Editing Probability interpretations (section)

==Propensity==
{{Main|Propensity probability}}

Propensity theorists think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind or to yield a long run relative frequency of such an outcome.<ref>{{cite book | last = Peterson | first = Martin | title = An introduction to decision theory | publisher = Cambridge University Press | location = Cambridge, UK New York | year = 2009 | page = 140 | isbn = 978-0521716543 }}</ref> This kind of objective probability is sometimes called 'chance'.

Propensities, or chances, are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate given outcome types at persistent rates, which are known as propensities or chances. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives (see "single case possible" in the table above).<ref name="de Elía" /> In contrast, a propensitist is able to use the [[law of large numbers]] to explain the behaviour of long-run frequencies. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will be close to the probability of heads on each single toss. This law allows that stable long-run frequencies are a manifestation of invariant ''single-case'' probabilities. In addition to explaining the emergence of stable relative frequencies, the idea of propensity is motivated by the desire to make sense of single-case probability attributions in quantum mechanics, such as the probability of [[Radioactive decay|decay]] of a particular [[atom]] at a particular time.

The main challenge facing propensity theories is to say exactly what propensity means. (And then, of course, to show that propensity thus defined has the required properties.) At present, unfortunately, none of the well-recognised accounts of propensity comes close to meeting this challenge.

A propensity theory of probability was given by [[Charles Sanders Peirce]].<ref name="Miller 1975 123–132">{{Cite journal| last= Miller|first=Richard W.| title = Propensity: Popper or Peirce?|journal =[[British Journal for the Philosophy of Science]]| volume=26| issue=2| pages=123–132| doi=10.1093/bjps/26.2.123 | year=1975 }}</ref><ref name="Haack 1977 63–104">{{Cite journal|title=Two Fallibilists in Search of the Truth| author-link1=Susan Haack | first1=Susan|last2=Kolenda, Konstantin | last1=Haack | first2=Konstantin |last3=Kolenda|journal=Proceedings of the Aristotelian Society|issue=Supplementary Volumes|volume=51| year=1977|pages= 63–104| jstor=4106816| doi=10.1093/aristoteliansupp/51.1.63 }}</ref><ref>{{Cite book|author-link=Arthur W. Burks|last=Burks|first=Arthur W.|year=1978|title=Chance, Cause and Reason: An Inquiry into the Nature of Scientific Evidence|publisher=University of Chicago Press|pages=[https://archive.org/details/chancecausereaso0000burk/page/694 694 pages]|isbn=978-0-226-08087-1|url=https://archive.org/details/chancecausereaso0000burk/page/694}}</ref><ref>[[Charles Sanders Peirce|Peirce, Charles Sanders]] and Burks, Arthur W., ed. (1958), the [[Charles Sanders Peirce bibliography#CP|''Collected Papers of Charles Sanders Peirce'']] Volumes 7 and 8, Harvard University Press, Cambridge, MA, also Belnap Press (of Harvard University Press) edition, vols. 7-8 bound together, 798 pages, [http://www.nlx.com/collections/95 online via InteLex], reprinted in 1998 Thoemmes Continuum.</ref> A later propensity theory was proposed by philosopher [[Karl Popper]], who had only slight acquaintance with the writings of C.&nbsp;S. Peirce, however.<ref name="Miller 1975 123–132"/><ref name="Haack 1977 63–104"/> Popper noted that the outcome of a physical experiment is produced by a certain set of "generating conditions". When we repeat an experiment, as the saying goes, we really perform another experiment with a (more or less) similar set of generating conditions. To say that a set of generating conditions has propensity ''p'' of producing the outcome ''E'' means that those exact conditions, if repeated indefinitely, would produce an outcome sequence in which ''E'' occurred with limiting relative frequency ''p''. For Popper then, a deterministic experiment would have propensity 0 or 1 for each outcome, since those generating conditions would have same outcome on each trial. In other words, non-trivial propensities (those that differ from 0 and 1) only exist for genuinely nondeterministic experiments.

A number of other philosophers, including [[David Miller (philosopher)|David Miller]] and [[Donald A. Gillies]], have proposed propensity theories somewhat similar to Popper's.

Other propensity theorists (e.g. Ronald Giere<ref>{{cite book |author=Ronald N. Giere |title=Studies in Logic and the Foundations of Mathematics |chapter=Objective Single Case Probabilities and the Foundations of Statistics |chapter-url=http://www.sciencedirect.com/science/bookseries/0049237X |doi=10.1016/S0049-237X(09)70380-5 |volume=73 |pages=467–483 |publisher=[[Elsevier]] |year=1973 |isbn=978-0-444-10491-5|author-link=Ronald N. Giere }}</ref>) do not explicitly define propensities at all, but rather see propensity as defined by the theoretical role it plays in science. They argued, for example, that physical magnitudes such as [[electrical charge]] cannot be explicitly defined either, in terms of more basic things, but only in terms of what they do (such as attracting and repelling other electrical charges). In a similar way, propensity is whatever fills the various roles that physical probability plays in science.

What roles does physical probability play in science? What are its properties? One central property of chance is that, when known, it constrains rational belief to take the same numerical value. [[David Lewis (philosopher)|David Lewis]] called this the ''Principal Principle'',<ref name=SEPIP /> (3.3 & 3.5) a term that philosophers have mostly adopted. For example, suppose you are certain that a particular biased coin has propensity 0.32 to land heads every time it is tossed. What is then the correct price for a gamble that pays $1 if the coin lands heads, and nothing otherwise? According to the Principal Principle, the fair price is 32 cents.