# What is the difference between a probability and a possibility?

I ask this in a fairly naive way. I understand that "probabilities" can be quantified in frequencies, degrees of belief, etc. with some defined "space" of probability.But I know little about modal logic or the uses of the term "possibility."

In one sense we talk about "the number of possibilities" when determining a probability. And we count or define "possible worlds." But in another sense "possibility" seems to be an uncountable continuum of, for example, all that is not logically contradictory, as I believe Leibniz uses the term.

Are the terms defined relative to one another, or do they simply appear in different contexts with different meanings? It would seem that quantifying a probability must entail counting the "number of possibilities." Which would seem to be like counting points on a line.

Perhaps I am only asking if "possibility" has some consistent definition. Or perhaps the best answer is: "go read some modal logic and resubmit the question."

• I don't see the inconsistency between "possible worlds" and "all that is not logically contradictory". Could you elaborate? – user2953 Sep 30 '15 at 14:47
• Don't know enough about "possible worlds" to answer, but I thought they were somehow quantifiable, as in a comparison of "six different possible worlds," whereas the "absence of contradiction" would be open. Would it help if I just took that reference out? Not really essential to my question. – Nelson Alexander Sep 30 '15 at 14:52
• Instead of modal logic, you could note that Possible(x) := Probability(x) > 0 and then read on the philosophy of mathematics (with respect to probability). – R. Barzell Sep 30 '15 at 15:32
• For probability, see Interpretations of probability : "some interpretations locate probability ‘in the world’ while other locate it in our heads or in logical abstractions." For possibility, see Varieties of Modality. – Mauro ALLEGRANZA Sep 30 '15 at 15:34
• In mathematics at least possible can not be identified with positive probability. Being a rational number on [0,1] has probability zero, but it is possible. – Conifold Oct 1 '15 at 1:02

On the prevailing extensional interpretation of modality the difference between possibility and probability is the diffference between quality and quantity, possibility is the quality quantified by probability, see Probability Distributions Over Possible Worlds by Bacchus. This interpretation can be traced back to Leibniz's determinate possible worlds, but it became prevailing after Kripke's extensional formalization of modal semantics in late 1950s. To get probabilities one needs a positive unit measure on the set of all possible worlds, which is more general than counting because an infinity of them can be admitted. For instance, if a dart is thrown at a dartboard the possible worlds will have it sticking out of a particular point on it, of which there are infinitely many. But probabilities can still be assigned to particular regions based on their area (measure). Of course, different measures can be put on the same set of worlds, even when there are finitely many of them, so quantification is not unique.

The trouble is that in less trivial contexts possible worlds can not be precisely specified or surveyed, so they do not form a set ("sample space"). Because of the vagueness of description, or because there are too many of them, or both. Is it possible that the sun won't rise tomorrow? Depends on the meaning of possible. Is it possible that Russell could have been non-human? Depends on the philosophy of modality. Can a possible world be constructed with a level of detail even remotely approaching the actual world? Not humanly. But complete and determinate possible worlds are reasoned about by analogy with the actual world nonetheless, despite the fact that no ways of constructing and/or accessing such things are available, let alone surveying their totalities.

Kauffman, a mathematical biologist, gives an interesting analysis of the impasse this creates in biology, where the "adjacent possibles" are indeterminate:"if we do not know all the possible preadaptations that might arise in the adjacent possible of the biosphere, then not only do we not know what will happen, we do not even know what can happen! Can we make probability statements about the evolution of the biosphere by preadaptations? Consider flipping a coin 10,000 times. It will come up heads about 5000 times, with a binomial distribution. But notice that we knew ahead of time all the possibilities, all heads, all tails, and so forth. We knew the sample space of the process, so could erect a probability measure on the frequency interpretation of probabilities for this coin flipping process. But we do not know the sample space of the evolution of the biosphere by preadaptations, so can make no probability statements about it".

Probability works when it narrows possibility to a manageable set of determinate outcomes at the expense of limiting their range and making them highly schematic. Modal logic tries to do the same, but trades quantitative precision for a wider applicability of qualitative conclusions. Even that may distort the notion of possibility. As Felt writes in Impossible Worlds:

"The shadow of Parmenides seems to lie over these discussions... the anti-Parmenidean (Aristotelian) notion of potentiality, as an intrinsic character of the actual, has tended to be supplanted by possibilities (in the plural), Lewis’s “ways things could have been,” purely formal and discrete patterns. The dynamism of potentiality has been exchanged for a dust of homeless forms... Bergson was right, then, in maintaining that the “possible” (understood determinately) arises only simultaneously with the real... The actualists are therefore right in denying an independence to the possible... On the other hand, to be potentially is really a way to be, even though it is not to be actually. And this of course is just what Aristotle said in response to Parmenides, who conceived of only one way of being, being in actuality".

So to refine the short answer, probability quantifies not possibility in general, but a somewhat impoverished projection of it.

• Thanks, excellent answer. And makes sense to me, especially your "final take" and Felt's distinction, as I read it, between uses of the term as continuum or multiples. Lots to chew on. At a "psychological" level I"m still wondering about the apparently willful operation that turns "possibilities" into "probabilities. And I am mulling possibility as a "quality." Interesting. Seems more like a non-quality. – Nelson Alexander Oct 1 '15 at 1:30

To say something is probable as opposed to possible, at least in everyday discourse means the former is more likely, and the latter less likely; and at its limit just conceivable.

Probability has a quantitative definition, intuitively through frequency of occurrence.

Possibility has a logical definition - modal logic - which leaves its degree of possibility indeterminate.

One can say something very specific here; what is not possible must be impossible; and what is not impossible must be possible; thus we have immediately a negation operator!

It's here that Liebniz enters; for what is logically contradictory, pace the centrality of the Law of the Excluded Middle in Aristotles Organon, is impossible; and this leads into his possible worlds: for what is not logically contradictory must be possible.

Thus one could concieve a possible world as a world in 'logical space' that contains no contradiction; but this may not be how Liebniz concieved it.

But why restrict to just possible worlds? Why not all worlds, that is include the impossible worlds; and then one can say that at the limit of the possible worlds lie the impossible ones - they are it's boundary.

• If "probability" is quantifiable and "possibility" is not, yet any "probability" depends on a limited or bounded set of "possibilities," is the relationship purely contingent? Is "a possibility" simply a flagrant imprecision of common language? Not being provocative, just thinking out loud. – Nelson Alexander Oct 1 '15 at 1:39
• @NelsonAlexander You could look at 'fuzzy logic' as an answer to that question. To merge with a previous answer, in that model possibility has the same relationship to probability that utility has to value in classical economics. – user9166 Oct 1 '15 at 16:13
• @nelson Alexander: all very good questions...this is why I pointed out that even when possibility is indefinite one can still say definite things about it. – Mozibur Ullah Oct 2 '15 at 22:23

In short, the probability of an event is the mathematical chance that some object will have some effect whereas the possibility of an event is the infinite outcomes of the effect on an object. For example, I might have a 40/60 chance of winning a drawing if I bought 60 out of the 100 available tickets but there is a possibility I could win or not win. I hope this helps.

• Thanks. I believe I understand probability, but I am still not sure if "possibility" is somehow quantifiable or well-defined. Or if it is used in a variety of ways. Perhaps it is better thought of as "possible outcomes." I get the sense I just stumbling with some syntax problem. – Nelson Alexander Sep 30 '15 at 19:06

In an easy way it could be taken like this (prior commentators made some of these points, but didn`t connect them):

Possibility means being able to be thought without contradiction at the same time (!). In the sense of probabilities it means a state (the total number of states = total number of possibilities in Laplace's sense) that can be thought as an outcome without contradiction.

Probability of an event then simply means a certain number of states 1. thought as causally invoked by or 2. conceptually thought within an event devided by the total number of states being able to be thought without contradiction.

This also means: no probability without possibility, but possibility without probability. Therefore it is (strictly speaken, see below) perfectly possible to become President of the United States for every American (= no contradiction), although it is thought improbable for nearly all of them.

The problem of thought arises when situations with a probability near or (mathematically) equal 0 are named impossible, althought they can be thought without contradiction. Like saying "My neighbor becoming the president is impossible." It often occurs when we are judging heuristically, which we are doing because we are simply not able to say what the probability of an outcome really is (otherwise we would have holy wisdom). It is an aequivocation, a slight move of sense ruining the scientific usability of a language. Becoming rid or at least revealing aequivocations is basically the main task of philosophy, Husserl would have said.

• Thanks, good elaboration of the prior answers. As noted in my comment above, still wondering about other ways to sort out definitions of "possibility," but this leads into other questions. Glad you mentioned Laplace. – Nelson Alexander Oct 1 '15 at 15:38
• For example, possibility can also be defined as "allowed by laws of nature", which is a quite common connotation, but derived from the logical one (since laws of nature are general ones, without exceptions). "It is not possible to be faster than light (in relation to any other standpoint)." Would be one sentence including this one. The probability is thought as not existing there, too, obviously. This has been source of some weird sentences from our standpoint of today, though. Like "impossible to destroy atoms" or "impossible to move faster than 30 mph". – Philip Klöcking Oct 1 '15 at 16:00

From a fuzzy logic point of view, there is only one difference, a probability is normalized to be zero to 100 percent, and a possibility is on an infinite scale. I think this is really representative of the general usage in a productive way, and is a better notion than considering quantity and quality distinct, or considering the more abstract model of possibility from modal logic.

Possibility is not 'simply qualitative' or binary. We do consider things to be a 'distinct' or a 'minimal' possibility, or to be 'just barely' possible (or red, for that matter). So possibility has a degree. Qualities with degree are, then, quantities with a variable, subjective basis, and the distinction is compromised, and therefore generally not useful.

Possibility is also not indicative of 'possible worlds' or some metric over them, as we often decide that things we would admit in a possible world are not possible in reality. We extend our construction of possible worlds by asking 'might' questions, but often decide that what is proposed is not, in fact possible. We still conceded that it isn't, but it might have been.

In computations like entropy, we do speak of the 'number of possibilities', but we mean the number of parameters underlying the measure of possibility. Each of the things we count is possible to a different degree, or we would not have to integrate over them to determine the entropy. This is just a convention of grammar. We do not really believe in a specific 'number of reasons' or a 'number of perspectives'. We know there are really always a continuum of such things, if only by being different mixtures of alternatives. But we are classifying them by a finite number of dimensional parameters.

By admitting that there is some unknown but monotone function, variable by situation, between possibility and probability (like that between utility and investment value) you can actually get some handle on vague intuitions not clear enough to translate directly into probability, for use in places like expert systems, besides simply having a reasonable model of day-to-day usage.

• Thanks, that does add clarity and succinctness. Once possibilities are limited, 0 to 100, their probabilities can be measured. I guess I am still wondering, though, about the nature or perhaps even ontology of "a possibility" or "all possible outcomes," and what sort of infinity we are talking about here. Since noncontradiction holds only "at the same time" is possibility another description of time? But this is edging towards another question and getting wooly. – Nelson Alexander Oct 1 '15 at 15:32
• I would cop out and declaim "statistics" at you. Everything that really might happen has a probability distribution with some countable number of degrees of freedom. Things that are not such that they really might happen but would require a variation in the rules that govern the world in order to be possible, put you in the meta-universe of 'possible worlds', which I would claim does not really exist. – user9166 Oct 1 '15 at 23:32
• As noted in the answer, I go with a 'Wittgensteinian' notion of modal logic as a functional interaction over a heuristic space, where the operation of 'might' is not enumeration over something imaginary, like Kripke's space of 'all possible worlds' but a request to introduce a new parameter into the model under consideration. We like to reify potential enumerations, but doing so is largely an exercise in rationalizing illusions. – user9166 Oct 1 '15 at 23:36