A probabilistic argument is one which concludes that something has some probability based upon information about probabilities given in its premisses. Such an argument is invalid when the inference from the premisses to the conclusion violates the laws of probability. Probabilistic fallacies are formal ones because they involve reasoning which violates the formal rules of probability theory. Thus, understanding probabilistic fallacies requires a knowledge of probability theory.
A Short Introduction to Probability Theory:
In the following laws of probability, the probability of a proposition, s, is represented as: P(s).
- P(s) ≥ 0.
The probability of a proposition is a real number greater than or equal to 0. In other words, zero is the lowest probability, and there are no negative probabilities. It follows from axioms 2 and 3, below, that P(s) ≤ 1, which means that a probability is a real number between 0 and 1.
- P(t) = 1, if t is a tautology.
The probability of any tautology is equal to 1.
- If s and t are contrary propositions, then P(s or t) = P(s) + P(t).
It follows from the axioms that for any two propositions, contrary or not, that: P(s or t) = P(s) + P(t) - P(s & t), where "P(s & t)" is the probability that both s and t are true.
A conditional probability is the probability of a proposition on the condition that some proposition is true. For instance, the probability of getting lung cancer is an unconditional probability, whereas the probability of getting lung cancer given that you smoke cigarettes is a conditional probability, as is the probability of getting lung cancer if you don't smoke. Each of these probabilities is distinct: The probability of getting lung cancer if you smoke is higher than the unconditional probability of getting lung cancer, which is higher than the probability of getting lung cancer if you don't smoke. The conditional probability of s given t is represented as: P(s | t).
- P(s | t) = P(s & t)/P(t) or, equivalently, P(s & t) = P(s | t)P(t).
The conditional probability of s given t is equal to the probability of the conjunction of s and t divided by the probability of t, assuming that P(t) ≠ 0. The equivalent part is often called "the multiplication rule" or "the multiplication law". If s and t are probabilistically independent―that is, if P(s | t) = P(s) and P(t | s) = P(t)―then the rule simplifies to: P(s & t) = P(s)P(t).
The above laws are logically sufficient to prove every fact within probability theory, including a theorem that is important for explaining probabilistic fallacies:
Bayes' Theorem: P(s | t) =
P(t | s)P(s) / [P(t | s)P(s) + P(t | not-s)P(not-s)].
Proof: From axiom 4, we know that P(s | t) = P(s & t)/P(t). Since "s & t" is logically equivalent to "t & s", P(s & t) = P(t | s)P(s), again by axiom 4, which is the numerator of the fraction in Bayes' Theorem. To get the denominator of the fraction, "t" is logically equivalent to "(t & s) or (t & not-s)", so P(t) = P[(t & s) or (t & not-s)]. Since "(t & s)" and "(t & not-s)" are contraries, it follows that P[(t & s) or (t & not-s)] = P(t & s) + P(t & not-s), by axiom 3. By applying axiom 4 again, we have that P(t) = P(t | s)P(s) + P(t | not-s)P(not-s), which is the denominator.
Mistakes in reasoning about probabilities are typically not treated as formal fallacies by logicians. This is presumably because logicians usually do not make a study of probability theory, and the mathematicians who do don't generally study logical fallacies. However, in recent decades, psychologists have discovered through observation and experiment that people are prone to make certain types of error when reasoning about probabilities. As a consequence, there is now much more empirical evidence for the existence of certain fallacies about probabilities than there is for most traditional fallacies. Again, logicians are often unaware of the existence of this evidence, and they usually do not discuss it in works on logical fallacies. It is about time that logicians broadened their intellectual horizons and began to take note of discoveries in the psychology of reasoning.
- The Base Rate Fallacy
- The Conjunction Fallacy
- The Gambler's Fallacy
- The Hot Hand Fallacy
- The Multiple Comparisons Fallacy
Resource: Amir D. Aczel, Chance: A Guide to Gambling, Love, the Stock Market, & Just About Everything Else (2004). About as untechnical an introduction to probability theory as you will find.
Acknowledgment: Thanks to Emil William Kirkegaard for pointing out a problem, which has subsequently been fixed, with the wording of the informal description of the first axiom of probability.
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