>[A priori insight is] also closely connected to the ontological status of relations of
>necessity. If we can grasp such relations a priori, they must really be
>“out there” in some sense. (Bosanquet held that logical implications were
>of the form, “This, or nothing.” If he’s right, we can go a long way toward
>justifying belief in a logically coherent reality by showing that we depend
>on a priori insight into necessity in every statement we make. Note that I
>said “justifying,” not “proving” – I don’t think there can be a “proof” of
>this belief that doesn’t beg the question.)
I think there are necessary truths, truths about counterfactuals, etc., but I’m wary of talking about a priori insight. I’m not sure I have any nor am I sure what I need it for. Naturally, some things seem obvious to me and I find myself unable to imagine alternatives, but that can be the case for things that are quite coherent. I find lots of people think it obvious that there can’t be more than three dimensions or that space has to be Euclidean.
I’m inclined to just view logic as a high-level theory – roughly, the theory of what’s true, no matter what else is. We can deal with necessary truths using the semantics of possible worlds and interpret that (for S5, for example) as logical truths ranging over maximal consistent sentence sets. Counter-factuals and laws of nature can also be handled in these terms using different accessibility relations. (I hope the last couple of sentences are not too opaque. They presuppose some familiarity with modal logic.)
>4. Our fallibility in principle isn’t always terribly important. If I were
>to acknowledge that I could be wrong about, say, the Law of
>Noncontradiction, I wouldn’t really be granting that there are any legitimate
>grounds for doubting it; indeed I’d be saying that any such grounds will
>automatically be illegitimate because (I can tell apriori that) they will
>assume the very Law at issue.
>
>(Of course if I can’t be wrong about the Law of Noncontradiction – as in
>fact I think I can’t – then it’s not true that we can always be mistaken
>in what we take to be a priori truths in the first place.)
It seems to me that I can, just barely, conceive of the falsity of the Law of Non-Contradiction. (I don’t think that’s any reason at all not to think it’s true, though.)
One thing that leads me in this direction is thinking about the Law of Excluded Middle. I used to think that was obvious. Indeed, in standard first-order logic, it’s equivalent to the Law of Non-Contradiction. However, I’ve come to think that it’s not true in the absence of a domain restriction of some kind – or, to put it differently, there’s more to logic than standard first-order logic.
You might consider this analogy: The best evidence we have indicates that space is non-Euclidean, though it very nearly approximates to Euclidean space under the conditions present on earth. However, people seem to be “wired” to interpret space as Euclidean. This is easy to understand in evolutionary terms. Since Euclidean space is a very good approximation to the truth on earth, wiring our brains to find the real structure of space intuitive might have been more trouble than it was worth – it would require too much processing power in exchange for too small a gain in reproductive success. So we find Euclidean space intuitive and, unless we’ve been beaten over the head with the relevant math and physics, we find ourselves unable to imagine any alternative. Nonetheless, there is an alternative (more than one, but leave that aside) and that alternative happens to be the truth about the way physical space is structured.
What if the Law of Non-Contradiction is like that? Almost always, it holds, at least in our neighborhood, but every now and then, nature allows a contradiction. Building and supporting the processing power to understand a system of logic that allows occasional contradictions may not have had enough reproductive pay-off to be worth it. So, we don’t see how it could be true. Nonetheless, it might be.
Again, I’m not saying we have the least reason to doubt or reject the Law of Non-Contradiction. But I don’t see that anything guarantees that we are right about it. It may be that we can’t see an alternative to it because there is no alternative. But it may also be that we’re not smart enough to see the alternative.
Best,
Rob
---
Rob Bass
Those who agree with me may not be right, but I admire their astuteness.
– Cullen Hightower
=============
Hi, Scott –
[....]
>> I think there are necessary truths and truths about counterfactuals,
>> etc., but I’m wary of talking about a priori insight. I’m not sure I
>> have any nor am I sure what I need it for. Naturally, some things
>> seem obvious to me and I find myself unable to imagine alternatives,
>> but that can be the case for things that are quite coherent. I find
>> lots of people think it obvious that there can’t be more than three
>> dimensions or that space has to be Euclidean.
>
>Part of the problem here is the ease with which we can conflate several
>different ideas: a priori knowledge (on any of several understandings),
>necessity of several sorts, and self-evidence. Forget self-evidence; I’m
>not defending that. Drop the term “a priori” if you like; I’m not married
>to it either. The key point is that we can grasp relations of necessity and
>thereby know truths that can’t be undermined by later experiences.
>
>I not only know that two plus two is four, I also know in advance that any
>apparent exception will admit of another explanation. I not only know that
>no specific surface is in fact red and green all over, I also grasp that
>none can be. And, relevantly to what follows, I not only know that no
>contradictions are in fact true, I also know that any apparent exceptions
>are just that. (I don’t think these examples are akin to the
>“intuitiveness” of Euclidean space, and I’ll say more about that below.)
>
>> I’m inclined to just view logic as a high-level theory – roughly, the
>> theory of what’s true, no matter what else is. We can deal with
>> necessary truths using the semantics of possible worlds and interpret
>> that (for S5, for example) as logical truths ranging over maximal
>> consistent sentence sets. Counter-factuals and laws of nature can
>> also be handled in these terms using different accessibility relations.
>> (I hope the last couple of sentences are not too opaque. They
>> presuppose some familiarity with modal logic.)
>
>You won’t scare me off with modal logic, but please note that the
>ontological character of logical law is already assumed in the idea of a
>“possible world.” Strip off the bells and whistles and a “possible world”
>is just one in which the laws of logic hold. Well, why are precisely
>those worlds to be regarded as “possible?” If we’re not just playing a
>language game, surely the reason is that we regard those laws as somehow
>constitutive of what it means to be possible. Why? Because we regard them
>as ontological laws – as you acknowledge in describing logic as “the
>theory of what’s true, no matter what else is.” And the key point is that
>we regard them thus because we see them to hold necessarily; otherwise
>why talk of “possible worlds?”
I didn’t expect to scare you off with talk of modal logic. On my view, of course logic is ontological – like any other theory, it says something about how things are, or in my words, what’s true no matter what else is. But I don’t “see” what “seeing” logical truths to be necessary has to do with it. If there are truths that hold, no matter what else is true (and I agree that there are), then of course there’s a clear sense in which those truths are necessary. What does “seeing” add except to say that we have a correct theory about those truths?
>I should also confess to my strong conviction – which I won’t defend here
> – that ultimately there’s precisely one “possible world,” and you’re
>sitting in it.
Now, this doesn’t seem the least bit plausible to me. I suppose I can conceive of all truths being necessary truths, but I don’t see anything to recommend it.
>That is, the Law of the Excluded Middle doesn’t actually fail; everything
>is still either “true” or “not true” in an absolute sense. But there’s a
>whole spectrum of “not-true” that ranges from “close but no cigar” all the
>way to “completely barking mad.”
Well, if “false” just means “not true,” then every well-formed statement will be either true or false. But in standard logic, “false” means not just “not true” but also “true that not.” There may, however, be cases in which an additional truth-value is needed, for example, in dealing with future contingents. (Remember Aristotle’s famous discussion of tomorrow’s sea battle. He appears to have concluded that bivalence fails to hold for that case. I agree.)
Other counter-examples to Excluded Middle come from fictional contexts (It is neither true nor false that John Galt had oatmeal for breakfast the day before he first ate lunch with Eddie Willers.) Still others can be constructed using Russell’s Paradox (which, by the way, I think is very deep and to which no fully satisfactory resolution has been provided). For example, we have to say that a set can’t be a member of itself even though it seems obvious that the set of non-coffee cups is not a coffee cup. Now, push that just a little bit further and you run into problems with Excluded Middle. On one hand, we have to deny that the set of all non-coffee cups is a non-coffee cup, but on the other, that doesn’t mean that the set of all non-coffee cups is a coffee cup! So, the set of all non-coffee cups neither is nor is not a coffee cup!
Avoiding examples like those may be possible if you put a lot of emphasis on the requirement that the statements to which Excluded Middle applies be well-formed – but we don’t have any good way in general of saying what well-formedness is. So, with respect to this range of cases, we can hold that Excluded Middle applies to every well-formed statement, but that won’t tell us that it applies to the statement in front of us. We can make it apply by defining a domain, but then we will only be making the more restricted claim that Excluded Middle applies within that domain, not that it always applies.
>> What if the Law of Non-Contradiction is like that? Almost always, it
>> holds, at least in our neighborhood, but every now and then, nature
>> allows a contradiction. Building the processing power to understand a
>> system of logic that allows occasional contradictions may not have had
>> enough reproductive pay-off to be worth it. So, we don’t see how it
>> could be true. Nonetheless, it might be.
>
>I see what you’re driving at here but I don’t think you can pull it off;
>the possibility you raise is not, strictly speaking, conceivable. What
>you’re maintaining, in effect, is that even though we can’t imagine that
>a contradiction could be true, nevertheless it might be.
>
>There are several problems here.
>
>1. If contradictions could be true, there’s no basis for limiting it to
>“every now and then.” Once we grant that any contradiction whatsoever could
>be true even though we can’t see how, we’ve undermined every possible
>belief we might have, since no belief would in principle exclude its
>negation.
That depends on what the correct law is. There’s no way of limiting it within standard logic, but it doesn’t follow that there’s no way of limiting it with some revisions of standard logic. Indeed, some people have claimed to be able to present a logical system in which non-contradiction doesn’t always hold. I know little about it, but they call it paraconsistent logic.
>2. On the other hand, I don’t think you can strictly conceive that a
>contradiction might be true even though your naturally-selected wetware is
>stuck thinking otherwise. If the possibility of a veridical contradiction
>has any meaning for you, then you are conceiving it despite the alleged
>limitations of your meat-based data processor; and if not, then in what
>sense can you be said to be considering that possibility?
It depends on what “conceivable” means. If it’s “conceivable according to standard logic,” then I can’t conceive either of a veridical contradiction or that I could be mistaken in thinking that there are no veridical contradictions. But that rather begs the question whether standard logic is correct. If it’s “conceivable” meaning “imaginable,” I may be unable to imagine a veridical contradiction, but able to imagine my being mistaken about that. (I have that problem with 11-dimensional spaces.) And there are other possibilities.
>3. The basic problem, I think, is that you’ve got the evolutionary cart
>before the ontological-epistemological horse. The evolution of a brain that
>accepts the Law of Non-Contradiction makes sense on the view that there
>are no contradictions; a reality-construing organ had in that case better
>operate accordingly. To put it the other way around is to treat the laws of
>logic as though they were laws of thought only, rather than also, and
>primarily, laws of reality.
But the evolution of a brain that accepts non-contradiction also makes sense on the assumption that there are very few contradictions and that programming the brain to handle the correct system of logic would cost more in terms of inclusive fitness than it would gain.
I don’t have any difficulty thinking that laws of logic are also laws of reality. In fact, I do think that. I also think we’re right about the Law of Non-Contradiction. But I still think we could conceivably be mistaken. (Remember that the denial of “there are no true contradictions” is not “all contradictions are true,” but “there are some true contradictions.” If a correct system of logic allowed some true contradictions, it might also have some way of keeping the ones allowed from infecting and subverting the rest of the system.)
==============
Scott –
I’m not sure we can get very far on the issues about logic. On the issues in dispute, we seem too far apart. Still, I have some remarks. Since that’s what we’ve been talking about mostly, this is a combined answer to (some things in) your last two messages.
On formality:
I entirely agree that we can’t capture necessity relations in a Principia-esque logical system. But I don’t see what the objection is to modal logic. You say:
>But it still doesn’t do the job. Here’s why: the only sort of
>necessity/impossibility that modal logic can discuss is the purely formal
>kind. In order to assert the mutual exclusivity of red and green, I must
>either (a) simply postulate a proposition of the form in the preceding
>paragraph, or (b) derive it by finding a formal contradiction between
>“red” and “green.”
[....]
>And case (b) won’t happen. The mutual exclusivity of red and green depends
>on the content of these terms, not on any purely formal relation between
>them. In order to have a formal contradiction, I have to postulate further
>that being green is a specific way of not being red; in this case I again
>have to rely on the relevant insight in advance of formalization, and that
>insight disappears once I do formalize. (For how would I formalize the
>statement that being green is a way of not being red, except by falling
>back on case (a)?)
My first thought is to wonder why you left out reference to possible worlds. To say that red and green exclude one another, that it is impossible for one object (at one time, etc.) to be both red and green is to say that there is no possible world in which it occurs – that is, that there is no consistently describable world (or state of affairs) in which an object is both red and green. But that means – doesn’t it? – that redness, fully (enough) characterized is logically incompatible with greenness, fully characterized.
I really don’t see what the problem is. You can’t, after all, demonstrate a contradiction or impossibility even in something like “James is a married bachelor” without filling in some definitions. Why should you expect it to be any different with the mutual exclusion of colors or musical notes or anything else you care to name?
I think it is not true that “[i]n order to have a formal contradiction, I have to postulate further that being green is a specific way of not being red.” What I have to do is to find some necessary feature of red things that is necessarily absent in green things (or vice versa). Then I build relevant definitions that incorporate those features ... and away we go!
The only sense I can see to “[t]he mutual exclusivity of red and green depends on the content of these terms, not on any purely formal relation between them” would be that there is nothing contradictory about asserting that an object is both red and green, but nonetheless, there is no possible world in which the combination occurs. I don’t think that’s intelligible. (Some people have supposed there was something that they termed ‘metaphysical necessity’ that was like that, but I think they’re confused.)
None of the above means we can’t recognize a necessary relation without formalizing it. Obviously, we can. But I see no reason for supposing there are necessary relations that are in principle beyond the scope of formalization. (The modal system I was assuming above was S5. Of course, all sorts of other neat things can be done with different accessibility relations. It’s even possible – and consistent – for there to be an accessibility relation such that “~ P” and “~ ~P” can both be true!)
In addition, you wrote:
>Finally, I should note that the concept of concrete necessity is precisely
>what keeps a coherentist epistemology tied firmly to reality so that it
>doesn’t drift off into anti-empiricism. On the view that concrete
>necessities are actually “out there” to be discovered, coherentist
>rationalism is empirical; the “bad” kind of rationalism consists
>essentially of trying to do ontology in a vacuum of pure formalism, when in
>fact we can’t grasp concrete necessities in advance of our particular
>understanding of the cases in which they arise.
Of course, concrete necessities are ‘out there’ to be discovered. That’s not the least reason for thinking they can’t be formalized. Is your concern that they can’t be fully formalized in a finite system? If so, that’s true enough, but as you know, we can’t do that even with arithmetic. There will always be more to be discovered and questions we can’t answer using the techniques and formal systems we’ve developed so far.
On A Priori Insight
>> On my view, of course logic is ontological – like any
>> other theory, it says something about how things are,
>> in my words, what’s true no matter what else is. But I
>> don’t “see” what “seeing” logical truths to be necessary
>> has to do with it. If there are truths that hold, no matter
>> what else is true (and I agree that there are), then of course
>> there’s a clear sense in which those truths are necessary.
>> What does “seeing” add except to say that we have a correct
>> theory about those truths?
>
>Primarily, it adds a consistent account of how we can know this.
You haven’t given any reason for supposing that other accounts are not consistent, so that doesn’t support your account over others. I think our ordinary logic, including non-contradiction, is correct. Since I view logic as a high-level theory, our knowing it to be correct is just a matter of its successfully passing every relevant test we can think of. Here’s something I wrote elsewhere about logic:
--- begin quote ---
Here’s an attempt to say a little more about how I look at logic. We can start from a characterization from Popper: a system of logic is a set of rules for transmitting truth from premises to conclusions and for retransmitting falsehood from conclusion to premises. In other words, if the premises are true (and the rules have been followed), then the conclusion has to be true as well. If the conclusion is false (and the rules have been followed), then at least one of the premises must be false as well. (This is just a starting point and Popper was mistaken if he thought that it was a full characterization. A full characterization would need to say something about systems with more truth-values than ‘true’ and ‘false,’ or that employ probability-metrics or degrees of confirmation, and so on.) That, so to speak, sets an ideal for a system of logic.
The question is: how do we bring the ideal down to earth? Where do our logical systems come from? I don’t think we start from intuition except perhaps in the blandest and most innocuous sense – namely, with things that seem obvious for which we do not at present see a further reason. Where we start, both in the life of an individual and in the history of the species, is with inferences or arguments, with taking one thing as a reason for something else. Plainly, both in the case of each person and historically, we did that before we ever formulated or considered principles of logic.
But when, on this level, we reflect a bit on the arguments and inferences we make and hear others making, we find that some seem better than others; one of the obvious things about arguments is that they’re not all equally good. We attempt to construct systems of rules that will both formalize our practice and discipline it, that will give us the “intuitively” right answers in the easy cases and help us to find right answers where they aren’t intuitively obvious. They help us to understand what good arguments have in common as well as to see where defective arguments (which may also seem good) go wrong. These are systems of logic. None of them, I think, is founded on absolutely self-evident or incontrovertible principles. They are, one and all, open to development and improvement. (Some, though, may not need to be improved; it’s possible we’ve gotten something right.)
Now, this perspective does depend on a kind of assumption, but it is not the assumption that any particular logical principle is a certain starting-point. Rather, it is the assumption that we are not hopelessly bad at reasoning. If we were hopelessly bad, then there’d be no more reason to accept what seems obviously right than to reject it nor would there be any reason to try to improve our logical systems. If you like the term, you might call this our basic cognitive act of faith. But if it’s an act of faith, it’s one to which we really have no alternative: If it’s a mistake, all bets are off – including the bets of anyone tempted to deny it. For even to make clear what is being denied, one has to rely on argument and inference. The doubter would be making an argument that there are no good arguments, providing a reason for not paying attention to reasons.
When people first hear of multiple systems of logic, I suspect they often feel dizzy or disoriented, as if the ground were shifting under their feet. But actually, there’s considerable convergence among different systems and such disagreement as there is is often at the margins where we are currently trying to extend the reach of our formal systems or is over the best or most perspicuous way to formalize facts and relations admitted on all sides. In short, there is a substantial consensus. Familiar principles like modus ponens or non-contradiction in their ordinary applications are in no danger whatever. But I think it’s a mistake to say that the validity of logic “comes from human agreement or consensus [or the way our brains happen to be structured].” If that were really so, we could have no good reason to change or alter what there was already agreement upon.
--- end quote ---
>>>I should also confess to my strong conviction – which I won’t
>>> defend here – that ultimately there’s precisely one “possible
>>> world,” and you’re sitting in it.
>>
>> Now, this doesn’t seem the least bit plausible to me. I suppose
>> I can conceive of all truths being necessary truths, but I don’t see
>> anything to recommend it.
>
>To pursue this point would take us onto a major sidetrack at this point,
>but it may come up again so here’s the short version: (a) to explain
>something is to see it in a context that renders it necessary, and (b)
>everything is explainable. I won’t try to defend either of those views
>here.
I’ll try to be almost as brief: I think (a) and (b) are both false. Note that if either one is, that’s enough for there to be more than one possible world.
About (a), there are explanations that do not involve seeing something in a context that renders what is explained necessary. Suppose you flip a fair coin a thousand times and come up with heads 503 times. You ask why you came up with 503 heads rather than 600 and I explain it by telling you that 503 heads are more likely than 600. That’s a perfectly good explanation, but it’s not an explanation that renders the result you got necessary. There isn’t anything necessary about your getting 503 heads nor even about your not getting 600.
About (b), explanations are normally in terms of something else, so there isn’t an explanation for everything unless there’s also an explanation for the “something else.” To avoid a lot of complications, it seems to me that there’s no logical impossibility in there being nothing that exists at all. (How could there be? If there’s no true statement of the form, (›x) Fx – i.e., there is something that has some property – what material do you have to derive a contradiction?) If an explanation has to be in terms of necessity, there is no explanation for why there is something rather than nothing.
>> That depends on what the correct law is. There’s no way of limiting it
>> within standard logic, but it doesn’t follow that there’s no way of
>> limiting it with some revisions of standard logic. Indeed, some people
>> have claimed to be able to present a logical system in which
>> non-contradiction doesn’t always hold. I know little about it, but
>> they call it paraconsistent logic.
>
>I’m not familiar with it enough to comment but you’ll find if you look
>closely that whatever it is, it assumes that the Law of Non-Contradiction
>still holds in its “primitive” sense even if not in its “standard” sense.
Briefly – and this just about exhausts my knowledge of paraconsistent systems – it is not a theorem of such systems that, for arbitrary values of “P,” ~(P & ~P). That is, there is no axiom or inference rule of the system such that ~(P & ~P) can be derived without some additional premise. But the very fact that the Law of Non-Contradiction cannot be derived in the system shows that the system is consistent – because, in an inconsistent system, anything can be derived.
>And I’m afraid we do find something of the sort going on if we get specific
>about “imagining” oneself mistaken about the Law of Non-Contradiction (or,
>more precisely, about its universal applicability, or having the entirety
>of reality in its domain). The question-begging actually occurs in
>considering alternatives, not perhaps to what you mean by “standard” logic,
>but to the “primitive” Ur-logic on which the validity of the former
>ultimately rests. The Law of Non-Contradiction, in its “primitive” version,
>basically says that things are this way rather than that, and you’ll find
>that you assume precisely this in imagining that there is an objective
>reality to which the Law fails to apply.
No, I don’t need to assume the Law of Non-Contradiction to think that there may be exceptions to the Law. The Law says that always (for well-formed statements), ~(P & ~P). I could believe that sometimes it is true that (P & ~P) without believing that this time was one of those times. That is, I could believe it true and not false that there are exceptions without thinking that this very belief is one of the exceptions.
In interpreting what I said above, please do not attribute to me the belief that the Law is not true or that it is only a law of thought, not of reality. Of course, I do believe in the Law and, just about anywhere outside of such abstruse discussions, I will take a contradiction in a position as sufficient to show that it can’t be true. I won’t – at least not because of this – disagree with you on other things at all. I just don’t think “a priori insight” gets us anywhere. It seems to me like an appeal to magic in epistemology.
==================
Hi, Scott –
>Okay. I think you’re right about our approaches to logic being different,
>but one more round may at least be instructive. I won’t try to hit every
>one of your points; I’ll try to limit myself to points that illustrate the
>differences.
[….]
>> I think it is not true that “[i]n order to have a formal contradiction, I
>> have to postulate further that being green is a specific way of not being
>> red.” What I have to do is to find some necessary feature of red things
>> that is necessarily absent in green things (or vice versa). Then I build
>> relevant definitions that incorporate those features ... and away we go!
>
>And this is exactly what I’m saying you can’t do. If this shade of red now
>before me is not further analyzable (as I think it is not), then there
>isn’t any nugget present in it that’s absent from that shade of green and
>there’s no way to define either one other than ostensively. And even if
>particular shades were further analyzable, you’d simply have the same
>problem again at the next level down: this shade of red has feature R,
>while that shade of green has feature G; now it’s R and G as such, rather
>than red and green as such, that are mutually exclusive. Do we now need to
>find a necessary feature of R that is necessarily absent from G? At some
>point we just have to see that two “specific universals” (Blanshard’s
>term) are necessarily mutually exclusive. I don’t see any epistemological
>magic in this, nor do I see that formal logic is adequate to capture such
>mutual exclusivity apart from my postulating it.
You’re familiar, I suppose, with the determinable-determinate distinction? Red and green are different determinates of the determinable, color. Crimson and scarlet are different determinates of the determinable, red. Square and round are different determinates of the determinable, shape. Pure heterosexual and pure homosexual are different determinates of the determinable, sexual orientation – and so on. The distinction can be applied on different levels – as the red-green and crimson-scarlet instances show. But as long as the determinates are on the same level – so we’re not comparing, say, crimson with red – different determinates of the same determinable always exclude one another. It seems to me that that’s all we need to know to get a straightforward logical contradiction out of saying that the same surface is both red and green. We need to know that they are different determinates of the same determinable. If you can’t tell that much about the difference between red and green, then I suppose there’d be trouble formalizing their mutual exclusion of one another – but if you’re not allowed that much, I’d wonder how you know that they really are mutually exclusive.
>> About (a), there are explanations that do not involve seeing
>> something in a context that renders what is explained necessary.
>> Suppose you flip a fair coin a thousand times and come up
>> with heads 503 times. You ask why you came up with 503
>> heads rather than 600 and I explain it by telling you that 503
>> heads are more likely than 600. That’s a perfectly good explanation,
>> but it’s not an explanation that renders the result you got necessary.
>> There isn’t anything necessary about your getting 503 heads nor
>> even about your not getting 600.
>
>1. It’s not a perfectly good explanation of why I specifically got 503 in
>this specific series of tosses, as your own account admits by acknowledging
>that after you provide it, the result “503” still appears to be contingent.
>A full explanation of this concrete series of coin tosses would involve at
>least a physical description of the history of each toss, and with such a
>description I think we would begin to see some necessity at work. The
>problem with your explanation is that it’s at the wrong level: I don’t want
>an explanation of a general and abstract proposition, I want an explanation
>of why this series of tosses eventuated as it did. Not that your
>explanation wouldn’t explain to any degree, but I would hardly call it a
>full explanation.
It’s a perfectly good explanation if there isn’t a better explanation. Or put it differently: If there are any fundamental laws of nature that involve chance irreducibly – so the most basic law has some form like “If A then B with probability p or C with probability 1-p” – then there will be no deeper explanation why B turned up on a particular occasion than (if that is the case) that p > 1-p. (Similarly, if C turns up on a particular occasion, there will be no deeper explanation than that it is allowed by the fundamental law.) As nearly as I understand what the quantum physicists say, most of them think that the most basic laws of nature do involve chance irreducibly.
I see no philosophical reason to object to that. Even if it should turn out that the physicists are mistaken – so the most basic laws of nature do not involve irreducible chance, that would still leave something unexplained if explanation has to be in terms of seeing something to be necessary. It would leave it unexplained that the laws of nature never involve irreducible chance. (Unless, of course, you think that’s a necessary truth! I know some people have claimed that, but all the arguments I’ve seen for it are patently unsound.)
There’s also the question: why these laws of nature rather than some others?
>(Of course we aren’t going to achieve such an explanation, but nothing
>less would be fully satisfactory. As long as we can still ask “Why?” we
>haven’t arrived at full understanding.)
We may have arrived at as full an understanding as is possible. We may have answered all the questions that have answers, but not all questions have answers.
>“In terms of” doesn’t mean “reduced to.” If the entire universe consisted
>of two facts and they could be shown, or seen, to be related to one another
>necessarily, a full explanation of this micro-universe would still not
>reduce to either fact; the explanation would be circular but not
>viciously so. The two facts would be explained by being included in a
>single system of which each was a necessary part.
But why is that system necessary? From “A is necessary given B, and B is necessary given A,” you cannot detach either A or B. It is entirely consistent to go on to say “neither A nor B.” It’s even consistent to go on to say “Necessarily, neither A nor B.”
>(By the way, it’s
>precisely the availability of such non-vicious circularity that allows
>“coherentism” to avoid an important critique of “foundationalism.” Your own
>account is vulnerable to this critique: how do you justify the foundational
>beliefs themselves?)
I’m puzzled. My account of what? What foundational beliefs?
>> To avoid a lot of complications, it seems to me that
>> there’s no logical impossibility in there being nothing that
>> exists at all. (How could there be? If there’s no true
>> statement of the form, (›x) Fx – i.e., there is something
>> that has some property – what material do you have
>> to derive a contradiction?) If an explanation has to be in
>> terms of necessity, there is no explanation for why there
>> is something rather than nothing.
>
>Hmm. Your argument doesn’t actually show that existence isn’t necessary; it
>shows only that existence wouldn’t be necessary if there weren’t
>anything. And of course it wouldn’t be, or there would be. If you follow
>me.
>
>Silliness aside, “existence” doesn’t seem to me to be analyzable in terms
>of anything else; neither does “truth” or “necessity.” Nor, as I said, does
>this particular shade of red. That doesn’t mean that those unanalyzable
>items don’t sit in necessary relations with other items. Existence wouldn’t
>be existence if it didn’t differ from nonexistence; this shade of red
>wouldn’t be what it is if it didn’t differ in just this way from that shade
>of green, and in just this other way from that shade of blue. Those
>relations of difference are both necessary and internal to the characters
>they relate.
I don’t see any answer here. Saying existence enters into necessary relations does not support any claim that it is necessary that there is something. That something exists is necessarily true if and only if its denial is self-contradictory. But its denial is not self-contradictory, therefore, it’s not a necessary truth (though, of course, it is a truth).
Look at it another way: For arbitrary values of “P,” “Necessarily, if P were the case, P would be the case” is true, but “if P were the case, necessarily P would be the case” is false. Inferring the second from the first is a well-known modal fallacy (briefly discussed in the paper on future contingents I sent). If, for the special case of “something exists” as the value of P, you want the second without inferring it from the first, you need some special reason for supposing it’s true. As far as I can see, it’s just not.
>Note, however, that by a realistic logic (as opposed to a “formalist”
>logic), it isn’t true that an inconsistent system allows “anything” to be
>derived. (Bosanquet and Russell had a famous argument about a closely
>related issue. I’m on Bosanquet’s side.) Again, formal “implication” isn’t
>the same thing as necessary inference by a longshot; from the fact that
>“the sky is green and the sky is not green,” I can’t derive “I am Julius
>Caesar” by any necessity at all. Or even “I am Scott Ryan.” Neither
>proposition considered in itself has anything to do with the color of the
>sky. And on the postulate of a world so different from our own that the sky
>could be both green and not green, I don’t see that any inferences would
>be possible. (Real inferences, I mean, not “material implications.”)
Sounds like you’re talking about relevance logics.
>Note also that you rely just as surely on Ur-logic in discussing these
>alternative systems as you do in discussing “standard” logic. The premises
>and rules of inference of a system commit you to all the valid derivations
>and not to the invalid ones. The relation between the premises and rules,
>on the one hand, and the valid derivations, on the other, is not one the
>developers of such alternative systems seem willing to drop.
Of course. That’s part of my point. Admitting some contradictions doesn’t mean that “anything goes.” What “goes” depends on the system.
=============
Hi, Scott –
I’ll only make a couple of remarks.
1. Somehow, it looks as if we’ve slipped into talking past one another. Sometime back you said you thought there was only one possible world and supported that with the claims that explanation involves setting the explanandum in a context in which it is seen to be necessary and that there is an explanation for everything. I agree that those would imply that there’s only one possible world, but I think they are both false.
What I was trying to do was to exhibit cases in which one or the other or both of those claims failed. My aim was to argue that this isn’t the only possible world. If you admit that we don’t know a priori that there are not irreduceably probabilistic laws or that it is necessary that something exists or that the laws of nature are what they are rather than something else – you’re at least agreeing with me that, as far as we know, this isn’t the only possible world.
>> As nearly as I understand what the quantum physicists say, most
>> of them think that the most basic laws of nature do involve chance
>> irreducibly.
>
>Most, perhaps, but not all. For whatever my own opinion is worth
>(admittedly not much), I think the minority view of e.g. Albert Einstein
>and Heinz Pagels is correct: something determinate is going on down there,
>we just don’t have observational access to it. But I certainly don’t regard
>that view as “provable.”
2. This is odd. I would have said they were on opposite sides. More precisely, they’re on the same side on one point – that something determinate is going on down there – but I never suggested that there wasn’t something determinate going on down there. But they are on opposite sides about whether what is going on does or does not include anything irreducibly probabilistic. Pagels is definitely on the side of chance (which is my side), while Einstein is not.
Best,
Rob
---
Rob Bass
Those who agree with me may not be right, but I admire their astuteness.
– Cullen Hightower
=========================
Back to logic:
>> What I was trying to do was to exhibit cases in which
>> one or the other or both of those claims failed. My aim
>> was to argue that this isn’t the only possible world. If
>> you admit that we don’t know a priori that there are not
>> irreduceably probabilistic laws or that it is necessary that
>> something exists or that the laws of nature are what they
>> are rather than something else – you’re at least agreeing
>> with me that, as far as we know, this isn’t the only possible
>> world.
>
>Not quite.
>
>1. What I think I said was that I didn’t think we could prove a priori
>that there are not irreducibly probabilistic laws. That doesn’t contradict
>either my conviction otherwise or my claim (below) that the alternative
>view would have to meet an apparently impossible burden of proof in order
>to render itself even plausible.
I see no reason for that conviction and therefore no reason the burden of proof is impossibly hard.
>2. That it is necessary that something exists: Here what I think I said was
>that we ultimately had to regard reality as given. If it’s true, as I think
>it is, that existence is irreduceably “given,” then I suppose we could say
>that as far as we can tell, there might not have been any universe in the
>first place. But the nonexistence of anything whatsoever hardly qualifies
>as a “possible world”; on the contrary, it would be the express absence of
>any world at all.
Sure it does. A possible world is a logically possible state of affairs. I can tell you how to produce an exhaustive (infinite) set of sentences for describing such a state of affairs and why that set is consistent. For every single-place predicate, ~(›x) Fx. For every two-place predicate, ~(›x)(›y) Rxy. For every three-place predicate ... and so on. The set is consistent because no properties or relations are attributed to anything, and therefore there is nothing to conflict with any properties or relations attributed. (Of course, to produce the set, you would have to have a list of all possible properties.)
>Nor do I think this “possible world” is strictly conceivable. I don’t seem
>to be able to conceive of absolute nothingness without depending on a prior
>conception of existence; there doesn’t seem to be any way to conceive of a
>sheer blank except by implicit contrast with something that isn’t there.
>If that’s correct, then strictly speaking we can’t regard “nonexistence” as
>a possible world.
I think you’re confusing conceiving with imagining. Lots of things I can’t imagine are perfectly possible. (Many, no doubt, are actual.) Whether I can conceive something, though, could be understood in two ways. First, I might be unable to conceive something because I lack the relevant concept(s). Plainly, that’s no bar to its being true. Second, I might be unable to conceive it because there are inconsistencies in the notion. As I pointed out above, there are not.
>Nor am I sure what it would mean for it to (fail to) be necessary that
>“something exists.” Necessity, as I conceive it, operates within the
>context of existence; this or that particular “thing” came into being
>through a causal process in which an element of necessity was present, and
>it sits in relations to everything else that are “necessary” in the sense
>that without them, the thing would have to be other than what it is. It is
>in this sense that I regard everything as in principle explicable.
>
>But by what sort of necessity could it be, or fail to be, necessary that
>“something exists”?
By logical necessity – or, in this case, the lack of it.
>Would it mean that the proposition
>“something exists” is “necessary” in the sense that its negation is
>self-contradictory? Again, in a limited sense this is true: the proposition
>“nothing exists” surely exists, and moreover could not do so as a
>proposition apart from the existence of a given reality to which it might
>refer or fail to refer. Would it mean that the judgment “nothing exists”
>is self-contradictory? This is also true, in a less limited sense: a
>judgment is a mental act, and mental acts exist; the judgment’s content
>would be contradicted by the fact that it was being made. Would it mean
>that the fact that “nothing exists” would be a self-contradictory state
>of affairs? This is true as well, in an even less limited sense: if nothing
>existed, there wouldn’t be any “state of affairs.”
Not true. See above.
>I can only think, then, that you mean none of these things. But then I’m
>not sure what you do mean. Is there a way to ask the question that
>doesn’t presume existence?
Of course, if nothing existed, there’d be no one around to say or think that nothing existed. And it’s obvious something does exist. That doesn’t make it necessary. For example we could conceive of the possibility that the universe is gradually wasting away. I don’t just mean energy being degraded and becoming unusable; I mean matter or energy or what-not actually ceasing to be. Surely, we can conceive of that happening to, say, a single proton. If to one, then why not to all? At some, presumably far future, date, the last particle will totally cease to exist. There will be no mass, no energy, no events of any kind. And since time and space are relations between events, there will be no time or space. Nothing – absolutely nothing – will exist.
>Then I may be misremembering Pagels’s arguments; my memories of The Cosmic
>Code are well over a decade old. Whatever his views may be, mine are as
>follows.
>
>To say “there is something determinate going on down there” that causally
>depends irreducibly on probabilities is to take away with one hand what
>one gives with the other. The meaning of a causal law is that under a set
>of conditions C, the result R occurs – invariably and, I would say,
>necessarily. And while I don’t regard a priori insight as utterly
>infallible, if the form of causal law does depend on such insight, as I
>think it does, there is a heavy burden of proof on any view that would deny
>it.
I disagree. There’s no reason to suppose that “if A, then B or C” isn’t a respectable causal law. (After all, it rules out D and, more fully stated, would assign definite probabilities to B and C.)
But even if you do insist that’s not a real causal law, I’ll just ask why you assume that there are causal laws everywhere. As far as I can see, the only case for it has to do with the pervasiveness of causal relations in our everyday experience. But those can be explained by underlying irreducibly probabilistic laws, so they provide no evidence that the underlying laws aren’t probabilistic.
>Now, if we find a law that says that under one and the same set of
>conditions C the results R and R’ occur with probabilities such-and-such,
>the natural presumption is that we haven’t completely specified the
>conditions; that is, if conditions C are not enough to determine whether R
>or R’ occurs, then conditions C must not be all the conditions. Result R,
>we would expect, has its own set of necessary and sufficient conditions, R’
>has its own, and we haven’t properly distinguished between them.
The “natural presumption,” again, is based on everyday experience which is consistent with irreducible underlying probability.
>And at the quantum level, there is no way to distinguish between those sets
>of conditions, because events at that level are unobservable; it is simply
>impossible for us to know what the sub-sub-microscopic conditions are that,
>other things being equal, lead to R in one case, R’ in another. In this
>sense, the physical laws of such events are “irreducibly probabilistic”
>because they are the best laws we can get, not because there’s something
>ontologically “probabilistic” going on. If so, then a complete explanation
>of such events may not be available to us, but it “exists” nevertheless.
Actually, there’s a proof – von Neumann’s proof – that the quantum statistics can’t be explained by underlying deterministic mechanisms. He made a mistake in adopting the “natural presumption” that the mechanisms would be local. Still, his proof shows that no deterministic local theory is consistent with the evidence. Bohm exploited the loophole to create a non-local deterministic theory. However, Bohm’s theory has other problems, among them, being incompatible with another extremely well-confirmed theory, general relativity. (It also adds exactly nothing to our predictive power – it yields the same predictions and the same probability distributions as other interpretations.)
>The alternative view is that the physical events themselves are
>irreducibly probabilistic. (That this is possible is, I take it, both your
>own view and the one you attribute to Pagels). But how might this
>conclusion be arrived at when it is admitted on all sides that the
>conditions are unobservable? If we don’t know what’s going on down there,
>then we don’t know, and that’s that. On what basis could we ever reify
>our ignorance into an ontological principle? (For that matter, on what
>basis could we rule out the possibility of ever finding any way to
>observe events at the quantum level, relying perhaps on as yet unknown
>physical laws?)
It’s not reifying ignorance. It’s accepting the best interpretation of the most rigorously tested and confirmed physical theory ever. Of course, we could be wrong – but so what? That doesn’t mean we aren’t justified in accepting it.
>So while I admit we can’t prove a priori that no physical laws are
>irreducibly probabilistic, I do think there is an a priori presumption
>against it that places the burden of proof on the alternative view. And by
>the nature of the problem, the alternative view can never meet that burden.
I see no a priori presumption for it at all. And the best interpretation of the best evidence we’ve got is against it.
Best,
Rob
---
Rob Bass
Those who agree with me may not be right, but I admire their astuteness.
– Cullen Hightower
