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In the literature about the epistemology of modality I stumbled upon various types of possibilities, e.g. epistemic possibility, metaphysical possibility. I have a rough unterstanding of these, but would like to know what it is exactly for something to be a metaphysical possibility.

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Some formal notes to complement Mauro's excellent answer. As one would expect in a discussion of modality, we're going to talk about modal models when defining things. Most will be familiar with logics K, S4, and so on. K and its superlogics are too sophisticated for a discussion of metaphysical modalities, so we'll begin with pre-Kripke modal models, going back to Carnap. We start with the language:

Definition 1. (Propositional modal language) Given a propositional letter p, the language of propositional modal logic is defined by the following grammar:

                                              φ   :=   p   |   φ′   |   ¬φ   |   (φ ∧ φ)   |   □φ.

This means that p with any number of primes is a formula, ¬φ is a formula if φ is a formula, and so on. To interpret this language we define Carnapian modal models (later we'll refine these; cf. Definition 6):

Definition 2. (Carnapian models) A Carnapian modal model is a singleton M = (V), where V is a set of valuations from formulas in the language of propositional modal logic to truth-values.

With these we can define what it means for a formula of propositional modal language (Definition 1) to be true in a Carnapian model with respect to a valuation (~world):

Definition 3. (Carnapian semantics) The truth of a formula φ of propositional modal language in a Carnapian model at a valuation v ∈ V is defined by induction on the complexity of φ as follows:

  1. M, v |= p               iff     v(p) = 1;
  2. M, v |= ¬φ            iff     ¬(M, v |= φ);
  3. M, v |= φ ∧ ψ        iff     (M, v |= φ) and (M, v |= ψ);
  4. M, v |= □φ            iff     ∀v′ ∈ M : M, v′ |= φ.

The only clause worth paying attention to here is the one for the box: □φ is true in a Carnapian model M = (V) at a valuation v ∈ V just in case φ is true in all valuations in V, with no restriction whatsoever! Because of this lack of restriction on accessibility, Carnapian models are a natural framework wherein to explicate the notions of metaphysical possibility and necessity:

Definition 4. (Metaphysical necessity □m). Formula φ of propositional modal language is metaphysically necessary (symbolically: □mφ) iff it is true in Carnapian models at all valuations.

Example. Let's see whether the formula (p ∨ ¬p) is metaphysically necessary. It is iff □(p ∨ ¬p) is valid with respect to Carnapian models. Recall that valuations are simply extensions of truth-assignments to the propositional letters occurring in formulas. In this case, we have only one propositional letter, viz. 'p'. There are therefore two possible valuations: v1 = {(p, ⊤)} and v2 = {(p, ⊥)}, so let's check whether (p ∨ ¬p) holds in both. Clearly it does: v1 satisfies p, so the whole disjunction is also satisfied; and v2 satisfies ¬p, so again the whole disjunction is also satisfied. Therefore, we know that (p ∨ ¬p) is metaphysically necessary. Metaphysical possibility is defined in an analogous manner:

Definition 5. (Metaphysical possibility ◇m). Formula φ of propositional modal language is metaphysically possible (symbolically: ◇mφ) iff it is true in Carnapian models at some valuation.

Example. Let's see whether the formula (p ∧ q) is metaphysically possible. It is iff ◇(p ∧ q) is valid with respect to Carnapian models. Since there are two propositional letters in that formula, viz. 'p', and 'q', there are 22 = 4 possible valuations: v1 = {(p, ⊥), (q, ⊥)}, v2 = {(p, ⊥), (q, ⊤)}, v3 = {(p, ⊤), (q, ⊥)}, v4 = {(p, ⊤), (q, ⊤)}. Now, ◇(p ∧ q) will be valid just in case there is at least one valuation among those four s.t. p ≡ q ≡ ⊤. Is there such a valuation? Yes, v4. Therefore, we know that (p ∧ q) is metaphysically possible.

Now, if we want to explicate the notions of physical, biological, epistemic, and so on, modalities, we have to somehow restrict these Carnapian models. Thankfully that has already been done by Kripke, who introduced the notion of relative modalities (in Carnapian models, modalities are absolute in that we evaluate them with respect to all models). The propositional language we defined above (Definition 1) we'll now interpret in Kripke models with the hope of defining non-metaphysical modalities.

Definition 6. (Kripke models) A kripke or relational modal model is a triple M = (W, R, V), where W is a set of possible worlds, R is a binary accessibility relation on W, and V is a valuation from formulas in the propositional modal language and worlds in W to truth-values.

With these we can define what it means for a formula φ of propositional modal language to be true in a Kripke model with respect to a world of evaluation:

Definition 7. (Kripke semantics) The truth of a formula φ of propositional modal language in a Kripke model M = (W, R, V) at a world w ∈ W is defined by induction on the complexity of φ as follows:

  1. M, w |= p               iff     V(p, w) = 1;
  2. M, w |= ¬φ            iff     ¬(M, w |= φ);
  3. M, w |= φ ∧ ψ        iff     (M, w |= φ) and (M, w |= ψ);
  4. M, w |= □φ            iff     ∀w′ ∈ W : wRw′ → M, w′ |= φ.

This last, fourth clause is the crucial distinction between pre-Kripke and post-Kripke modal logics: a formula is necessary iff it is true in all accessible/related worlds to w, not in all worlds whatsoever. It is this notion of relative accessibility that we can exploit to define non-metaphysical notions of possibility and necessity. To stay focused, let's define the notion of physical necessity/possibility, following Mauro's definition. Let's suppose a set L(w) of physical laws is given for any given world w. We define the notion of physical accessibility Rp as follows:

Definition 8. (Physical accessibility Rp) World v is physically accessible from world w (symbolically: wRpv) with respect to the space of physical laws iff L(w) and L(v) are consistent.

Example. Let w be our world, and L(w) our laws of physics. An arbitrary possible world v is physically accessible from our world (wRpv) just in case the laws of physics L(v) at v don't violate any of our L(w).

With this notion of physical accessibility we can define physical necessity as follows:

Definition 9. (Physical necessity □p) Formula φ of propositional modal language is physically necessary (symbolically: □pφ) at a world w in a Kripke model M iff it is true in all physically accessible worlds, i.e., for all v ∈ M s.t. wRpv, M, v |= p.

In an analogous manner we can define the notion of physical possibility:

Definition 10. (Physical possibility ◇p) Formula φ of propositional modal language is physically possible (symbolically: ◇pφ) at a world w in a Kripke model M iff it is true in some physically accessible world, i.e., for some v ∈ M s.t. wRpv, M, v |= p.

A similar explication can be done for the other non-metaphysical modalities. All that has to be added is an appropriate definition of accessibility (e.g. epistemic, doxastic, etc.).

I should mention that we didn't have to go into Kripke models to explicate non-metaphysical modalities; we could have added restrictions to Carnapian models directly using valuations. Similarly, we could obtain (something extremely close to) Carnapian models by considering Kripke models with Rs that are equivalence relations (corresponding to S5); because for such models, since every worlds is accessible from every world, the notion of accessibility is rendered useless. So consider it as my own personal preference to use Carnap's models for metaphysical necessity (as opposed to something like S5) and Kripke models for non-metaphysical ones.

Hope this helps. Suggestions/corrections are welcome, as always.

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Excellent. Thank you for your work. –  iphigenie May 18 at 12:57
    
@iphigenie Thank you :) –  Hunan Rostomyan May 18 at 13:02

See SEP entry about The Epistemology of Modality :

Φ is metaphysically possible if and only if Φ is true in some metaphysically possible world. Example: It is metaphysically possible that some physical particle moves faster than the speed of light.

Compare with :

Φ is physically possible with respect to physical laws L if and only if Φ is logically consistent with L. Example: Given the actual laws of physics, it is physically possible for a train to travel at 150 mph.

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But that just moves the burden to metaphysical possible world. What is it for a world to be a metaphysical possible world? –  Lukas May 17 at 12:23
    
Here's a way to help. it is physically possible in the actual world w to $\phi$ iff there is at least one accessible possible world w* that has the same physical laws as the actual world, and in w* $\phi$ is true. So, if we think of the laws of metaphysics as being like the laws of physics, then we could generalize. $\phi$ is metaphysically possible in the actual world w, iff there is a possible world w* accessible from w such that w* has all the same metaphysical laws as w, and in w* $\phi$. –  shane May 17 at 13:06
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@Lukas - I agree with you; but all "follows from" the assumption that "metaphysics" is something ... meaningful. –  Mauro ALLEGRANZA May 17 at 14:15

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