Wednesday, February 10, 2010

Asserting a conjunction

One might think that to assert a conjunction is the same as asserting the conjuncts. However, the lottery paradox shows that this is false. I can relatively unproblematically say: "One of x1,...,xN will win. x1 won't win. ... xN won't win." But if I said "One of x1,...,xN will win and x1 won't win and ... and xN won't win", then I would have said something I know to be necessarily false.

21 comments:

enigMan said...

I think that to say "Bill is good, Bob is bad, and Ben is ugly" is to say the same as "Bill is good and Bob is bad and Ben is ugly" and that, in general, to assert a sequence of propositions at one time is to have asserted their conjunction. I therefore think that the Lottery paradox must have a different resolution...

enigMan said...

...e.g. your first saying in the post is only relatively unproblematic: You said "x1 won't win", but I suspect that you meant that x1 probably won't win, or that you would bet that it won't (although presumably not at worse odds than the lottery gives you:)

Alexander R Pruss said...

No, I do mean that x1 won't win. Let's say N=1000. Then the probability that x1 won't win is 999/1000. That's pretty strong evidence that x1 won't win. I am quite willing to assert things outright on much weaker evidence. I am willing to say: "There is a horse in that field", even though at that distance, the reliability of my visual recognition of horses is only 98%. (And I really am saying that there is a horse in that field rather than that there probably is one there, because if you point out that it's a large donkey, then I will have to say I was wrong.)

If that doesn't convince, just increase N.

Consider also the fact that we're quite willing to say, upon seeing the cream put into the coffee, that in five minutes the cream will be blended. But that's just a future-directed probability.

Alexander R Pruss said...

About the first point: A comma can act as a conjunction in written English. In spoken English, one may pause differently for a comma or a period, or the context may determine which one has said--or, maybe, it's indeterminate. But there is still a distinction.

Here's another argument for my conclusion. Suppose I plan to say: "p and q". But while saying q, I realize that p is false. I am inclined to think that the prohibition on lying requires me, if I am able to do so, not to finish my sentence but to say something like: "... no, wait, actually I withdraw what I said about p". But if I say: "p. q.", then by the time I'm speaking about q, p's already been asserted. Now, there is a prima facie requirement that one correct one's mistakes. However, that prima facie requirement is significantly weaker than the duty not to lie. So, the requirement to withdraw p is significantly weaker in this case. So the two cases are different.

Another way to see the difference is that temporal indexicals work differently. In "F(now). G(now).", the default rule is that the first "now" refers to the time of saying "F(now)" and the second "now" refers to the time of saying "G(now)". But there is no such default in the case of "F(now) and G(now)." In fact, the default in conjunctive case is that the two "now"s have the same reference (unless verbal emphasis changes that--"Now, this, and now, that.")

Yet another way to see that is that the argument:
1. p and q
2. if (p and q), then r
3. therefore, r
is simply a matter of modus ponens. But the argument:
1. p
2. q
3. if (p and q), then r
4. therefore, r
requires another step between 3 and 4, namely:
3.5. p and q (conjunction-introduction)

enigMan said...

You may mean that it won't win, but you surely don't mean that it definitely won't win, because you presume, with the Lottery paradox scenario, that there is a chance of it winning. So you must have meant that one of them will definitely win (although see below) and that w1 will probably not win, and that w2 will probably not, so on. Now, it may be that none of the tickets of a real lottery win, for some unusual reason, but such uncertainty is of a different order to that which gives rise to the Lottery paradox. One could precisify what one meant even more, but the relevant distinction for the Lottery paradox is between one ticket (relatively) definitely winning, and each ticket probably losing.

An ideal Lottery could involve arbitrarily large numbers of tickets, and so the subjective probabilities for seeing a horse in a field may well be further from 1. And there are probabilities of unknown value (and hence not necessarily small ones) for all sorts of sceptical scenarios; but the context of the Lottery paradox is sufficiently different. Such perceptual probabilities do not stop you rationally believing in a horse in a field, but if you were informed of them, and then asked which of the following best describes your belief -- you have probably seen a horse, and you have definitely seen a horse -- it would be most rational to pick the former. And with the Lottery paradox you are informed of the probabilities (and little else).

enigMan said...

Regarding the general point, I tend to agree that asserting a conjunction is not always the same as asserting the conjuncts, most obviously when the asserting is done over time, but most importantly when subjective probabilities should be taken into account. But even so you could, I think, hand someone your book and say that it is all what you still believe, even had you had said in the Preface (as all philosophers should) that something in the text is probably wrong.

enigMan said...

...incidentally, I find the Preface paradox most paradoxical. I do think that if I say "Bill is good, Bob is bad and Ben is ugly" then by asserting each conjunct (about a situation that I take to be constant) I have asserted the conjunction. I find that self-evident. And yet I know from elementary probability theory that I ought not to (and I find the statement in the Preface quite a reasonable one to make).

enigMan said...

Another way to look at why you are wrong about the Lottery paradox: Since you do mean that x1 won't win, you mean by "won't" both definitely won't and probably won't. There is therefore no contradiction in the Lottery paradox, because there is that equivocation. Many paradoxes can be resolved by the (partial) clarification of a term. And in general, in language-use, a contradiction often makes us engage in some clarification procedure. There are usually pragmatic reasons why we do not begin with perfect clarity, and a prima facie contradiction shows us that we need to be more careful. So prima facie contradictions can be useful.

I think that the prima facie contradiction in the Preface paradox is like that. The preface statement shows that one is engaging in an on-going discussion. Indeed, I think that this is the more interesting paradox because to assert some propositions is to assert their conjunction. What is going on, I think, is that the conjunction is not asserted in the same way. But I think it is being implicitly asserted. Consider some statements, p, q, r and so forth. One could assert them like that, or one could say p. Furthermore, q. Furthermore, r (and so forth). Or one could say p, and also q, and also r, and so forth. I don't see that there is any real difference.

enigMan said...

And one could argue that asserting a sequence of propositions is not the same as asserting those propositions. Suppose (for the sake of a simple argument) that you know that I can see the future, and can show it to you too. And suppose that xn is the winning ticket. You say "xn won't win" and I show you the future with xn winning and you are very surprised. But if you say "x1 won't win. x2 won't win. x3 won't win. ... xn won't win. ... xN won't win." and then I show you the future then you would hardly be surprised at all. Now, there is the difference that the former case only involved you asserting one proposition, but that is rather my point. Asserting the whole sequence affects the nature of the individual assertions. To make the cases more similar, you would say "x1 won't win" in the former case. But you might be quite surprised if n = 1 in the latter case (cf. if the winning lottery numbers were 1, 2, 3, 4, 5, 6). (And presumably surprise correlates with degree of belief.)

enigMan said...

And I could also argue that asserting a proposition in English is not the same as asserting it in German, on the grounds that I would only know what I was saying in the former case, since I can neither speak nor understand German, at all. But what would that prove?

enigMan said...

Regarding your last argument, conjunction-introduction is always valid (and so the two formal arguments are very similar, more a matter of style than substance). And in particular, if "One of x1,..., xN will win and x1 won't win and ... and xN won't win" is necessarily false then "One of x1,..., xN will win. x1 won't win. ... xN won't win." necessarily includes a falsity. So the semantic difference between those two statements is a matter of what you know, rather than of what you say (whence my previous comment about English and German).

Regarding temporal indexicals, it may be better in such cases to consider propositions rather than sentences, as otherwise it would again be a matter of style rather than substance. But even so, I can say quite straightforwardly (and without emphasis): "This frog is moving, and now it is stationary, and now it is moving; frogs move by jumping."

Regarding the difference between correcting mistakes and not lying, one might assert p&q by saying: "The following statements are both true: Firstly, p. And secondly, q." By the time I am speaking about q, p has been asserted, and the conjunction is bound to be false if p is.

Research Writer said...
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enigMan said...
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Alexander R Pruss said...

I deleted a spam comment in this thread, and also deleted a response to the spam comment which contained a swipe at continental philosophy and no other philosophically substantive claims.

enigMan said...

Hi again; I was wondering if you still think that to assert a conjunction is not the same as asserting the conjuncts? I think that to assert a conjunction is precisely to assert the conjuncts, and that to assert the conjuncts in one go (so to speak) is to implicitly assert the conjunction. Furthermore, the implication is so elementary that there would have to be something seriously wrong with one's grasp of one's language for one to miss it. Could one say relatively unproblematically: "The cat is nowhere; the cat is on the mat"? But to say "The cat is on the mat" is not the same as asserting that the cat is somewhere, except by implication. (And regarding temporal indexicals, although saying "It is midnight" at midnight is not the same as saying "It is midnight" at noon, it is of course false that assertions are not the same as themselves:)

Alexander R Pruss said...

I am inclined to stand by my original argument here.

1. Here's another line of thought.

A sentence consists of a way of indicating the content plus a way of indicating the illocutionary force. (Many natural languages have a very simple way of indicating the illocutionary force of assertion, and the assertion is the base form, but a null can still be an indicator.) Now, think of a language where this works as follows. You indicate the content by expressing it in FOL. You indicate the illocutionary force by surrounding the expression of the content with square brackets, and preceding the square brackets by an abbreviation of the illocutionary force. Thus, we have things like:
Assert[p] ("I assert: p.")
Query[p] ("I query: p?")
Promise[p] ("I promise: p.")
Command[p] ("I command: p.")
Prohibit[p] ("I prohibit: p.")
Permit[p] ("I permit: p.")
Deny[p] ("I deny: p.")
Present[p] ("I present for consideration: p.")

Now, in all of these, the content is the same--the proposition that p--but the illocutionary force differs. Now, in some of these there is no plausibility at all to the thought that
I[p&q]
is the same speech act as the pair:
I[p] & I[q].
For instance, if I=Query, I=Deny, I=Prohibit, I=Permit or I=Present, these are clearly different speech acts. I also think we have different speech acts if I=Promise or I=Command, though this is somewhat less obvious. (If I promise to do both A and B, and I only do A, I haven't kept any promise of mine. The normative consequences of promising a conjunction are different from those of promising each conjunct separately. Ditto for commands.)

In all of these cases, there is one speech act per expressed propositional content. It would be surprising if in the case of assertion you suddenly had two speech acts. Consider such simple dialogues as:
A: Query[p & q].
B: Assert[p & q].

A: Assert[p & q].
B: Deny[p & q].

A: Command[p & q].
B: Promise[p & q].

A: Command[p & q].
B: Assert[p & q].

A: Command[p & q].
B: Deny[p & q].

It seems that in all these cases, the second speech act concerns exactly the same content as the first. But not so if Assert[p & q] is just Assert[p] Assert[q].

2. Here's another idea. There are many ways of formulating rules of inference, and we should not say that two people who have different habits of use of rules of inference use different connectives just for that reason. For instance, maybe your internalized rules of inference for conjunction are conjunction-introduction and conjunction-elimination. But Bob does it differently. He has three rules: conjunction-introduction (same as yours), conjunction-commutation (from "p&q" infer "q&p") and leftward-conjunction-elimination (from "p&q" infer "p"). Thus, for him it's two steps from "p&q" to "q", and it's not at all obvious. But, for him, it's obvious to go from "p&q" to "q&p", while for you this isn't obvious, because you have to apply conjunction-elimination twice and conjunction-introduction. (Here "obvious" means: one-step.)

enigMan said...

Re 1: To deny p&q is not necessarily to deny p and to deny q, but it is to deny either p or q (or both). So if the context includes a commitment to p, then there could, in a sense, be two speech acts here: the denial of p&q, and the denial of q. Of course, I would say that there was only the one speech act, which denies both q and p&q.

Now, if I deny pVq (p or q) then I am denying p and denying q, because denying pVq is not the same as either denying p or denying q, and so & and V form a naturally symmetrical pair.

Furthermore, by keeping part of a promise I may have kept the most important part of it. And if the literal promise turned out to be well nigh impossible to keep, I could be regarded as keeping to my promise if I put my effort into keeping that part at the expense of trying to do the well nigh impossible.

Re 2: I find it likely that logical conjunction aims to capture, in a language related to that of mathematics, part of what we naturally do when presented with a descriptive list of assertions. The commutation therefore comes most naturally to those used to logical and mathematical notation, and to those who are primarily describing the world.

I agree that those who use connectives differently might not be using different connectives, but I don't see how that helps your original argument.

enigMan said...

...incidentally, thinking about your second idea gives me the germ of an idea for an argument for your view. When we assert several things. we're not just asserting all of them, we are most naturally giving them in a list. The second assertion is naturally given in the context of the first having been asserted, and so on, whereas conjunction (unlike the English "and") is always commutative. (Still, that's only the germ of an idea because I'm not sure how it prevents the assertion of a conjunction being the same as asserting all the conjuncts in one go:)

enigMan said...

...upon reflection, that is just another thought against your original argument: each assertion being made in the context of the previous assertions, you assert "xN won't win" in a context which makes it contradict some of your other assertions (if there is a contradiction there, rather than some vagueness), and hence just as problematic as the conjunction. You could of course be unaware of the former contradiction, but the only way to get a substantial difference would be if you were no longer committed to your previous assertions when you said "xN won't win". But then you would not have simply asserted the conjuncts (e.g. you might have later retracted some of them, or changed your identity). So I would say that to assert a conjunction is the same as simply asserting the conjuncts.

enigMan said...

Indeed, I think that to be asserting a conjunction is to be asserting the conjuncts. It may not seem so, but that is I think because of the ambiguity of the English "and". If the fact that one is using "and" conjunctively were made explicit, e.g. "The cat is on the mat and, conjunctively, the dog is..." then it would be clear that part way through asserting the conjunction one had asserted the first conjunct.

And while there is that ambiguity to "and", there is a similar ambiguity in simple assertions, i.e. knowing when one has ended and the next has begun. (Indeed, I think that logical conjunction is more like asserting a set of propositions than the use of "and".)

Consider the following 2 responses to "Where are the cat and the dog?" Firstly, "The cat's on the mat (pause) and the dog's..." Secondly "The cat's on the mat (pause) the dog's..." The first might be to begin to say "The cat's on the mat, and the dog's by the door." or it might be "The cat's on the mat, and the dog's mat, not the other one, and..." Similarly the second might be to begin to say "The cat's on the mat. The dog's by the door." or it might be "The cat's on the mat, the dog's mat, not the other one, and..."

James said...

I'm not sure if this is a relevant here or not. But last night my housemate and I had the following conversation, and I thought it was quite interesting.

What are you up to now?
I'm going to watch a bit of this film?
Oh, you're not going to watch all of it?
No, I don't really fancy watching it all: maybe just half an hour's worth.
Are you going to watch it from the start?
Of course, I don't want to start it half way through.
Oh. Odd that you just want to watch the first 30 mins of this film.

It occurred to me that while it was true that

(1) I wanted to watch 30 mins of film F, and that
(2) I wanted to watch F from the beginning,

it wasn't actually true that

(3) I wanted the conjunction of (1) & (2), namely to watch the first 30 mins of F--at least not specifically.

I'm not sure quite what's going on here though, and wondered if anyone had any interesting thoughts on it.