ON BILL’S NOTES FOR HIS 730 CLASS, FALL 2007  

 



On Bill's posts for his Phil 730 class, Fall 2007.


I have to confess that there are a couple of points where I disagree with Bill. I won’t get nit-picky, but focus on the crucial ones instead.



First, some replies to Bill's "Thomas and Substitution".


The main thing I disagree with is Bill's main objection at the end. Of course, I agree with Bill that his (3) does not follow from (1), and that those who think they are equivalent are assuming that a man did it, and given this they are equivalent. That is all fine. But the numbers case is different. Note that Bill is not objecting that the alleged equivalence between


(A) Jupiter has four moons.


and


(B) The number of moons of Jupiter is four.


as follows: (B) assumes that four is a number. But that is not contained in (A), etc., which would be the analog to his case of Oswald being a man. That four is a number, rather than a color, say, is a conceptual truth, and although it is not made explicit in (A), it doesn’t add anything when made explicit in (B). Instead,  Bill’s objection is that (B) assumes that there are numbers at all. That is different. And I disagree with Bill's assessment. (A) and (B) are equivalent, I hold, and (B) does not contain any more semantically singular terms than (A) (on standard uses like in trivial inferences). The reason for the latter claim is the following: only under the assumption that (B) is like that can we explain why it has a focus effect at all. If (B) were as Bill seems to think it is then it is not clear why there is a focus effect in (B) at all. I have tried to make this case in my paper "Innocent statements and their metaphysically loaded counterparts". The simple idea is that identity statements formed with descriptions and names don't have a focus effect by unless it comes from intonation. But standard uses of (B) have a focus effect without special intonation. This can only be explained if (B) is as I think it is (without extra semantically singular terms). And then (B) is indeed equivalent to (A), full stop, not just given that there are numbers.


Some smaller points:


- on the "not enough names" issue: I didn't mean to introduce demonstratives into the substitution class in my paper, as Bill seems to suggest. Rather the idea is a bit more complicated. When we quantify over properties the internal quantifiers still are defined by their inferential role, but now it is crucial to allow for the quantifier free sentences, to which the quantified ones are inferentially related, to contain context sensitive expressions. (think of the property of being his brother). This is so exactly because not every object is denoted by a name in our language. Once the inferential role is spelled out this way it turns out that the truth conditions of sentences with internal quantifiers over properties and propositions are still equivalent to an infinitary version of the quantifier free language, but now the infinitary language is more complicated in that it combines infinitary disjunctions and conjunctions mixed with infinitary (external) quantification over objects. Note that the involved external quantifiers only range over objects, not properties or propositions. The details of all this are in the paper on inexpressible properties. This way all the cardinality objections are avoided.


- on more thing on paraphrase and reduction. I don't see myself as being in that ballpark, as Bill might be seen to suggest. I do not claim that talk about properties can be paraphrases away or reduced. In the simplest cases like (A) and (B) above this might be seen to be the case, but in general I believe this is not so. Instead I want to argue that what appears to be a semantically singular term, like a number word, or a that-clause, really isn't one. That it has a different semantic function than to refer doesn't mean that is can be reduced or paraphrased away.



Now on Bill's alternative line about numbers:


The main thing I disagree with Bill about is the general setup of his position. Bill asks whether he can consistently hold that there are numbers, but they don't exist. I am happy to agree that there are a number of views about either natural language, or the use of natural language, such that if they are true then this can be held consistently. But that is not the question. What is the question is this: what are we in fact doing when we talk about numbers? Does it require an ontology of mathematical objects for its literal truth? Why is such talk so important? etc. These are not questions about what one can hold consistently, but questions about what is in fact the case. As Bill says, he is not trying to answer these latter questions, only to outline a strategy, a consistent way, to avoid commitment to numbers. But I disagree with Bill that this is an important question. Whether you can avoid commitment is one thing (unimportant), whether what we in fact do takes on this commitment is another (important). Avoiding commitment is no goal in itself, and it is easy to achieve by simply rejecting all talk about numbers as false, say. Sure, Platonism about numbers is problematic, but it might be true nonetheless, or at least it might be required for our mathematical talk to do what it is supposed to do. Whether or not what we in fact do requires Platonism (and thus in fact commits us to numbers as abstract objects) is the crucial question. I have tried to argue, on partly empirical grounds, that this is in fact not so. The literal truth of arithmetic does not require the existence of any objects whatsoever. This is in fact so. And the way it is so explains why talk about numbers occurs in different ways, why talking of how many moons there are is tied to talk about the number 4, and so on.


Bill can stipulate that from now on he quantifies over numbers only in an innocent way, but that doesn't account for why talk about numbers appears in different ways in natural language, why it is so useful and all that. To understand that we have to look at the details of how we in fact do it, and how we always did it before we started worrying about ontology.


I also disagree with Bill on the issue of what impact it would have for his view if it turned out that number words are referential expressions. Bill says that it wouldn't bother him much since fictional names are referential, but their bearers don't exist. But of course that means that subject predicate sentences with such names in it aren't literally true. And if the analogy is supposed to carry over to numbers then it would mean that "5 is a prime" isn't literally true. But that is bad math, not good philosophy, and so more of an issue than Bill makes it out to be.


Other than the above, I completely agree.