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As usually happens, a statement (p) and its negation (~p) contradict each other. So, e.g. God does not exist, the negation of, God exists, together form a contradictory pair. A statement (p) and its negation (~p) are such that they're mutually exclusive (if one is true the other has to be false and if one is false the other has to be true) and jointly exhaustive (at least one of the two has to be true).

Take now the statement God may exist and its negation, the statement God may not exist. They do not contradict each other. The compound statement that goes God may exist AND God may not exist is not a contradiction.

  1. What does this have to do with the principle of bivalence?
  2. What are the implications for the law of the excluded middle (LEM)?
  3. What does this say about the law of noncontradiction (LNC)?
MJD
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Agent Smith
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    Let P be "god may exist". Then not P is "not the case that god may exist". under standard modal semantics, this is "impossible that god exist" and not "god may not exist". So maybe you mean something different by negation? – emesupap Sep 07 '23 at 04:51
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    Not much to do with your 3 classic laws of thought but your modal scope. Even weirder consider the correct claim from elementary arithmetic that 'it's necessary that 8 is less than 10', also we know '8 equals the number of planets of our solar system', but you cannot substitute 8 and confidently claim as before 'it's necessary that the number of planets of our solar system is less than 10'... – Double Knot Sep 07 '23 at 05:13
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    Modality; the first one is "It is possible that..." Negating it, we get "It is not possible that..." that is equivalent to "It is necessary not..." The second instead, is "It is possible that not..." and this, as you say, is not the contradictory of the first one. Contradictory pairs are produced putting the negation IN FRONT of the statement. – Mauro ALLEGRANZA Sep 07 '23 at 06:01
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    @MauroALLEGRANZA, Please read my reply to Bumble – Agent Smith Sep 07 '23 at 06:15
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    See also de dicto vs de re. – Mauro ALLEGRANZA Sep 07 '23 at 13:42
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    This is nothing to do with logic or the principle of non contradiction and everything to do with the fact that English does not precisely express logical propositions well. – Karl Knechtel Sep 08 '23 at 03:16
  • When I was in elementary school, there were 9 planets in the solar system. – JosephDoggie Sep 08 '23 at 16:44
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    may or may not is redundant, yes, and already implied by may. – Eric Duminil Sep 09 '23 at 18:09

2 Answers2

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This is one of those cases where it helps to use some formal logic, because one of the things formal logic is especially good at is exhibiting the scope of operators.

Suppose we use the propositional symbol G to represent the proposition "There is a god". We can overlook the issue of how to make sense of attributing existence to a thing. And let's use ◇ as a modal operator that we will understand as "possibly". We can overlook exactly what kind of possibility for the present purposes.

When you say "God may exist" this is naturally understood de dicto as "It is possible that there is a God". So we might write this as:

◇G

When you say "God may not exist" the negation may sit inside or outside the possibility operator. So,

◇¬G

means "it is possible that there is no God", while

¬◇G

means, "it is not possible that there is a God".

The sentences ◇G and ¬◇G together form a contradiction, so they satisfy your criteria that they cannot both be true and cannot both be false. This is entirely consistent with bivalence, excluded middle and non-contradiction.

The sentences ◇G and ◇¬G are not a contradiction. It is possible there is a God; it is possible there is no God.

We might also add that ◇(G ∧ ¬G) would be a weird sentence stating that it is possible for the contradiction "there is a God and there is no God" to hold. This sentence would never be true in what are called normal modal logics.

I think that answers your question. Things get more complicated if we combine modal logic with quantifiers and write things like:

◇(∃x)Divine(x)

or even

(∃x)◇Divine(x)

This introduces all kinds of additional problems concerning how to understand potential existence and potential properties, and how to quantify into referentially opaque contexts.

Bumble
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  • In colloquial usage I think the word "possible" means "we cannot at this time rule it out." In other words, X is possible if ~X is not "easily" proved from the current state of knowledge. Is there any logic that tries to model this concept? – causative Sep 07 '23 at 05:44
  • So what is the logical relationship between God may exist and God may not* exist? Both can't be false (as when God exists/doesn't exist). So, can both be true? This is a tough nut to crack because the claims seem to be more/entirely epistemological* than metaphysical i.e. If it was me utterimg those words, I'd be merely fessing my ignorance rather than making a statement about reality. – Agent Smith Sep 07 '23 at 06:21
  • Food for thought: Assuming, M = God may exist. Claiming M is logically equivalent to claiming God may not exist i.e. they mean the same thing. So the negation of M should be ~M, which is 1) not possible that God exists OR 2) not possible that God does not exist. – Agent Smith Sep 07 '23 at 06:30
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    @Causative That sounds like 'epistemically possible'. On this understanding □P means P is known to be true. □¬P means P is known to be false. ◇P is equivalent to ¬□¬P so it means we don't know P is false, hence P might be true for all we know, or P being true is consistent with what we know. Some accounts of epistemic logic dispense with the □ symbol and use K. There is disagreement about whether S4, S5, or some other modal system provides the best logic. https://plato.stanford.edu/entries/logic-epistemic/ – Bumble Sep 07 '23 at 06:38
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    @AgentSmith You are correct that ◇G and ◇¬G cannot both be false. We would need a little modal logic to prove it, but ¬◇G ∧ ¬◇¬G entails □¬G ∧ □G which entails □(G ∧ ¬G) which entails a contradiction in normal modal logics. As per my response to Causative, we could take 'possibly' to be epistemic. – Bumble Sep 07 '23 at 06:38
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    @AgentSmith It is not correct to say that God may exist is logically equivalent to God may not exist. Possibility does not exclude necessity. ◇P does not rule out □P. It is analogous to saying that some ravens are black does not rule out that all are. – Bumble Sep 07 '23 at 06:39
  • @Bumble, varrum? – Agent Smith Sep 07 '23 at 07:14
  • @Bumble But those modal operators don't actually model the process of gaining knowledge, do they? I am thinking of something like: a knowledge base of formulas is maintained, and expanded with each proved theorem, and ◇P if ~P is not in the knowledge base, and ~P is also not deducible from the knowledge base in 3 deductive steps. Therefore with every deductive step, the set of formulas P for which we can say ◇P decreases. – causative Sep 07 '23 at 13:57
  • @causative I doubt you can make counting the steps work. Depending on the proof system you will get different numbers of steps. And the set of ◇P's is potentially infinite, so it doesn't really decrease. I get the idea that it would be nice not to assume deductive closure, since in effect that assumes logical omniscience. You might be able to get close to the desired effect by using a formal model of belief revision. The most commonly used one is the AGM model. https://plato.stanford.edu/entries/logic-belief-revision/ – Bumble Sep 07 '23 at 14:19
  • @causative If ◇ represents this knowledge base, then things get weird if symbol ◇ is part of its own logical system, since it "depends on time", ie the truth value of ◇P might change as we prove more stuff (so we could prove ◇P, then prove more stuff, then ◇P becomes false even though we've proven it earlier). Even worse, it could sit at an edge of knowledge, where ◇P is true, but as soon as you add ◇P to the knowledge base, it becomes false. – Stef Sep 07 '23 at 15:49
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    @AgentSmith: Your intuition that “might” could imply “might not” comes from human language pragmatics, and particularly implicature, subtly different from implication. If you say “I’m hungry!” and I say “Oh, I’ve got some chocolate here”, the conversational context suggests I’m about to offer you some, but my statement doesn’t logically imply that. Similarly, if I say “It might rain today”, you can normally infer I also think it might not (because if I was sure, I’d have said so); but that’s implicature, not logical implication. – Peter LeFanu Lumsdaine Sep 09 '23 at 19:59
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  1. Nothing.

  2. There are none.

  3. Nothing.

May already entails perhaps, or perhaps not. If you are uncomfortable with perhaps, perhaps not, that is absolutely hunky-dory, but then best stay away from may (and similar indicators of modality).

Deipatrous
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  • Leaks assume ex arguendo that none of us disputes that 1+ 1 = 2. Can one then dispute the proposition that one plus one may equal two? In any case, I, for one, would dispute the proposition that one plus one may not equal two. – Paul Tanenbaum Sep 08 '23 at 19:26