do we ever need nomic necessity?
Nov. 16th, 2005 12:31 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslAccording to the "received view" in formal philosophy of science, we need a notion of nomic (law-like) necessity in order to formulate scientific laws.
For example:
"all lions that ever drowned in the North Atlantic were female"
forall x ( (lion(x) /\ drowned-in-NA(x)) -> female(x) )
is not considered a law because it's a contingent fact,
while
"all bodies with mass have a gravitational attraction to the sun"
forall x ( ( body(x) /\ has-mass(x) -> has-grav-attraction-to-the-sun(x) )
is.
There is a sense in which the latter statement is necessary: if we learn that X is a body with mass, we say that X *must* have a gravitational attraction to the sun. While you could still say the word "must" when drawing a conclusion from the first sentence, you would be less likely to.
I would say that this is because the first sentence is only quantifying over actual lions in actual observed situations, whereas the second quantifies over all possible bodies, giving it the generality required for being a scientific law.
The necessity expressed by the "must" can be formalized by adding the so-called "nomic" modality (nomos(gr.) = law). There are many things that are nomically necessary that are not logically necessary: in fact, scientific laws are never logically necessary. Any statement that is logically necessary is unfalsifiable, and thus fails to be "scientific", at least in Popper's view.
Determinism can be seen as the view in which all true statements are necessarily true (different modes of "necessary" corresponding to different brands of determinism). While determinism is an irrefutable view, one should not simply discard the nomic modality: there exists an important difference between the two kinds of sentences exemplified above, even if it's only a cognitive difference: the second sentence allows us to draw conclusions about all potential massy bodies (or future situations involving massy bodies), while the first does not allow us to draw conclusions about all possible lions (or future lions).
My thesis has been about formalizing scientific reasoning. I think my formalization is safe, even though it doesn't use a nomic modality, because my laws always quantify over all possible situations.
So for example,
My system already distinguishes laws (tagged "
So while statements like
For example:
"all lions that ever drowned in the North Atlantic were female"
forall x ( (lion(x) /\ drowned-in-NA(x)) -> female(x) )
is not considered a law because it's a contingent fact,
while
"all bodies with mass have a gravitational attraction to the sun"
forall x ( ( body(x) /\ has-mass(x) -> has-grav-attraction-to-the-sun(x) )
is.
There is a sense in which the latter statement is necessary: if we learn that X is a body with mass, we say that X *must* have a gravitational attraction to the sun. While you could still say the word "must" when drawing a conclusion from the first sentence, you would be less likely to.
I would say that this is because the first sentence is only quantifying over actual lions in actual observed situations, whereas the second quantifies over all possible bodies, giving it the generality required for being a scientific law.
The necessity expressed by the "must" can be formalized by adding the so-called "nomic" modality (nomos(gr.) = law). There are many things that are nomically necessary that are not logically necessary: in fact, scientific laws are never logically necessary. Any statement that is logically necessary is unfalsifiable, and thus fails to be "scientific", at least in Popper's view.
Determinism can be seen as the view in which all true statements are necessarily true (different modes of "necessary" corresponding to different brands of determinism). While determinism is an irrefutable view, one should not simply discard the nomic modality: there exists an important difference between the two kinds of sentences exemplified above, even if it's only a cognitive difference: the second sentence allows us to draw conclusions about all potential massy bodies (or future situations involving massy bodies), while the first does not allow us to draw conclusions about all possible lions (or future lions).
My thesis has been about formalizing scientific reasoning. I think my formalization is safe, even though it doesn't use a nomic modality, because my laws always quantify over all possible situations.
So for example,
(IMPLIES (PRED1 x) (PRED2 x)) should be interpreted as saying that all potential objects (are these the same as Zalta & Fitelson's abstract objects?) satisfying PRED1 will satisfy PRED2. You could put a nomic necessity box in front of this statement if you like, but I don't think it adds anything.My system already distinguishes laws (tagged "
LAW TH" for some theory TH) from contigent statements (boundary-conditions, tagged "BC"). While laws in the corpus (a corpus is log of what has been seen before: the idea is that it represents the scientist's experience) can get reused, boundary conditions should not (although they are still true, as long as names are kept unique), except when the same condition remains across problems. Better idea: we could have libraries of boundary-conditions, for reuse in problems that share the same BC's. Each library contains statements a set of BC's, and you could possibly use several libraries simulatenously (e.g. one library has information about the sun's radiation, one has information about the Itaipu Dam).So while statements like
(IMPLIES (UCM B1 B2 (UCM-PERIOD B1 B2)) (= (acc B1) (/ (^ (vel B1) 2) (distance B1 B2))))) should get reused, statements like (= (height wall) (* 3 m)) should not, unless there exists only one wall in the universe, whose height is 3 meters. A statement like (= (height wall78942396) (* 3 m)) seems perfectly fine, however, as long as there is some name management (generating large random numbers seems like a fine solution).![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
Recognizing Nomic Necessity
Date: 2005-11-18 09:50 am (UTC)However, this argument doesn't really seem to hold up because I can also fairly easily imagine a world in which "all bodies with mass have a gravitational attraction to the sun which is inversely proportional to the *cube* of it's distance". Imagine a game of Quake, where the physics of gravitation are different.
Although intuitively there does seem to be at least a cognitive difference between accidental generalizations and nomic necessities, I currently don't see any reliable method/algorithm for distinguishing between the two. And as Quine would say, "no entity without identity". If we have no way of distinguishing between the two, we might as well get rid of these concepts.
Re: Recognizing Nomic Necessity
Date: 2005-11-18 12:29 pm (UTC)You seem to be saying that the law of gravitation is not logically necessary, and I agree. It is merely nomically necessary.
Re: Recognizing Nomic Necessity
Date: 2005-11-18 01:53 pm (UTC)I agree of course that the law of gravitation is not logically necessary. But, I currently unwilling to give it the status of nomically necessary either, because I don't believe we currently have any good semantics for nomic necessity.
My point boils down to if we don't have a good notion of nomic necessity, we should either:
a) Find a clear semantics for nomic necessity.
b) Get rid of this notion altogether.
Because, I am currently unable to see how to do a), I feel inclined to go for b) and argue that talking about nomic necessity is gibberish.
My Quake example was meant as a counter-example to what seemed to me as your implicitly defined notion of nomic necessity:
I would say that this is because the first sentence is only quantifying over actual lions in actual observed situations, whereas the second quantifies over all possible bodies, giving it the generality required for being a scientific law.
I read you roughly as saying the following:
Quantification in the first case is restricted to actual and observed phenomena.
Quantification in the second case is more general.
And this larger scope of quantification determines what nomic necessity is.
Now if quantification is truly more general in the second case, it would mean that you are quantifying either over things which are not actual or over things which have not been observed.
Things that have not been observed cannot play any role in determining what is true or what is nomically necessary in our scientific theory because by definition, we have not yet observed them. Thus, we can forget about these things.
Things that are not actual seems to refer to things in other possible states of affairs for the world. Differently said you are quantifying over things in other possible worlds.
Now, if you allow your quantification to range over ALL possible worlds, you basically end up with logical necessity. This is bad, since according to you we want logical necessity to be different from nomic necessity. Thus, your quantification must include more than just the actual world but also not all possible worlds.
The hard question is what are these other possible worlds? Do you have any criteria for determining what these possible worlds are? Why does the scope of your quantification include a possible world in which a male lion has drowned and not a world in which the law of gravitation doesn't hold as we know it?
If there is a nice way of separating out those worlds which have the same law-like generalizations as our own from the collection of all possible worlds, then we might have meaningful notion of nomic necessity. Otherwise, we might as well stop talking about nomic necessity.
Re: Recognizing Nomic Necessity
Date: 2005-11-21 10:20 am (UTC)Which "possible" are you talking about?
Not all *logically* possible worlds. Ok.
We are quantifying over all physically possible worlds. Or even just the nomically possible ones. Do I sound circular?
We can still talk about nomically possible worlds, and you know what I mean. What kind of answer could satisfy this hard question of yours?
You know exactly what I mean by "nomically necessary" and "physically possible". Is there some reason why I need to come up with some sort of semantics? What would satify you?
Re: Recognizing Nomic Necessity
Date: 2005-11-21 04:14 pm (UTC)Yes, you are being circular. I am challenging you to give me a good semantics for nomic necessity and all you are giving me is a definition in terms of nomically possible worlds.
"We can still talk about nomically possible worlds, and you know what I mean."
No, I don't know what you mean.
"What kind of answer could satisfy this hard question of yours?"
Give me criteria for determining what are nomically possible worlds.
"You know exactly what I mean by "nomically necessary" and "physically possible". Is there some reason why I need to come up with some sort of semantics? What would satify you?"
I strongly disagree here. I do NOT know what is meant by "nomically necessary".
"Physically possible" is only meaningful once we have agreed upon what the laws of physics are. All the worlds which satisfy these laws of physics are the physically possible worlds. But before we have decided on what will count as a law of physics, the notion of "physically possible" is meaningless.
Re: Recognizing Nomic Necessity
Date: 2005-12-07 11:09 am (UTC)For the zillionth time, this would require me to know all the laws of physics. Agree or disagree?
Do you agree that there exist laws of physics that are still undiscovered?
Re: Recognizing Nomic Necessity
Date: 2005-11-26 11:04 am (UTC)These worlds are the physically possible worlds! Is this good enough? What kind of answer are you hoping for?
Would you like me to give you a list of all physically possible worlds? Or a complete decision procedure?
If so, this would require me to know all the laws of physics...
And yet, the concept of nomic necessity is useful!
We can define the abstract set of all physically-possible worlds. Then a sentence like [n]phi means that phi holds in all physically-possible worlds. Do note that phi cannot be checked deductively. We are now busy with induction. (forgive my Denglish)
You're asking for too much.
"separating out those worlds which have the same law-like generalizations as our own from the collection of all possible worlds" requires knowing all the laws of physics. The whole point of using abstract concepts is that we can talk hypothetically about different sentences being laws, without having to give an extensional definition of the set of physically-possible worlds.
Re: Recognizing Nomic Necessity
Date: 2005-11-27 02:11 am (UTC)Yes, that is axactly what I mean!!!
Many philosophers of science seem to take the distinction between true "accidental generalizations" and actual "laws" very seriously. You can take a look at Alexander Bird's "Philosophy of Science" book. He spends quite a few pages trying to discredit the idea that laws are simply true generalizations. He believes there are many true generalizations which are simply not laws.
To me this position doesn't make sense. How could we possibly distinguish between a true accidental generalization and a law?
<< If you want to destroy this concept, the burden is on you.>>
Why?
<<And yet, the concept of nomic necessity is useful! We can define the abstract set of all physically-possible worlds. Then a sentence like [n]phi means that phi holds in all physically-possible worlds.>>
Once we have committed ourselves to a theory of physics T, I have no problem talking about the class of all physically possible worlds. This is analogous to talking about the class of all models that satisfy a first-order theory. In this case, any statement phi is nomically necessary if and only if it belongs to T.
What bothers me is to believe that there some metaphysically special and unique theory T. And here is why...
Suppose that T is an adequate theory of physics (ie. the actual world A is a model of T; or in symbols T |= A).
Now suppose, that phi is a so called "accidental generalization", which happens to be true in A.
Then define T* := T U { phi }. Clearly, T* |= A. Whence, T* is also an adequate theory of physics. Now notice that the class of physically possible worlds and our notion of nomic necessity are different depending on whether we define it terms of T or in terms of T*.
I do not mind speaking of nomic necessity and of possible worlds in which the same laws of physics apply, so long as we keep in mind that categorizing phi as being an accidental generalization or a law is simply a pragmatic choice. The choice is done for sociological or psychological reasons within the scientic community and NOT because there is some sort of metaphysical difference between these two categories as some philosophers of science try to claim.
Re: Recognizing Nomic Necessity
Date: 2005-11-27 02:37 am (UTC)By the scope. You can sometimes distinguish laws from accidental generalizations from the logical form of the statement: as Dale Jacquette pointed out, it could be said that we use an informal modal logic, where the word "must" corresponds to nomic necessity.
Of course, for every accidental generalization, there exists a potential law: just generalize it to all physically-possible situations, e.g. "all lions that could ever die in the North Atlantic must be female"
(if you're a determinist, everything is necessary. But that's not enough to make all truths laws: law need to be general too)
Re: Recognizing Nomic Necessity
Date: 2005-11-28 03:01 pm (UTC)"all lions that will ever die in the North Atlantic will be female" (E1)
"all lions that could ever die in the North Atlantic must be female" (E2)
or more formally as Dale Jacquette would write...
"for all x, (Lion(x) /\ Death_in_North_Antlantic -> Female(x))" (F1)
"for all x, (Lion(x) /\ Death_in_North_Antlantic -> [n] Female(x))" (F2)
Let A be the actual world.
I guess your point is that F2 is a stronger statement than F1 because
A|= F1 does not imply that A|= F2.
In other words, it is not because some accidental generalization happens to be true for our world that it will be true in other physically possible worlds.
I do not find this convincing because it presupposes a notion of "physically possible" that comes prior to us determining what is a law and what is merely an accidental generalization.
I prefer to define what it means for something to be physically possible by referring to what we have chosen to accept as our laws of physics. And this requires, sorting through all true generalizations and determining which ones are to be considered laws and which ones are to be considered accidental.
Stipulating that the difference between accidental generalizations and laws can be made by checking whether we feel psychologically at ease by introducing some modality like "must" is indeed a useful heuristic. But once again, the distinction seems to be psychological and not metaphysical. There seems to be no difference between these two types of statements other than a psychological willingness for human scientists to insert a modality into some sentences and not into others.
My take on this phenomenon is the following...
We have a web of beliefs à la Quine.
Those beliefs that are very central like mathematics and logic are more immune to belief revision because they support the rest of the structure. Scientific hypotheses that have not been well-established are at the fringes of the web. These beliefs can easily be changed with the arrival of new empirical facts because changing them will not require us to make any major modifications to the overall structure of the web of beliefs.
Now, what I claim is that statements so called "laws" are beliefs in the web which are fairly deeply entrenched but not quite as much as say mathematics. A lot of what we know about physics and chemistry would fall under this category of beliefs. As for "acciendental generalizations", these are also true statements to the best of our knowledge, but we usually do not refer to them as laws because they are not essential in providing structural support for the web of beliefs.
For example, "all lions that will ever die in the North Atlantic will be female" might be true but we are somewhat unwilling at a cognitive level to ascribe to this statement the status of law because we realize that it is not essential for our web. We can easily modify our web if this belief turns out to be false. On the other hand, it would require a lot of effort to modify our web if we had to give up on something like the law of gravitation.
Logical Necessity points towards those beliefs which we are most unwilling to give up on.
eg: "2 + 3 = 5"
Nomic Necessity points towards those beliefs on which we have placed quite a bit of faith to support the structure of our web of beliefs, but not quite as much as mathematics because we are still willing to revise these beliefs given enough empirical data to warrant a change.
I do not believe that these categories form a clear-cut partition, but I do believe that having these notions might be somewhat useful when we are engaged in belief revision.
We could establish different degrees of Nomic Necessity depending on how much support those beliefs provide for the rest of the web.
In any case, *necessity* and *possibility* have much more to do with our cognitive abilities and our conceptual framework than with the state of affairs in the world or other metaphysical considerations.
Re: Recognizing Nomic Necessity
Date: 2005-11-28 06:08 pm (UTC)<< I see no difference between:
"all lions that will ever die in the North Atlantic will be female" (E1)
"all lions that could ever die in the North Atlantic must be female" (E2) >>
The form is different: I would quantify E1 over all physically-possible worlds, and E2 only over the actual world. So E1 is law-like, while E2 isn't.
<
<< I see no difference between:
"all lions that will ever die in the North Atlantic will be female" (E1)
"all lions that could ever die in the North Atlantic must be female" (E2) >>
The form is different: I would quantify E1 over all physically-possible worlds, and E2 only over the actual world. So E1 is law-like, while E2 isn't.
<<I do not find this convincing because it presupposes a notion of "physically possible" that comes prior to us determining what is a law and what is merely an accidental generalization.>>
What is not convincing?
<< I prefer to define what it means for something to be physically possible by referring to what we have chosen to accept as our laws of physics. And this requires, sorting through all true generalizations and determining which ones are to be considered laws and which ones are to be considered accidental. >>
Right. So "law" is a cognitive term, not a metaphysical one.
<<On the other hand, it would require a lot of effort to modify our web if we had to give up on something like the law of gravitation.>>
This sounds a lot like Kuhn!
<<I do not believe that these categories form a clear-cut partition, but I do believe that having these notions might be somewhat useful when we are engaged in belief revision.>>
I agree.
<< We could establish different degrees of Nomic Necessity depending on how much support those beliefs provide for the rest of the web. >>
Sure.
<<In any case, *necessity* and *possibility* have much more to do with our cognitive abilities and our conceptual framework than with the state of affairs in the world or other metaphysical considerations.>>
Exactly.
For me, nomic necessity is a useful concept for *cognitive* reasons. An AI scientist would need similar concepts in order to predict and "understand" the world.
But there is a sense in which the world behaves as if it followed metaphysical laws, and we can refer to an abstract set of physically possible worlds: it might even be useful! But I think that this will be indistinguishable from the cognitive notion of "physically possible".
Re: Recognizing Nomic Necessity
Date: 2005-11-28 07:48 pm (UTC)Lol
:P
Why is that?
<<The form is different: I would quantify E1 over all physically-possible worlds, and E2 only over the actual world. So E1 is law-like, while E2 isn't.>>
Yes, I understand that.
But as I said before, E1 presupposes we already possess a notion of "physically possibility".
<<What is not convincing?>>
That we can distinguish beteween laws and accidental generalizations by simply looking at the structure of a sentence: "X will happen" vs "X must happen".
The problem is the following...
Suppose we are scientists trying to come up with a nice theory of physics.
We gather a whole bunch of empirical data and we try to find an adequate theory that fits the data. Now, there is a generalization G made from the data which Reut would like to include as a law, but Samson argues that we shouldn't include it because it is only an accidental generalization. So you come along and say: "Don't worry guys I know how to determine whether this is a law or just an accidental generalization. You just need to check whether or not G can be expressed with a modal 'must'. It all depends on whether G is quantifying over just the actual world or over all physically possible worlds." Reut believes that G definitely holds true in all physically possible worlds and Samson believes that G only holds in a small subset of all physically possible worlds which happens to include the actual world. So I come along and say: "Hey guys, there is no point in arguing about this. We cannot distinguish between laws and accidental generalizations. Thus, we might as well forget about this distinction, include G as a law and there is no harm done."
Reut and Samson's argument boils down to whether they should flag G as being a statement they are willing to give up easily.
<<<
<<In any case, *necessity* and *possibility* have much more to do with our cognitive abilities and our conceptual framework than with the state of affairs in the world or other metaphysical considerations.>>
Exactly.
For me, nomic necessity is a useful concept for *cognitive* reasons. An AI scientist would need similar concepts in order to predict and "understand" the world.>>>
Great. I am happy to see we have finally converged to an agreement.
Once again confirming that one cannot rationally agree to disagree.
<<< But there is a sense in which the world behaves as if it followed metaphysical laws, and we can refer to an abstract set of physically possible worlds: it might even be useful! But I think that this will be indistinguishable from the cognitive notion of "physically possible". >>>
Of course they would be indistinguishable. What sort of experiment could you set up that could discriminate between "physically possible" in a metaphysical sense and "physically possible" in the cognitive sense.
It MUST be the case in all possible worlds that the metaphysical nomic necessity and cognitive nomic necessity are the same!
:P
Re: Recognizing Nomic Necessity
Date: 2005-11-28 08:29 pm (UTC)Because if it were me, I would easily get excited trying to find a good comeback... and if I found one, I wouldn't want to let go of it, even after I realized that you already agreed with me. This, of course, is a waste of time in the path to our goal of converging to an agreement.
<< But as I said before, E1 presupposes we already possess a notion of "physically possibility". >>
We do have such a notion.
<< So you come along and say: "Don't worry guys I know how to determine whether this is a law or just an accidental generalization. You just need to check whether or not G can be expressed with a modal 'must'. >>
I would not do that. Reut and Samson have a serious and legitimate problem in their hands, namely, (again) the problem of induction.
<< Thus, we might as well forget about this distinction, include G as a law and there is no harm done." >>
If Samson turns out to be right (i.e. G is refuted), then harm has been done!
But of course, Samson and Reut will disagree what the laws are. Samson, being a skeptic, is being epistemically conservative, whereas Reut probably thinks that it's ok to take G as a working hypothesis about which she can change her mind later.
<< It MUST be the case in all possible worlds that the metaphysical nomic necessity and cognitive nomic necessity are the same! >>
I didn't think I knew anyone who was more logical-positivistic than me! But anyway, you seem to be assuming completeness here. WHAT IF there are laws that we simply can never discover? These laws would still be physically necessary, but not necessary in the sense of epistemic logic (i.e. known). How do you even define "cognitive nomic necessity"?
Furthermore, shouldn't it be possible for us to talk about the laws of physics, even before we know what they are? If you insist in knowing the laws before defining nomic necessity, then this is not possible.
Re: Recognizing Nomic Necessity
Date: 2005-11-18 12:50 pm (UTC)"no male lions could ever drown in the North Atlantic", on the other hand, could be a law because it has the law-form.
The problem of induction (i.e. distinguishing laws from non-laws) doesn't come into it. If we make an accidental generalization, then we will simply be mistaken about laws.
Re: Recognizing Nomic Necessity
Date: 2005-11-18 02:03 pm (UTC)And that "no male lions could ever drown in the North Atlantic" is law-like.
And it is obvious why the former isn't and the latter is. Namely, the latter allows us to make predictions and the former doesn't.
At the risk of sounding like a broken record...
My point is that we have no good method of distinguishing between so called 'accidental generalizations' such as: "no male lions could ever drown in the North Atlantic" and so called 'law-like generalizations' such as: "all bodies with mass have a gravitational attraction to the sun".
Re: Recognizing Nomic Necessity
Date: 2005-11-21 10:00 am (UTC)Again, this is the problem of induction.
Re: Recognizing Nomic Necessity
Date: 2005-11-19 07:46 pm (UTC)There's no reason to distinguish them here. They are both law-like.
You seem to be saying that we shouldn't have a concept of law because of the problem of induction.
Ok, now you're telling me over Skype that we should treat law-like generalizations like laws... for the purposes of prediction. I agree.
Re: Recognizing Nomic Necessity
Date: 2005-11-21 04:24 pm (UTC)Absolutely not!
I am saying we should have no concept of "nomic necessity" because there is no good way of distinguishing between "laws of nature" and "accidental generalizations". Empirically, there is no way of making the distinction.
I do have an idea as to how such a distinction might still be feasible... but it is a bit vague and it would require talking about webs of belief.
Re: Recognizing Nomic Necessity
Date: 2005-11-26 10:47 am (UTC)Again, this is the problem of induction. Do you agree?
P.S. please don't be anonymous.
Re: Recognizing Nomic Necessity
Date: 2005-11-26 10:51 am (UTC)In any case, we already use a concept of nomic necessity, even if it's only intuitive. If you want to destroy this concept, the burden is on you.
(no subject)
Date: 2005-12-07 06:13 pm (UTC)I claim to be able to distinguish law-like from non-law-like statements when they are expressed in clear English sentences, because law-like statements have universal (counterfactual) scope. But using a naive FOL formalization, this is indeed not possible.
However, if we formalize those predicates further, then this distinction should be straightforward. Law-like statements need to quantify over all time, and over all possible circumstances in some way.