives can express all true propositions. No finite language can be used to give a complete true description of the world. In particular if the a language L has only n place truthfunctional connectives then there will be tautologies using n+1 place truthfunctional connectives that cannot be expressed in L.

"Just what I mean by the term"

Whether or not you think there are philosophically interesting facts about analyticity, normal people do respond to certain a) epistemic challenges and b) calls for scientific/philosophical explanation by saying 'that's just what I mean by the term'.

So I think any philosopher who doesn't want to accept massive error about justification and argument owes some account of what is going on here. What does this expression do?

Maybe one thing the claim 'x is just part of what I mean by the term' maybe does is diffuse the expectation that there is an interesting natural kind or scientific fact in the neighborhood. Here's what I mean...

Suppose I had a made a map with mountain ranges labeled and you said, how come all mountains have a slope of at least 1 percent (cf. wikipedia definition). If you said "how do you know that mountains all have a slope of less than 1percent?" or "why do all mountains have a slope of less than 1 percent" I might say `that's just what I mean by the term mountain'.

I *think* hearing this might helpfully lead us to rule out possibilities like the following: 99% of paradigm mountains are caused by a certain geological process, and this process always produces things that have a steep slope, and then reliably maintain this steep slope through the process of erosion. If there was some more natural kind in the neighborhood of "mountain" then there would be a less trivial answer to questions like 'why do all mountains have at least a 2% slope?' and 'how do you know that all mountains have at least a 2% slope.

Tuesday, November 1, 2011

Pictureability and Definiteness in Mathematics

A fairly popular position in philosophy of mathematics is the following: there are definite right answers to questions about arithmetic, but not to questions about set theory (especially not to questions like the Continnum Hypothesis whose answers can be changed by forcing), because we have a clear notion of what the numbers are supposed to be like but not what the sets are supposed to be like.

This claim about clarity is sometimes supported by appeal to notions of mental picturing. The idea would be that we can picture all the numbers, but that we can't picture things like `all the subsets' of the numbers, or the hierarchy of sets as an item that contains, at each level, all subsets of the levels below. But I want to ask: In what sense can we picture all the numbers? In what sense CANT we picture the sets?

All the ways I can think of of picturing all the numbers involve some kind of "..." e.g. you might picture all the numbers by having a mental picture like the physical picture I have drawn below:

| || ||| |||| ....

When we use this picture to conceive of the structure of the numbers, we are employing a highly non-trivial method of picturing, which lets us represent the state of there being a countable infinity of numbers by considering some finite collection of drawings like the above.

And people certainly do *in some sense* draw pictures of the hierarchy of sets. At the beginning of a set theory class (and at the beginning of my college analysis class) the professor will frequently draw up a V on the board. In a way that is a picture of the heirarchy of sets. And, presumably, one could mentally picture the heirarchy of sets in the same.

For this reason I am (a little) skeptical about justifying the claim that we have a definite idea about the sets but not the numbers by appeal to considerations about what we can mentally picture. We can picture the numbers if we are willing to let "..." represent the idea of all finite successors to a stroke symbol. We can picture the hierarchy of sets if we are willing to let the increasing girth of the V represent the idea that each level of the hierarchy of sets contains all possible subsets of the level below. What (mathematical) objects one can picture depends on what methods of picturing we will accept, as allowing one to genuinely conceive of a situation by entertaining a mental picture.

At the very least, I think that if someone wants to argue that claims about numbers are definite and claims about sets are not on the basis of appeals to mental pictures, they owe us some kind of story or explanation for why one method of picturing is OK and the other is not.

Sunday, October 23, 2011

Hammond Questions for Philosophy

I was just taking a break by rereading this heartwarming little essay , which you should definitely check out. This made me think about what some "Hammond Questions" for philosophy are (i.e. big questions, which we nonetheless seem to be in a position to reasonably attack now)

1. Is there a meaningful distinction between statements that are "true by convention" and those that "state substantive claims about the world"? If not (I would guess there is not) what genuine distinction are people getting at when they say:
-it's a matter of convention that mountains have to be 100 feet
-the fact that I am a US citizen is a social/conventional truth (?)
-deliciousness and disgustingness are not something independently in the world but something that we project onto the world (???)
I suspect different things are at stake in each of these cases.

2. Why, despite our ability to quickly learn new words like `table', `policeman' and `gouche' has it proved to be so hard (impossible?) for philosophers to produce elegant necessary and sufficient conditions for the application of these terms using more primitive ones. Is it...
-that in learning new words we learn paradigm cases not definitions? (but cf. the known problems with e.g. trying to account for understanding "green apple" in terms of composing paradigm cases of "green" and "apple")
-that there is an adequate definition for being a policeman etc. to be found but they are ugly enough to be considered by linguists not philosophers (e.g. maybe they typically have many special clauses)?
-that external facts about natural kinds help determine the extension of terms like policeman, in a way that either gives these terms a very messy relationship to the extensions of other terms that we understand and/or makes it epistemically difficult for us to figure out what these relationships are?
-that there are clean necessary and sufficient conditions to be given, but `atoms' of these definitions turn out to be metaphysically wild and wooly notions like `purpose' `agent' `blameworthy' so that providing necessary and sufficient conditions for claims about policemen, tables, etc. in terms of primitives like these doesn't feel like (and maybe isn't) making philosophical progress.

2' Why has it proved so hard for philosophers to paraprhase away ceterus paribus clauses, despite their apparently unproblematic use in the sciences.

3. Do facts about forcing independence and large cardinals have any consequences for the trilemma about how to think about the height of the hierarchy of sets. If mathematical facts can't decide this issue, what can?

4. What are the truth or assertability conditions for claims about literary "function" e.g. x foreshadows y, x alludes to y, x raises questions about whether y? Provide a metaphysical story about what makes statements of the above kind true, plus corresponding "logic" of literary criticism e.g. a formal algorithm that captures many if not all truth/assertability preserving inferences about literary function? Does this
-bonus: add axioms and inference rules to your logic of literary function until you get something that captures all intuitively valid forms of argument, or prove that no recursively axiomatizable logic can do this.
-bonus: determine what if any relationship there is between claims about literary function and biological function (I guess Kant thought there was a connection but he clearly likes cute solutions too much to be trustworthy on a matter like this!)

5. In virtue of what does a piece of music count as expressive of sadness, excitement etc.? If these facts are relevant to some parameter like species or prior musical tradition state the relevant parameter. This is an old and daunting question but...
Currently fashionable psychological research into how pieces of music produce similar *judgements* about expressing sadness excitement etc. in different people may suggest promising new proposals with regard to the philosophical question of what features of music make it *count* as expressing sadness, excitement etc. What kinds of lower level features do people seem to be causally responding to when they say that a piece of music is sad, and what if any general-purpose causal reasoning is involved in these judgements?

Tuesday, March 29, 2011

Justifying Logic and the Normal Role of Proof in Justification

Some philosophers aim to show how we can be justified in accepting certain basic logical truths by giving "rule circular" proofs of the soundness of these basic logical truths. They admit that most people will never have considered the proofs in question, and they admit that these people still count as justified in using logic. But, they say that they showing that such proofs can in principle be given makes sense of how we can be justified in believing the basic logical claims established by the proofs right now.

That idea seems prima face implausible. In general the fact that someone 100 years later will prove P from premises that I accept (like the ZF axioms) doesn't suffice to show that I am justified in believing that P now. So why should the case be any different for the proofs of logical principles?

[I would rather say that we are prima facie justified in believing these logical principles in a way that has nothing to do with the possibility of giving further argument; coming up with more or less circular ways of proving the soundness of our logical principles is (at best) a way of improving our justification]

Saturday, March 19, 2011

are stipulative definitions a source basic knowledge?

Random thought:

Whether or not its OK to make a certain stipulative definition can depend very messy questions - and not just mathematically messy questions like questions about harmony.
For example: it would seem that it's OK to stipulate that people are to count as "gleb" whereas bodies are not to count as "gleb" if and only if people are distinct from their bodies.

This suggests that knowledge by stipulative definition is not a source of basic knowledge. (basic knowledge= justified belief that doesn't depend on any other beliefs for justification) For, you can say 'of course people are gleb and bodies aren't, thats just what I mean by the term! remember when I stipulatively defined it...'. But (it would appear) the justificatory buck doesn't stop when you say this. If you are unjustified in thinking that bodies are distinct from people, this would seem to poison your justification for making and appealing to this stipulative definition.

However, perhaps we should say that only some stipulative definitions do have prima facie warrant, and the above stipulation about glep is just not one of the ones that does.

p.s. if we say that stiplative definitions aren't basic knowledge, we will probably want to say that analyticities aren't either.