Daily Dialectic

Probabilities as single case propensities being non-additive-, non-normal and negative—the modal resolution logical impossibility argument. It is those such as Appleby, Hajek and Hitchcock that postulate such a possibility against propensity theories of probabilities.

Suppose per assumption that propensities, which are defined as a form of single case chance, at least under the single case propensity variant, can be negatively valued. Then it follows, it should seem, that so long as negative propensities are ‘considered’ logically possible, which is the claim that is used against them, an abstract infinite sequence can be constructed whose consequences disproves their logical possibility.

This is even if they are empirically or physically so determined such that their negativity cannot be discovered or falsified because, either accidentally or in virtue of the laws of nature, such worlds that have these chance set ups with negatively valued outcome propensities are such that those chance set ups that do have them, (1) either, in these worlds, never occur infinitely or long enough to entail a negative relative frequency, or (2) are such that these propensities in these worlds, change throughout time so that the average propensity of these events remain positive, despite some of them being negative, so that, by the law of large numbers, the outcome relative frequency they produce must be positive.

However, even if there are no logically possible worlds in which there exist an infinite number of certain chance set ups with negative propensity outcomes, and thus not having the incoherence of possessing a negative outcome relative frequency, this does not forestall the possibility of, at least abstractly, constructing such an infinite sequence. This is so long as we hold to the doctrine that if something is logically possible, there are an infinity of logically possible worlds in which it inhabits (which seems possible by virtue of the different, degrees and kinds, cardinalities and orders in which we alter the combinations and permutations of other external events).

Again, this is depends, of course, that by definition such negative propensities have no connotations about the connections they have with such combinations of possible external events, which they will not have, unless we deliberately add, as part of the definition of negative propensities, that they have such connections and inconsistencies; but of course this is of course ad hoc, and in any case without influence or importance, because of the tautological nature of such an addition; and Again, this argument is therefore sufficient for there being an infinite number of such worlds where such things exist, so long as we forestall those arguments that lead via the magic of metaphysical necessity and other forms of non-analytic necessities that have the force of logical necessity despite not being forms of logical necessity.

Given this requirements, we can nonetheless construct an infinite sequence of such deviant chance set ups across worlds, where each set up has this negative propensity that it allegedly possesses, so that we have an example of what must, in general be a negative relative frequency, an impossibility; and thus an argument against the possibility of negative propensities.

To make matters easier there needn’t even be an infinite number of worlds where this same event has this same negative propensity. Now of course there seems to be no argument to support this possible objection, so long as nonetheless, there are (1) an infinite number of these set ups with SOME negative propensity for the outcome, whatever value it is,even if different in each world, as the average is will still be negative, or (2) that in a great many of these worlds the outcome propensities of these chance set ups are negatively valued to the extent that even if there are only a finite number of these negatively valued instances across all worlds, the addition of enough instances of these same chance set ups across other worlds, so that an infinite sequence is achieved, despite the great majority having positive outcome propensities, is such that their average across this infinite sequence is still negative; or (3) even if so long as there are an infinite number of chance set ups, some of which may be completely distinct from each other (so that there isn’t infinite number of the same individual chance set up across worlds) but such these many and varied distinct chance set ups have negative propensities for the SAME outcome event, or an average negative propensity (given my previous two points) for that same event across this infinite sequence. Or to use licence, so long as even (4) holds,that is the negative propensitizied outcomes are distinct for each of these chance set ups, but such that they can still be considered the same outcome, that outcome being (I) This is as each of these ‘distinct’ outcomes of the chance set ups logically entails (I), the disjunction of all them. That is, the disjunction of all these same distinct individual non-disjunctive outcomes, the outcomes these chance set ups individually possess in each world.

Then (that is given, (1), (2) or (3) (or (4) if we are willing to accept it), if we even insist on these three (or four) points, which I needn’t think we must, but even if so, we can construct an infinite sequence of these chance set ups ACROSS worlds, which must, or at least in all likelihood, produce, by the laws of large numbers, a negative relative frequency- and this is logically impossible by virtue of either contradiction or incoherence. Thus, (A) propensities must be positive. A being only contradicted unless the fact that there is no such a world where within it itself, there is an infinite sequence of negative propensitized chance set up, follows by virtue of something inherent in the meaning of negative propensities- and not somewhat external like the law of large numbers, insofar as mathematical facts can be external)

Thus, the failure of the possible world approach to defining or indicating, or revealing possibility insofar that it suggests that by their being no infinite sequence of negative propensities in any possible world, that such a sequence is logically impossible, yet as demonstrated it is POSSIBLE; That is ACROSS WORLDs Rather then Within world. Perhaps intensional impossible worlds could resolve these inter-world\ versus intra-world problems by allowing one not to identify alleged impossibilities with impossibility This is done by denying the meaning or logical equivalence of things, even intensional things, that have neccesary co-extension. The point to made that is similar to the approach and stimulus againt necessary coextension as meaning, as used by impossible world theorists, is the externality or lack of relevance of the law of large numbers to the notion of probability intrinisically or analytically, just like the lack of relevance of some tautology (B) to all and sundry statements, be they contingent or tautologous themselves, unless they are that very same statement(B) itself; this being despite all tautologies being allegely true of logical necessity given any antecedent, and thus connected by logical necessity, and thus equivalence to every other tautology.

Of course one may attempt to remove this problem of negative relative frequencies (and thus the argument against the negativity or propensities), or the likelihood of this problem of negative relative frequencies(and thus the validity of that same argument against propensity negativity), by saying that such an infinite sequence could still fail to converge to that negative value(C), this value (C) being that of the probabilities thus associated with these set ups. And in so doing argue that the infinite sequence still demonstrates a logical possibility (as the negative probability value either fails to be converged to at all, or a positive relative frequency value is converged to instead. The mere the probability of one on which the the law of large numbers may allow this- probabilities of one do not imply necessity).

This response can most be avoided because the infinite sequence can be arranged and ordered in different ways such that from any infinite sequence, an infinite sequence of different (or at least differently ordered) infinite sequences can be produced, and thus moreover, an infinite sequence of infinite sequences of infinite sequences can be produced, ad infinitum. So for any meta-probability of one (as produced from every extra iteration of the law of large numbers when substituted into a previous instance of ),extra order of infinite sequences can be produced to combated to nullify its effect

Now Using this argument to Combat the other alleged discrepancies at the start of the argument

Likewise, propensities cannot not add to one (not be normalized), for likewise given this supposition, the relative frequency of the conjunction of their antecedent and that antecedent’s negation must be more or less then one (roughly speaking, once a few clarifications must be made), so that the relative frequency of this conjunction is also more or less then one- which would seem to violate bivalence, or the law of non-contradiction because their instantiations are non-disjoint on certain occasions, or again, as before, the concept of relative frequency itself.

Thirdly, they cannot be non-additive (for similar reasons, we will have relative frequencies of A being such and such, and B being such and such, which, despite their independence, will not have additive relative frequencies which is impossible given what relative frequencies mean).

Now of course, one may deny these conclusions by stipulating that my theory of propensities wherein they are connected to relative frequencies, could be false, but this has merit only insofar that single case propensities actually exist and exist and operate according to a contrary theory (thus disqualifying the very purpose for which these arguments were created; namely that propensities may be nonsense and not exist); or (2) by arguing that the law of large number fails when negative probabilities are involved, which may well be possible because it was clearly designed only with positive probabilities in mind—but if this failure is only in virtue of the fact that it fails simpliciter because negative probabilities are impossible, then propensities insofar that they are an interpretation of probability, must by definition, be positive anyway- and moreover, if it is not in virtue of this fact that the law of large numbers fails (namely that negative probabilities are are not impossible)- then the fact that such propensities and therefore probabilities could be negative, given what we know about propensities, is no claim whatsoever against their impossibility. Perhaps, one could argue that from what we know about propensities, they could possibly be negative, but negative propensities lead to the logical impossibility of negative relative frequencies, then as we have as a possibility an impossibility, therefore propensities are impossible simpliciter.

But I see we know no reason for thinking that such an assertion can be made, as its formulation is based on no more or less then that single case probabilities could, from what we know of them, be negative, and single case probabilities are such that insofar as they are by definition probabilities, must be positive. Moreover, the possibility that they could be negative is implicitly based on the fact they could be possible, which is the same thing as being possible (possibly possible reduces to possibility in those modal logics that most think govern the logics of logical consequence), as relative frequencies within such worlds in which they inhabit could remain positive; thus being possible.Moreover, ‘the notion of given what we know of such propensities’, cannot be used to mis-appropriate the negative relative frequency argument i have given for the opposite purpose thus described (namely to disprove propensities rather then to support them by showing that they cannot be negative), as a logical argument insofar as it based in definition, cannot be based on premises of an epistemic, and therefore a posteriori nature(namely that which we do not know, that we must discover)— as definitions are not empirical, necessities are not empirical; unless kripke is correct, which I doubt, and even so, this would apply to metaphysical and not logical necessities in any case