Examples of logical possibility

Question

According to Wikipedia, something is logically possible if it doesn't imply a contradiction. In that case, how could a mathematical statement be false but possible? Wouldn't a false statement be false because it contradicts a mathematical axiom, making it logically impossible? What are some examples of such statements?

Answer 1

Here is a concrete example. Let our base system be Euclidean geometry without the parallel postulate. Then, in our ordinary understanding, the parallel postulate is true (it is true in the standard Euclidean plane) but its negation is possible (because there are other planes that satisfy the rest of the axioms of Euclidean geometry, but not the parallel postulate).

The key point is that "possible" is always with respect to some base set of axioms - the "possible" statement does not imply a contradiction from those axioms.

Answer 2

Something may be logically possible while it is actually false.

It is possible you might have met the queen of England at a football match last Sunday.   Have you actually done so?

It is not possible you might have met the queen of England at a football match while she was actually sailing on a royal yacht.   That would be a contradiction.

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Basically, something is possible if there is some alternate universe in which it can happen.   Though we usually make some restrictions on what type of alternate universe we consider; in modal logic these are known as "accessible worlds".

It is possible that the queen could go to the same football match as you do. It's not a likely scenario, but it is possible.

It is not possible that the queen can be in two places at once; unless we consider universes where she has superpowers.

Answer 3

You may consider the table of truth. Consider any false sentence $P$, a sentence $Q$ and the sentence $P\to Q$. The truth table is:

$$\begin{array}[t]{c|c|c|} P & Q & P\to Q \\ \hline F & T & T\\ F & F & T\\ \end{array} $$

Here's an example:

Consider the axioms: "Every triangle is not a real number" and "Geometry is a field of Mathematics".

• If a triangle is a real number, then Geometry is a field of Mathematics.

• If a triangle is a real number, then Geometry is not a field of Mathematics.

Both of the above propositions $(P\to Q)$ are logically possible, because there is no contradiction in their content. However, the premises contradict the first axiom and for that reason they are always false. For the same reason, we can say that the second parts are true and false, respectively.

Logical possibility has nothing to do with the truthfulness or the falsity of the premises, in the sense we don't care if the premises are true or false. We just demand no contradiction in our axiomatic system.

A contradiction in our example would be of the form:

• If a triangle is not a real number, then Geometry is not a field of Mathematics.

Answer 4

Mathematical truths are generally held to be necessary truths, so there shouldn't actually be any examples.

I notice that Carl Mummert's answer claims to have an example, so let me say what I think is wrong with it. The question "is the parallel postulate true?" is not a question at all until you specify the context -- are we talking about Euclidean geometry? Hyperbolic geometry? Or maybe the real world? The answers to the questions become necessarily true, necessarily false, and contingently false, in those cases respectively. (At least, according to the standard paradigm in philosophy.)