Appeal to Probability

The appeal to probability is a logical fallacy. It assumes that because something could happen, it is inevitable that it will happen. This is flawed logic, regardless of the likelihood of the event in question. The fallacy is often used to exploit paranoia. While not considered a "true" fallacy by some (because it is rarely used by itself), the appeal to probability is a common trend in arguments, enough for many to consider it a fallacy of itself.

This has the argument form:

Possibly P.
Therefore, P.

Equivalently, using modal logic and logical connective notation:

\Diamond P → P

Some examples are:

* "There are many hackers that use the internet. Therefore, if you use the internet without a firewall, it is inevitable that you will be hacked sooner or later."
o While using a firewall is a prudent and sensible measure, it is not "inevitable" that a hacker will attack an unprotected computer. The argument does have some backing logic, but overstates the worst case scenario.
* "It doesn't matter if I get myself into debt. If I play the lottery enough, some day I'll win the jackpot, and then I can pay off all my debts."
o A reversal of the previous argument, as it is dependent on the best case scenario coming true. This is a dangerous argument, hinged on the relatively small (usually in the thousands or millions to one) odds of winning the jackpot, or at least enough to make a large enough difference to someone's debts, and then using these small odds to justify excessive amounts of debt.
* "When soccer becomes popular in a town, hooliganism will become a major problem. Thus, if we allow a soccer team in our town, we will be overrun by hooligans."
o This is another argument that assumes a worst-case scenario, without any other backing logic. In a addition, it also falsely implies that correlation means causation, as it assumes that soccer is the direct cause of hooliganism, without taking into account other socio-economic factors.

The logical idea behind this fallacy is usually that, if the probability of P occurring is approaching 1, it is best to assume that P will occur, since it will (almost) almost surely happen. The fallacy incorrectly applies a common tenet of probability: given a sufficiently large sample space, an event X of nonzero probability P(X) will occur at least once, regardless of the magnitude of P(X). This is derived from the definition of probability. The operative term is "given a sufficiently large sample space". Virtually all events are considered for probability within a finite number of samples, and the chance that X will occur in a given finite space S is directly proportional to S. Given a finite number of events S, each of which is X or not X, a sample space Y = 2PrS exists where one possibility is that all events in S are not X. Therefore, P(X in Y) = (Y-1)/Y. Because Y-1/Y < 1 for all finite Y, P(X in Y) < 1 regardless of P(X) or Y. There is thus always a chance that X will not occur, and therefore, no proof that X will occur given its probability.


1 comment:

  1. This is brilliant reasoning and these are very relevant conclusions, the Democratic race to the nomination being what it is now. I've found that one the factors that clouds this principle is that most people regard the statement of these situations within a framework of "infinite recuurence" or a boundless set of time. Your lottery and hacker examples are only inevitabilities if someone lives "forever" and even so, the case can be made for the possibility of the predicted/probable "eventuality" or "probability" not ever happening:
    If it can be be held that no thing is impossible, then it must consequently also be true that there is at least some thing that is impossible...
    -such as winning the lottery, something I have personal experience with.