# Should truth entail possible truth?

It is a well-accepted axiom of modal logic that truth implies possible truth.

Is there any philosophical argument against this conclusion? In other words, should truth entail possible truth?

If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, we reject that for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. Not-possibly A is equivalent to necessarily-not-A. So we have A and necessarily-not-A, meaning the necessity of not-A doesn't imply the actual truth of not-A.

However formally within modal logic itself, you can mess around with axioms and frame conditions in whatever way you want. Rejecting "if P then possibly P" amounts to rejecting reflexivity as a frame condition. See https://en.m.wikipedia.org/wiki/Accessibility_relation for more about frame conditions and their corresponding axioms. (EDIT: Frame conditions tell us what worlds we "see" when evaluating possibly P and necessarily P at a world w. If at least one world that w "sees" satisfies P, then w satisfies possibly P. If every world w "sees" satisfies P, then w satisfies necessarily P. Reflexivity tells us that w always "sees" itself when evaluating statements of possibility and necessity. It may be that P is true in the actual world, but if we reject reflexivity then we're not looking at the actual world to determine the truth of possibly P! And maybe every other world we "see" indeed fails to satisfy P.)

(Noah Schweber's comments below should be heeded as well. The box and diamond operators can be interpreted in different ways for different modalities!)

• +1. For the OP, keep in mind that "should 'p is true' imply 'p is possible'?" is a very different question from "should 'p is true' imply '<>(p)'?" There are many ways to interpret the modality <> (and its dual, []) - 'is possible' is one, but others include 'is possibly true in the future' (and the present isn't the future!), 'is permitted' (and life isn't fair!), and 'is consistent' (and Godel's theorem makes this surprisingly subtle!). (contd) Apr 12, 2019 at 22:32
• This answer's second paragraph reflects this: modal logic isn't just about the modalities 'is possible'/'is necessary' (and for that matter, frames aren't the only way to provide semantics for modal logic, and often aren't even appropriate for a given task!). This is all an aside, since your question really does focus on possibility specifically, but it's a point worth mentioning given the "modal-logic" tag. Apr 12, 2019 at 22:36

Obviously truth implies possibility. So let me make a case for truth not implying possibility.

Let's start with an "applied logic" example. Suppose I'm trying to reason about the world using imperfect information (i.e. my senses and informal induction). At any given moment, I'll have some idea of what the world is, but that idea will probably be contradictory in subtle ways. For example, I may "accept" - for some meaning of the word - two physical theories which each work extremely well in their appropriate contexts but which as currently posed contradict each other (think about general relativity versus quantum mechanics). I believe each of a set of statements the conjunction of which is not possible. This is a situation in which I might want a formal system in which <> is interpreted as "is possible" but I don't have the rule "from p, infer <>p." And this issue also arises, with somewhat more urgency, in the context of artificial intelligence and more generally any situation where a machine is "making decisions" based on data about the world around it, and we're modeling that process (either in implementing it or in analyzing it after-the-fact) with a logical system.

Of course, what's true and what's currently believed are different (duh!), and so this isn't really an example of the phenomenon you're interested in. But implicitly invoked in our bringing this up is the principle that there are no true contradictions, and this is not universally held; the rejection of this principle is called dialetheism.

Now we get into a very interesting mess: how should a dialetheist think of possibility? I don't know of anyone who's argued - within the dialetheist context - that possibility entails consistency, and hence that there are true impossible facts as well as true contradictions, but I can sort of see how an argument for this might go. Since I think producing "original research" here isn't really appropriate (you asked "is there any argument" not "can there be any argument," after all) I won't go into this, but I do think it's worth mentioning in this context: that dialetheism puts us in a situation where the question becomes at the very least not trivially trivial.