# Logical notation

I'm currently debating with someone whether something could be considered possible because we have no evidence that it's impossible. He made the statement that: "All things are possible which are not proven to be impossible", and attempted to justify it by claiming it's a truism (everything that is not ~A is A). I replied that his statement is not a truism because it doesn't have A/~A as his options, but A and "proven to be ~A".

He then offered me this:

"That objection doesn't seem to hold much merit given that the sentence structure indicates that two concepts are being included. First, you have the two clauses "A and ~A" then you have the contextual modifier "proven to be."

The arguments:

1) ∀X which are ~~ A -->A

2) ~W for ~A -->A

and:

∀X ~W ~A ---> A

Are logically equivalent."

Since I'm new to logical notation, I'm having a hard time expressing the above arguments in normal language.

Can anyone help?

• There is a ball in a box : the ball is either black or white, but we do not know which one. The only way to know if the ball is balck is to open the box and see (this is the proof). Conclusion (wrong) : the ball in the box is white because it is not proven that it is black. – Mauro ALLEGRANZA Mar 15 '19 at 19:40
• Standard symbols are ⊢ for "proven", and ◇ for "possible". Not proven impossible is ~⊢ ~◇A, proven possible is ⊢◇A. Your interlocutor's suggestion amounts to moving ~ across ⊢, which is fallacious. Not proven impossible gives us no information on the possibility of A, only on what we can prove. So it is consistent with the system of possible worlds containing a single world where A is false, which is a counterexample to ◇A, so it is not provable. By the way, even ~~ A → A (double negation law) is not a truism and is denied by constructionists. – Conifold Mar 16 '19 at 3:44
• @Conifold This is quite helpful, thank you! Could you direct me to somewhere I could read more about the standard symbols you mention and why my interlocutor's expression of the notation is incorrect/fallacious? – John Weston Mar 20 '19 at 13:02
• See SEP Modal Logic, especially sections 2 and 6. The most commonly used modal logic is S5 (it is also the strongest of the ones they describe). – Conifold Mar 20 '19 at 17:46

I'm not sure what the notation you quote is trying to achieve. It would help to point out that when speaking of something being possible, there are many different kinds of possibility. At the very least there is:

1. Physical possibility. Things that are compatible with our best understanding of the laws of nature. In this sense, a perpetual motion machine is impossible, meaning that if our understanding of the physics of our universe is correct then none can exist in our universe.

2. Actual possibility. Things that are compatible with the laws of nature together with some statement of the boundary conditions of the actual universe. It is not physically impossible for our solar system to have had seven planets rather than eight, but given the actual boundary conditions, eight is what we get. It is not physically impossible for unicorns to have evolved on our planet, but they didn't.

3. Epistemic possibility. Things that are compatible with what we know to be true, or things that might be true for all we know. This is a much broader kind of possibility than the previous two, because we don't know all the laws of nature or all the boundary conditions of the actual universe. For all we know, Arcturus might have twenty planets in orbit around it, so it is possible in this sense, though with future discoveries we might find that there are ten or a hundred or none. We tend to grade propositions by how credible they are, so we might still say of an epistemically possible proposition that it is highly unlikely.

4. Logical possibility. Roughly speaking, things that do not imply a contradiction. This is also a very broad understanding of possible. It is logically possible that there are unicorns, perpetual motion machines and honest politicians, though there may be none in our universe.

Your disputant appears to have either the third or fourth of these in mind. If some proposition is not proven to be impossible then it is epistemically possible: i.e. it might be true for all we can prove. And if there is no way to prove it false, i.e. there is no way to demonstrate that a contradiction follows from it then it is logically possible.

• What if my counterpart is talking about whether it's possible for souls to exist and leave the body? – John Weston Mar 15 '19 at 20:06
• In the absence of a definite theory of what souls are and how they are related to bodies, one would have to say that it is logically possible that souls exist and that they can exist independently of bodies. If we entertain it as a logical possibility but have no way to know that it is false then we might have to say it is epistemically possible too. But the important question here is: do we human beings in our universe actually have souls that can exist independently of our bodies? This is really a question about how plausible such a claim is given our other knowledge... – Bumble Mar 15 '19 at 20:34
• Considering that souls are typically thought of as intangible things, such a claim is difficult to assess. To a physicalist, souls are perhaps incompatible with a monistic understanding of the universe, or at the least their existence is an unwarranted hypothesis. But that just shifts the debate into the question of whether physicalism is correct. The problem with debates of this kind is that it is not obvious what even counts as evidence for or against physicalism, or souls for that matter. – Bumble Mar 15 '19 at 20:34

I think it's important to get clear on what kind of possibility you're talking about. On one way of understanding the kind possibility in question, your friend's claim is trivially true. On another, it's false. Take for example epistemic possibility, which is expressed in statements such as "for all we know, X". On this interpretation your friend's claim is: if we haven't proven ~X, then for all we know, X. This is trivially true we since "we haven't proven ~X" is just taken to mean "for all we know, X".

But if we take "possibility" to be a more objective kind of possibility, then your friend's claim is false. As of now, no one has proven Goldbach's conjecture nor has anyone proven its negation. But, at least on commonly accepted views, Goldbach's conjecture is either necessarily true or necessarily false. (To take another example: it was always necessarily true that water is H2O, so it was never possible that water wasn't H2O, even before this was proven.) Take epistemic possibility again: it's true that "for all we know, Goldbach's conjecture is true" and also true that "for all we know, Goldbach's conjecture is false", since no one has proven either. You and your friend may just have different notions of possibility in mind.

• What if my counterpart is talking about whether it's possible for souls to exist and leave the body? – John Weston Mar 15 '19 at 20:06
• @JohnWeston I would say it's hard to tell what he has in mind, as you can meaningfully apply different kinds of possibility to that question. However, it seems to me not very productive to talk about whether such a thing is possible or not. If you think it's false you may still admit that it's possible. Usually we're more interested in what is rational to believe, not in what is merely possible. You may say, for instance: sure, it's possible, but there is no good reason to believe it. – Eliran Mar 15 '19 at 20:11
• @JohnWeston Also, you may even accept your friend's claim that "if we have not proven it's impossible for souls to exists, then it's possible for souls to exist" and deny that we haven't proven that it's impossible for souls to exist. – Eliran Mar 15 '19 at 20:30