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The question is about the logical form of the sentence, x can be any object or process, actual infinity, tachyon, philosophical zombie, telepathy, etc. To conclude that x is impossible we are supposed to survey all possible worlds and verify that x does not occur in any of them. But there comes "it is possible" again, which seems to require something like meta-possible worlds, each being a collection of all(?!) possible worlds. Things get even more interesting with "it is contingent that x is impossible", in this case x would have to be missing from some meta-possible collections but occur in others.

But doesn't the second stage mean that we did not properly survey all possible worlds at the first stage? It seems that this meta-modality collapses, "it is possible that x is impossible" trivializes to "x is impossible", and "it is contingent that x is impossible" is a contradiction (if x is not to be found in any possible world then it is necessarily impossible, nothing contingent about it). However, colloquial meaning of the sentence seems to be distinct from the trivial one. If, on the other hand, the meta-modality is retained how are we supposed to distinguish between possible worlds of the first stage, and worlds that only become "visible" at the second stage?

Is there a system of logic that can interpret such sentences without trivializing them?

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Yes, there is the framework which is known as two dimensional semantics, a concept first developed by Robert Stalnaker in the 1970s. The related two modal dimensions are not modal in the same sense. According to a common interpretation (following the terminology of Saul Kripke in Naming and Necessity), one modal dimension is considered "metaphysical", subject-independent possible worlds, while the other modal dimension is considered "epistemic", subject-centered possible worlds. A classical example is "water is H2O". It is said to be epistemically contingent (our watery substance might not have been H2O), yet metaphysically necessary (once we discovered that our watery substance is H2O, it has become the very definition of water (for us)).

Here, from the preface of Laura Schroeter's SEP article on two-dimensional semantics:

Two-dimensional (2D) semantics is a formal framework that is used to characterize the meaning of certain linguistic expressions and the entailment relations among sentences containing them. Two-dimensional semantics has also been applied to thought contents. In contrast with standard possible worlds semantics, 2D semantics assigns extensions and truth-values to expressions relative to two possible world parameters, rather than just one. So a 2D semantic framework provides finer-grained semantic values than those available within standard possible world semantics, while using the same basic model-theoretic resources. The 2D framework itself is just a formal tool. To develop a semantic theory for someone's language, a proponent of 2D semantics must do three things: (i) explain what exactly the two possible world parameters represent, (ii) explain the rules for assigning 2D semantic values to a person's words and sentences, and (iii) explain how 2D semantic values help in understanding the meanings of the person's words and sentences.
The two-dimensional framework has been interpreted in different ways for different explanatory purposes. The two most widely accepted applications of two-dimensional semantics target restricted classes of expressions. David Kaplan's 2D semantic framework for indexicals is widely used to explain conventional semantic rules governing context-dependent expressions like ‘I’, ‘that’, or ‘here’, which pick out different things depending on the context in which the expression is used. And logicians working on tense and modal logic use 2D semantics to characterize the logical implications of operators like ‘now’, ‘actually’, and ‘necessarily’. Such restricted applications of 2D semantics are intended to systematize and explain uncontroversial aspects of linguistic understanding.
Two-dimensional semantics has also been used for more ambitious philosophical purposes. Influential theorists like David Lewis, Frank Jackson and David Chalmers argue that a generalized 2D semantic framework can be used to isolate an apriori aspect of meaning. Roughly, the idea is that speakers always have apriori access to the truth-conditions associated with their own sentences. On the face of it, this apriority claim seems to conflict with the observation that certain necessary truths, such as ‘water = H2O’, can be known only on the basis of empirical inquiry. But proponents of generalized 2D semantics argue that the 2D framework undercuts this objection, by showing how such aposteriori necessities are consistent with apriori access to truth-conditions. The positive reasons to accept generalized 2D semantics, however, are bound up with larger (and partly disjoint) explanatory projects. As a consequence, debates over the merits of generalized 2D semantics touch on broader controversies about apriority, modality, semantic theory and philosophical methodology.
The two-dimensional framework can also figure in a theory of ad hoc language use, instead of a theory of literal meanings. Robert Stalnaker's influential 2D account of assertion falls in this category. His “metasemantic” interpretation of the 2D framework is intended to characterize what is communicated when conversational partners are partially ignorant or mistaken about the literal meaning of their own words. Although it is formally similar to generalized 2D semantics, Stalnaker's use of the 2D framework avoids apriori accessible truth-conditions of the sort posited by generalized 2D semantics.

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Another option using a Possible Worlds account is to interpret "possible" and "impossible" slightly differently, making use of alternative Modal Logics for the box and diamond operators. These can still be worked into discussions about possible worlds, but the semantics of how these worlds are related to one another becomes a little more intricate.

The analysis of "impossible" you've used employs a quantification over every world in our domain of worlds. Some modal logics, however, don't treat necessity and possibility as absolutely general quantifiers in that way. Saul Kripke's weak modal logic K, for instance, adds only three basic rules to an underlying propositional logic which seem to capture some basic intuitions about necessity and possibility:

Necessitation: If A is a theorem of K, then so is □A

Distribution: □(A→B) → (□A→□B)

Duality: ◊A = ~□~A

(Distribution can be read as "If it's necessary that A follows from B, then if A is necessary then so is B")

In this logic, thinking of "Necessity" as something being true in all possible worlds doesn't quite match up perfectly. For instance, if that were the case, then we should expect that:

Localization: (□A)→A

That is, that if something is necessary then it is in fact true. But system K can't prove this, since its axioms aren't exclusive and exhaustive about what kinds of things are necessary (merely affirming that theorems are among the necessities)

What do such logics really mean, in a metaphysical sense? Well, Kripke's way of interpreting the weaker logics builds on the possible worlds model through the use of Frames. In Frame semantics, we have as before a set of possible worlds, but also a binary relation over the set of worlds which we call Accessibility. The idea here is that at each base possible world, some of the other worlds are accessible to it. What is possible at a base world is something that obtains at one world that is accessible to that base, and what is necessary at a base world is something that obtains at every possible world that is accessible to it.

In the case of standard (S5) Modal logic, we use a trivial accessibility relation such that every world is accessible to every other world, and so possibility and necessity quantify over all worlds. We can also use Frame semantics to account for weaker modal logics by trimming down what is accessible to any given world, or justifying the addition of new axioms by arguing for certain qualities for an accessibility relation. For instance, if we stipulate as a requirement that our accessibility relation is Reflexive (that any possible world w is always accessible to itself) then we'd have a model satisfying Localization.

In this way, we can see some models where something might be possibly impossible, but not actually impossible. Consider for instance the case of three possible worlds, w1, w2 and w3, where all worlds are accessible to themselves, and w2 and w3 are both accessible to w1 but not to each other. If we are at w1 where a proposition p is false, and p is also false at w2 but true at w3, then we can see that p is possible at w1 (because it's true at w3). However, since w3 is not accessible to w2, then at w2, p is impossible. So at w1, p is both possible and also possibly impossible.

  • Thank you, very interesting. What intuitive idea is accessibility supposed to represent? Is it based on "closeness" for worlds, accessible worlds are "similar" to the base, is accessibility transitive? – Conifold Jun 25 '15 at 17:27
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    One intuition behind accessibility is the idea that those worlds that are possible are in some sense proper variations of the actual world, rather than wholly distinct entities. For instance, one kind of possibility we might be interested in is possibility consistent with physics as we know it. What would it mean for, say, the sky to be green? Given what we know about how light refracts in the earth's atmosphere, could we sketch a possible world in which the laws of physics remain as they are, but where much less of the sun's white light is reflected off the earth's surface? – Paul Ross Jun 25 '15 at 19:01
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    Another way of thinking about it might be that sometimes we face possibilities that are in a sense exclusive, such as in temporal modal logics where we think of there being distinct futures. Depending on how we think of the identity of individuals, for instance, it might be possible that I could be a great doctor or a great financier, but it might not be possible that the me that is a great financier in one alternate possibility could be a great doctor - being the great financier (maybe due to character or training) is in some sense in conflict with the possibility of being the great doctor – Paul Ross Jun 25 '15 at 19:16
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    In itself, I guess Accessibility is much more of a technical tool than a metaphysical grounds for discussion about necessity, and we might think that some kinds of possibility would lean towards some kinds of accessibility (for instance, the physical possibilities might seem to all be transitively and symmetrically connected, whereas temporal possibilities might be asymmetrical. Stepwise computational derivation, as another option for a kind of accessible possibility, need not be transitive - it might take two distinct steps to reach some new state.) The application's the thing! – Paul Ross Jun 25 '15 at 19:22
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How to interpret "it is possible that x is impossible"?

As Paul Ross explains there is a certain calculus S5 of modal logic, i.e. a set of axioms and rules, where the two sentences "it is possible that x is impossible" and "x is impossible" are equivalent.

Using the possible worlds interpretation one can interpret the result as follows: If in at least one world the sentence "x is impossible, i.e. false in all worlds" is true, then x is false in all worlds.

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