Supposing
Let’s say I ask you to suppose that p. Naturally, in supposing that p, you won’t at the same time suppose that not-p, even if you’re inclined to believe the latter. But is there anything in the concept of “suppose” such that supposing p means not supposing not-p? I’m inclined to answer `No’. The request that you suppose p and also that not-p might well make sense. Say, I make the request because I want to show you that from a contradiction, anything follows. And if I then request that you suppose that p, without also supposing that not-p, I need not be saying anything redundant. Say, I make this request because the next thing I want to show you is that p makes q very likely, and I no longer need you to suppose that not-p. But the first request won’t make sense, and the second request will be redundant, if the answer is `Yes’. How then do we explain why it’s so natural that when asked to suppose that p, one doesn’t also suppose that not-p?
Assuming ex contradictione quodlibet, it’s not surprising that in the majority of cases, when I ask you to suppose that p, there is a strong implicature that you don’t also suppose that not-p—the point of asking you to suppose that p is defeated if you also suppose that not-p, and that which I want to show follows from p, or is made likely by p, now follows trivially from “p and not-p”.
October 12th, 2007 at 2:11 am
hey wenghong,
i’ve been thinking a lot about supposing lately, and what exactly one is doing when one supposes p. there are a lot of possibilities, but let me ask you this: on what kind of conception of supposition would it be possible to suppose both p and not-p at the same time? (and i mean supposing them explicitly, stated in that fashion — not supposing a bunch of stuff that turns out, unbeknownst to the supposer, to entail a contradiction after a bunch of complex logical manipulations.) in other words, what am i supposed to be *doing* when i suppose both p and not-p? i’m not sure a good way to show me that anything follows from a contradiction is to ask me to suppose a contradiction and see what follows. that’s for a number of reasons, but the first is that i’m not sure you can ask me to suppose a contradiction to begin with.
October 13th, 2007 at 7:35 am
Another thing that I’ve considered about supposition is whether your suppositions have to include all your actual beliefs as well. There seem to be at least two different types of supposition - one at play in purely logical proofs, where your actual beliefs don’t seem to play a role (though interestingly, anything that has already been established in the proof is allowed to be brought in still) and ordinary ones, where suppositions don’t cancel out things you believe. I don’t think I can coherently suppose right now that I’m standing up, because it’s plainly clear to me that I’m not standing up. I can imagine what would be the case if I had been standing, but that’s a counterfactual sort of attitude, and not a supposition, in the sense that they ordinarily seem to me.
October 13th, 2007 at 12:11 pm
I thought a little about this as well. As Kenny, it seems plausible to me that suppositions do not have to include all one’s actual beliefs. Intuitively, we can equip agents with a supposition faculty in addition to a belief faculty. Basically, by associating with an agent two sets of possible worlds, one for beliefs and one for supposition, we can treat supposition formally like belief: truth in all possible worlds. Call the set of beliefs world W, and call the set of supposition worlds S. To suppose p is to have p true at all worlds in S. Of course, S cannot only be individuated by the p in question. We also need to include in S all other propositions, which are relevant for the supposition in question. Presumably, this will be context dependent and, depending on the supposition, have significant overlap with the actual beliefs in W.
To enable us to suppose contradictions, we could be liberal about the worlds in S. If we want to suppose that paraconsistent logic is true, for instance, we could include contradiction-tolerant worlds in S. Or if we want to persuade someone that ex falso quidlibet is true, we could ask them to assume that p & ~p is true at a world in S, take them through a piece of classical reasoning and show that q follows. Noting, it seems, prevents us from taking the worlds in S to be radically different from the worlds in W, though of course we are not forced to.
On this kind of idea, beliefs and suppositions would be distinct entities, but nevertheless receive the same formal treatment. Also, it seems clear that what goes on in S can affect what goes on in W. Assume that I do not believe q and do not believe ~q. In W, then, there will be worlds at which q is true, and worlds at which q is false. However, I have a feeling that p is relevant for the truth value of q, but I do not believe p and do not believe ~p. I suppose p to see if q follows. In S, then, all worlds are such that p is true. In S, I keep some of my beliefs from W fixed – those that are consistent with and relevant for the truth of p — and check whether q follows from p. Assume q does indeed follow from p, and so, classically, ~q follows from ~p. I can now ask how well q fits with whatever else I believe. If q, in contrast to ~q, is consistent with whatever else I believe, then that is prima facie evidence that I should believe p rather that ~p (or raise my credence in p). But of course, if the supposition that p itself is in conflict with most things I hold dear, then maybe I should believe ~p instead and modify whatever else I believe to accommodate the truth of ~q. But in either case, it seems that the supposition can and should affect what I believe.
October 14th, 2007 at 8:49 pm
Hi Mike and Kenny,
Thanks for the comments. Unfortunately, I don’t have a positive account of supposition to offer. But as I see it, supposition comes pretty cheaply. I can suppose that there are purple cows (even though I believe there are none), that it’s both raining and not raining (even though such a thing is impossible), that it’ll be sunny in Canberra tomorrow (even though I don’t know whether it’ll be sunny or not), that I’m standing up right now (even though I’m sure that I’m not), and that I’m a zombie (even though I’m inclined to think that zombies are inconceivable).
It does seem strange to be asked to suppose that an explicit contradiction is true, or that I’m standing up when I’m sure that I’m not—how could we suppose things believed to be patently false or impossible? I wonder, however, if the strangeness is due to our reading “Suppose p” as something like (1) “Believe p hypothetically” or as (2) “Imagine p to be the case”. I find it hard to say what I mean by “supposition” exactly, but on my conception, it’s neither (1) nor (2). In a logical or mathematical context, it doesn’t seem that when we ask someone to suppose that it is both raining and not raining, or that the biggest prime exists, we’re asking her to believe such impossibilities hypothetically, or to imagine such impossibilities obtaining. To see what follows from “It is both raining and not raining”, we don’t need to believe hypothetically or to imagine that it’s both raining and not raining. (It’s doubtful that we can imagine such a thing). In an everyday context, “Thomason cases” show why “Suppose p” should not be understood as (1). Supposing that my crafty business partner is siphoning money off the company, I don’t believe that he is. (He hides his traces well.) But hypothetically, if I believe he’s siphoning money off the company, then I believe that he is.
I’ve been taking the kind of supposition that goes on in logical proofs to be the same as the kind that goes on in ordinary life. They could well be different, but I would like to toy with the idea that they are the same; perhaps, which of one’s actual beliefs enter into the suppositions depends on context.
If I’m certain that I’m not standing up, it sounds strange to say, “Suppose I’m standing up…”, but it seems all right to say, “Suppose I had been standing up…”, or “Suppose it were the case that I’m standing up…”. Following James Joyce, there seems to be indicative supposing, as well as subjunctive supposing. Or is subjunctive supposing not supposing as ordinarily understood?
October 18th, 2007 at 1:41 am
wenghong, your last comment brings up something interesting for me: though i’d never really thought explicitly about it, i guess i’ve been tacitly thinking that “the kind of supposition that goes on in logical proofs” is *different* from “the kind that goes on in ordinary life”. when i make some “suppositions” in a logical proof, i’m engaged in a formal action of writing some things down and then seeing what “follows” according to some logical rules. when i suppose something in ordinary life, i’m engaged in some sort of epistemic action. it strikes me that, just as we need principles to explain how what goes on in formal proofs should relate (normatively) to our epistemic lives, we should not take the formal action of supposing to be identical to the ordinary epistemic action. they’re probably related, but it’s not clear to me how. by the way, i do take the ordinary notion of supposing to be what’s involved in subjective conditional credence assignments.
October 18th, 2007 at 11:31 am
Perhaps we can distinguish between the two kinds of supposition in the way that Stalnaker, in Inquiry, distinguishes between beliefs concerning necessary propositions and beliefs concerning contingent propositions. Let’s say we can neither believe nor suppose what is logically or metaphysically impossible. Then, when we purportedly suppose that p & not-p, we’re really engaging in some sort of metalinguistic supposition: we’re supposing the contingent proposition that the sentence, “p & not-p”, expresses a logical truth.
October 25th, 2007 at 6:34 am
i’m not very familiar with the Stalnaker — how would that help?
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