# Why does this reductio ad absurdum work?

A is something which we do not know the truth value of, for example: A = It will rain tomorrow. It might happen, it might not. So possibly A is true, and possibly ~A is also true. What is wrong with the below reductio ad absurdum? Where I will take A as the assumed premise, show that it leads to a contradiction and then assert the negation of A as the conclusion.

Assume A. Possibly ~A (which we know is true). A & possibly ~A is a contradiction. Therefore ~A. If this works, I have proved that it will not rain tomorrow...which of course cannot be...so where does the problem lie? If we assume A, can we not say 'possibly ~A'? (I looked at some modal logic....but I don't think possibly ~A is the same as '~A is true in some possible world'.) Thanks very much guys

• "A and possibly not-A" is not a contradiciton. A contradicition is "A and not-A. – Mauro ALLEGRANZA Sep 7 '17 at 12:42
• According to the (more-or-less) precise definition of possible in modern modal logic, "possibly A" means: "there is a world, maybe different from the current one, where A holds". – Mauro ALLEGRANZA Sep 7 '17 at 13:21
• There is NO way to prove "by logic" that it is raining (or not). This is an empirical fact and no empirical facts can be known without the "resources" of experience. – Mauro ALLEGRANZA Sep 7 '17 at 13:41
• According to modal logic, "possibly not-A" is the same as "not-necessarily A". The semantics of "necessarily A" is: "in every possible world A holds". Thus, the semantics of "not-necessarily A" is: "there is a possible world where A does not hold". Thus, in this world it rains but in some other world "out there" in the multiverse of possibilities it does not. – Mauro ALLEGRANZA Sep 7 '17 at 13:58
• Think of "Possibly not-A" as "A or not A". Then your formulation becomes "1) A, and 2) A or not A" which has no contradiction. – kbelder Sep 7 '17 at 15:44

A and possibly not-A are not contradictory in modal logic.

In your question, you use epistemic possibilities: what is possible or not given our current knowledge. Now A could be the case while it would still be possible, according to our limited knowledge, that not-A, so A and possibly not-A are not contradictory.

You seem to mean by "let us assume A" that we would know that A is the case. Then of course it will be contradictory, but this should be expressed in modal logic by "necessarily A" which is indeed contradictory with "possibly not-A". If you had assumed, say, nomological possibilities, whether something is possible according to the laws of nature, there would be no contradiction either: "tomorrow it will rain" could be true while at the same time, "tomorrow it will not rain" could also conform to the laws of nature.

However, there's indeed a paradox with knowledge, called Fitch's paradox, which more or less resembles yours. You might be interested in that: https://en.m.wikipedia.org/wiki/Fitch's_paradox_of_knowability

Also related is Aristotle's discussion on fatalism: https://plato.stanford.edu/entries/fatalism/#1

Even without involving possible-world semantics, this is just an equivocation. English is really bad about the locations of negations in modal constructions.

Worst case: "You may not do that" means two completely different things depending upon the timing of the delivery -- 1) that you are not allowed, or 2) that you cannot be expected -- to 'do that'.

You need something more rigid to separate the ideas that A might possibly have be false, and that it is false that A might possibly have been true. Once you separate those, it is kind of clear that there is no contradiction with the idea that A might possibly have been false, even when it is true.

From one philological take, there is a good reason that English subjunctives expressing possibilities use 'would', 'should' or 'might', the past tenses of the words 'will', 'shall' and 'may', which express different conditions on predicting the future (for instance, doing so certainly, certainly if proper social rules are followed, or uncertainly). Possibility is not a state of reality, it is the potential future of a past state of reality. And even when used of the present, it is implicitly future perfect in tense.

In that sense, there is no way in which you can have both possibility and eternality in the same epistemology. If anyone can predict the future, things are possible only if they happen.

• Thanks for your response, if I don't know something, does that mean that 'it might possibly have been false' ? – Rob Hv Sep 7 '17 at 23:56
• That depends on what kind of thing it is, and that falls back on your basic ontology. By standard notions of logic, to the degree that the law of contradiction is now true, it is not true that it might possibly have been false. In the framing taken here, possibility is about prediction, which includes a theory of causation, and comes in flavors that go along with different standards of prediction. – jobermark Sep 8 '17 at 12:30
• In the same way, the 'should' modality depends upon a theory of ethics or politics, etc. And the 'could' modality depends upon a physics. Modalities apply theories to alternative facts. Otherwise, there is no point in having modality at all. You should just say what you do and do not know. – jobermark Sep 8 '17 at 12:35