# Is Arithmetic more Extensional than Probability?

One of the views of probability is that it should be viewed as a multi-valued logic where p(A) represents the probability that a proposition A is true.

In a discussion of this, I once read that probability cannot be considered a proper logic because it is not extensional (let's ignore for the purposes of this question whether a logic has to be extensional). For example, if A and B are events, the probability of A or B is p(A)+p(B), but only if the two events are mutually exclusive. You need to know what the events are in order to apply this rule. But why does that make it non-extensional? Arithmetic is the same way, and arithmetic is regarded as extensional. Let #A be the number of items in the class A. If you have two classes A and B, you can't say that the number of items in both classes together is #A+#B unless you know that the two classes are mutually exclusive.

• What does is mean that one domain of mathematics is more extensional than another domain? Please give a definition. - I do understand your two examples, and I understand what the extension of a set is. Dec 2, 2023 at 19:31
• I know the meaning of extensional via intensional. But I do not see any relation to your question about discriminating between arithmetic and probability theory on the basis of extensionality. Dec 2, 2023 at 20:18
• Are you sure it was "non-extensional" rather than "non-truth-functional"? Truth-values of connectives depending only on truth values of their terms is usually called truth-functionality rather than extensionality. Extensionality, in contrast, is a vague term, it usually means expressing just "how things are", not "representations" or "attitudes". Modal logic, for example, is "intensional", but Kripke's PW semantics extensionalizes it. Similarly, Kolmogorov's measure-theoretic semantics extensionalizes probability. Dec 2, 2023 at 22:28
• Who says logic has to be extensional? Modal logic is not. Intuitionistic logic is not. Relevance logics are not. In the context of logics that include quantification, extensionality is often understood in terms of whether there are 'intensional contexts'. Arithmetic and set theory are extensional by that definition. Many logics are not. Probability is a tricky case, since we don't usually combine probability theory with quantifiers, though there is some research in that direction. Dec 3, 2023 at 2:42
• Of course, `P(A or B) = P(A) + P(B) - P(A and B)` is true whether they're mutually exclusive or not. "A and B are mutually exclusive" is just a short way of saying `P(A and B) = 0` (for certain values of "short"). Similarly for most (all?) other identities: you can state them without making assumptions/knowing details about the specific events, but they'll have a few extra terms that would zero out if we made certain assumptions.
– Ray
Dec 4, 2023 at 20:51

In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements.

Since an event is a set of outcomes, the external properties are reduced to internal definition, ie. the A set/event must be the same with the B set/event. So according to the axiom, probabilities cannot be extensional.

Although I do have a university degree in math, I do not consider myself a professional mathematician, but I believe the above is a proof of your question.

I would like though to present another way of thinking too:

According to my view of probabilities - ontologically speaking - probability denotes our lack of knowledge, or inability to evaluate something. So, I cannot see how we could compare different lacks of knowledge. We could only compare probabilities as numbers, not their (not fully known or speculated in a specific case) ontological representations.

• Extensionality has a broader meaning in philosophy and logic that goes back centuries. The terminology of set theory came from this older and broader meaning. Dec 3, 2023 at 22:46
• @David Gudeman, ok, I guess it's a private club then. Dec 3, 2023 at 23:21