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Paradox about Laws

  1. Jul 21, 2006 #1
    The following claims each strike me as intuitively plausible:

    I. For any event e, if e occurs that it is possible that the laws of nature made it the case that e occurs.

    (This is supposed to capture the intuition that it is possible for every happening to be law-governed. The laws of nature making something the case is understood to be an event/happening, like more ordinary events such as eggs breaking and toast burning.)

    II. For any event e, if the laws of nature make it the case that e occurs, then e occurs.

    (This is supposed to capture the intuition that the operator "the laws of nature make it the case that" is factive.)

    III. For any events e1 and e2, if the laws of nature make it the case that both e1 and e2 occur, then the laws of nature make it the case that e1 occurs and the laws of nature make it the case that e2 occurs.

    (This is supposed to capture the intuition that the operator "the laws of nature make it the case that" distributes over conjunctions.)

    IV. There is at least one event e such that e occurs at some point in the history, present, or future of the world and it is not the case that the laws of nature make it the case that e occurs.

    (This is supposed to capture the intuition that not everything is in fact governed by a law -- e.g., some happenings are chancy.)


    Question #1: Do people agree that these are each intuitively plausible?


    Here's the problem: These four claims are jointly inconsistent.

    1. Assume that E is the event of a happening/ event occuring but not being made the case by the laws of nature. (instance of IV)

    2. Then it is possible that the laws of nature make it the case that E. (from 1 and I)

    3. Go to the nearest possible world in which the laws of nature make it the case that an event occurs but the laws of nature do not make it the case that such an event occurs.

    4. Then, in that world, the laws of nature make it the case that the event occurs; and the laws of nature make it the case that the laws of nature do not make that event occur. (from 3 and III)

    5. Hence, the laws of nature make it the case that the event occurs; and the laws of nature do not make it the case that the event occurs. (from 4 and II on the rightmost-conjunct of 4). Contradiction.

    6. So there is no such possible world as the one mentioned in #3: it's an impossibility for the laws of nature to make it the case that an event occurs but the laws of nature do not make it the case that such an event occurs.

    7. In other words, it's an impossibility for the laws of nature to make it the case that E occurs.

    8. This contradicts the result derived at line #2.

    9. Therefore, by modus tollens on I, there is not at least one event e such that e occurs at some point in the history, present, or future of the world and it is not the case that the laws of nature make it the case that e occurs. This contradicts IV.

    Question #2: Is this reasoning valid? If not, what are the flaws? If so, which claim (I - IV) is it most plausible to reject and why?
     
  2. jcsd
  3. Jul 21, 2006 #2
    Note that you form Claim I as a "possibility"--thus, it is also "possible" that Claim I is false, that is Claim I may be true, may not be true (as you present it). However, Claim IV is formed with 100 % certainty, and thus, IMO, is not intuitive vis-a-vis science since by definition science is defined as knowledge with uncertainty.
     
  4. Jul 21, 2006 #3
    It seems you assume that nature has laws and then follows them.

    "Laws" in nature are nothing more than human models and approximations of the measurements we make of nature.
     
  5. Jul 21, 2006 #4
    If nature didn't have Laws, then we would be in a spot of bother. The Laws seem to exist, only, we are not always too sure about their exact description. This is what Physicists look for. :cool:

    desA
     
  6. Jul 21, 2006 #5
    How do you know "so sure" that it does. For instance what excludes the possibility of nature being an "infinite Russian doll" of reductionism?
     
    Last edited: Jul 21, 2006
  7. Jul 21, 2006 #6
    Anything is possible, but, just take a closer look at the designs woven into nature... simplicity in complexity... a beautiful tapestry, indeed... Laws do seem to be at work. :approve:
     
  8. Jul 21, 2006 #7
    The designs woven into nature? Sounds more like some idea of Platonic design. :smile:

    What I see is that the deeper you go the more complex and weirder it gets. :biggrin:

    Anway, I think this is diverting too much from the subject, sorry just wanted to comment on the "assumption of laws".
     
  9. Jul 21, 2006 #8
    You need to identify the appropriate Laws operating at the scales you are observing. Then, all of a sudden, things don't appear so complex... :biggrin:
     
  10. Jul 21, 2006 #9
    Yes, and then wait half a century and find out that the old laws were wrong.... but good news, we got brand new ones and this time we are quite sure we got it down.... but then 50 years later .... :biggrin:

    By the way, what do you think about those "laws" that say "there is a 50% chance it goes up and a 50% chance it goes down". I am sure judges would love the discretion if we were to apply the same in common law.
     
    Last edited: Jul 21, 2006
  11. Jul 21, 2006 #10
    Reply

    There is a typo in my statement of I. It should read: for any event e, if e occurs THEN it is possible that ....

    I don't intend claim I to be a possibility claim: I intend for it to be a claim about what is true in the actual world (or even in every possible world -- a necessary truth?). Just because the word "possibility" occurs in the claim does not make the claim something about what is possible. (E.g., the claim "If you go to the store, then it's possible I won't be here when you return" is not a claim about what is possible.

    In any case, if it's claim I that's false, I wonder whether there is an argument that it is false (independent of the reasoning I gave).

    Also, I take claim IV to be pretty well supported by the scientific evidence (think quantum mechanics -- e.g., the radioactive decay of a particular piece of uranium seems to be merely chancy: although there are laws governing the probability of whether decay occurs, there don't seem to be laws that govern the actual event of decay.) Maybe QM is incorrect about this: but until some better theory comes along I think it's reasonable to accept claim IV on the basis of current scientific evidence. What do you think?

    -----------

    When I say "the laws of nature make it the case that", I think this is compatible with the laws of nature not playing an "active" role in what happens, so to speak -- in the same way that saying "the axioms of arithmetic make it the case that 2+2=4" doesn't endow the axioms of arithmetic with special causal powers.

    As for the existence of laws, I think there is good scientific evidence: flip through any textbook and you'll find scientists talking about laws -- the laws of newtonian mechanics and electrodynamics and thermodynamics and quantum mechanics, etc.
     
  12. Jul 21, 2006 #11

    loseyourname

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    The inconsistency arises due to an ambiguity in the word "possible." Take the following set of analogous statements:

    1. For any suspect X in murder case Y, it is possible that X was the actual murderer.

    2. There is at least one suspect Xn that was not the actual murderer.

    Both sound plausible, but as you point out, there is no possible world in which Xn was not the murderer but could nonetheless have possibly been the murderer. The problem is that the first statement is modal and the second is not. The first statement implies that there is at least one possible world in which Xn was the murderer, while the second statement says that, in this world, Xn was not the murderer, and these two statements are not inconsistent with one another.

    Returning to your example, we can say that e did not occur because of any natural laws in this world, but there is at least one possible world in which e did occur because of natural laws. Translating I modally removes the ambiguity.
     
  13. Jul 21, 2006 #12
    What if I make the claims about possibility more specific and say that I mean possibility in the sense of physical (nomological) possibility, so that the range of possible worlds I am considering are all and only those possible worlds that have the same laws of nature as this world?

    Also, I'm not sure this reply is effective (this is written in the way of asking for clarification, not criticizing the reply). "E" (upper-case) is the event of there being the occurrence of an event "e" (lower-case) that the laws of nature do not make the case. By IV, it is (physically) possible for the the laws of nature to make E (upper-case) occur. But the reasoning of steps 3 - 7 show that it is not (phycially) possible for the laws of nature to make E occur, because supposing (for the sake of argument) that it is physically possible leads to a contradiction. How does saying there is an ambiguity in claim I show that the reasoning of steps 3-7 is mistaken?

    In the murderer example, there is no inconsistency between claims 1 and 2 because there are no analogues of my claims II and III; and this means that reasoning analogous to my steps 3-7 is not possible for the murder case. Also, I'm having trouble understanding that example: it seems wrong to say that

    Suppose I'm not the murderer but I am a suspect. Still, it's possible that I am the murderer -- maybe if the circumstances had been slightly different, I would have been the murderer. So there is a possible world in which I'm not the murderer but could have been the murderer -- namely, the one in which I didn't do the murder but could have. I'm not seeing the analogy you want to make.
     
  14. Jul 22, 2006 #13

    loseyourname

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    Well, in that case you might actually be right. There is still an ambiguity in the word "possible" that is causing some trouble, but it's slightly different now. In statement I:

    For any event e, if e occurs that it is possible that the laws of nature made it the case that e occurs.

    There are at least two different modal propositions that can be intended by this sentence. The first is 'If e, then possible(natural laws caused e).' The second is 'possible(If e, then natural laws caused e.' [In both cases, we restrict the domain of possible worlds to those with the same natural laws.] You're actually right that, in either case, the statements are inconsistent with one another, as we can see from my less than rigorous explication below:

    1. [tex]\forall x \diamond (Ox \rightarrow Nx)[/tex]
    2. [tex]\exists x \square \neg (Ox \rightarrow Nx)[/tex]

    1 implies 1a: [tex]\forall x \neg \square \neg (Ox \rightarrow Nx)[/tex]

    Applying universal and existential instantiation, we get:

    1b. [tex]\neg \square \neg (Oe \rightarrow Ne)[/tex]
    2a. [tex]\square \neg (Oe \rightarrow Ne)[/tex]

    From these two premises, we can validly infer their conjunction, from which we can validly infer the conjunction of their non-modal counterparts (from axiom M of modal logic T - [itex]\square \varphi \rightarrow \varphi[/itex]), which is a contradiction, therefore the two premises are inconsistent with one another.

    Now, for the other translation:

    3. [tex]\forall x (Ox \rightarrow \diamond Nx)[/tex]
    2. [tex]\exists x \square \neg (Ox \rightarrow Nx)[/tex]

    3 implies 3a: [tex]\forall x (Ox \rightarrow \neg \box \neg Nx)[/tex]

    Following the same procedure:

    3b. [tex](Oe \rightarrow \neg \square \neg Ne)[/tex]
    2a. [tex]\square \neg (Oe \rightarrow Ne)[/tex]

    3c. [tex](Oe \rightarrow \neg\neg Ne)[/tex]
    2b. [tex]\neg (Oe \rightarrow Ne)[/tex]

    The conjunction of these is also a contradiction, so these statements are also inconsistent.

    So hmm, maybe you've proven that Mill was correct when he claimed that the principle of the uniformity of nature followed from the law of causation. There is another sense of the word "possible," though: that of uncertainty. To say that e could possibly have been the result of natural laws can be interpreted to mean that there may or may not exist possible worlds in which e was the result of natural laws, in which case making the further statement that e was not the result of natural laws removes that uncertainty. In other words, I is a reasonable hypothesis to posit until IV can be demonstrated, which falsifies I as a hypothesis.

    You don't need the reasoning. Statements I and IV are inconsistent with each other on their own merit, as I hope I've demonstrated. You don't need II and III.

    The application of possible world semantics to counterfactuals doesn't work that way. When you say you're not the murderer but could have been the murderer, you are saying you are not the murderer in this world, but could have been the murderer in another possible world.

    I apologize the lack of rigor in this post. I've glossed over some of the derivation, but I hope it's enough. I thought that if I did everything fully, it would take forever because I'm no good with LaTex, but it ended up taking forever anyway. If it's not clear enough, I'll edit it and make it rigorous if requested.
     
    Last edited: Jul 22, 2006
  15. Jul 23, 2006 #14
    (1) and (2) seem to be contradictory. (1) seems to say that the laws of nature do NOT make it the case that E, whereas (2) clearly says it is possible that the laws of nature make it the case that E. If the laws of nature do not make it the case that E, how can it be possible that they do make it the case that E? Am I misunderstanding (1)?

    Best Regards
     
  16. Jul 23, 2006 #15
    I think you're understanding -- the rest of the argument after step 2 is trying to show in a rigorous way that, given the further assumptions of claims I, II and III, these claims are contradictory. But I think that if one were to reject one of those other claims, (1) and (2) would be consistent.

    loseyourname seems to disagree with me on this -- but loseyourname's post replaces what I called claim IV with a similar claim, namely, claim IV prefaced with a necessity operator after the existential. loseyourname's claim is much stronger than the one I was using -- and so I think its incompatibility with claim I is much less surprising than the incompatibility I was trying to argue for.
     
  17. Jul 23, 2006 #16
    ? Not sure I understand. Radioactive decay follows laws of nature: see this link:http://en.wikibooks.org/wiki/Basic_...dioactive_Decay_Law#The_Radioactive_Decay_Law
     
  18. Jul 23, 2006 #17
    It seems to me that these claims (1) and (2) are not referring to the same possible world – that is why they seem contradictory, and there is no need for any further argument. Given a possible world where it is the case that the laws of nature do not make it the case that E, it follows that is is not nomologically possible in that particular world that the laws of nature make it the case that E.

    Of course there are other logically possible worlds where the laws of nature make it the case that E, but in those particular worlds it follows that it is not the case that the laws of nature do not make it the case that E. I believe the confusion arises because you are mixing the generic idea of possible worlds on the one hand (in 2) with a particular possible world on the other (in 1).

    In other words, in all possible worlds, the proposition L = “the laws of nature make it the case that E” is a contingent proposition in those worlds (ie it is true in some, but not all, possible worlds).

    Now (1) assumes that you are referring to one particular instance of a possible world in which L is (contingently) false.

    But (2) assumes that you are referring to all possible worlds, in some of which L is (contingently) false, and in some of which L is (contingently) true.

    You are not comparing apples with apples – that is why (1) and (2) seem contradictory.

    Best Regards
     
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