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There is a clear example I have in mind of a modal transformer. The primary winding is in possible world, W1. The secondary winding is in possible world, W2. W1 and W2 are not causally linked in any way. There is no magnetic flux that flows between the worlds. The transformer halves, primary and secondary, are at arbitrary space-time locations in each of the worlds. There is no law of conservation of energy in W1 or W2. However, both halves act in the way we would expect a transformer in our world with this difference. In W1, the primary coil loses energy. In W2, the secondary coil gains energy. They act as if they are causally linked but they are not. The W1 coil can best be described as an energy losing inductor. The W2 coil can best be described as an energy creating inductor. Suppose, they gain and lose energy in the proportions we would expect if the two halves were connected in our world.

Now, the following is true of this trans world modal transformer:

Across both worlds, no energy is lost by this system.

The energy lost in W1 = the energy gained in W2 without ANY causal links between W1 and W2.

The question is:

Is the modal transformer described above consistent with our best possible world semantics?

Does this example or one's like it suggest that we modify our definition of a necessary truth in modal logic?

The counterfactual truth involves W1 and W2, that when combined it matches a transformer in a world W3 which has a physical law for transformers. W1 does not have W3's law, nor does W2, so a statement of the relevant physical law is false in W1 and false in W2, and also false in W3, but analogous to a true statement in W3. The analogy breaks down specifically because of a lack of causal continuity between W1 and W2. So, the W1+W2 truth is necessarily false taking the worlds on an individual basis, but contingently true of W1+W2 taken together.

  • Can you make clearer what your question is other than checking your thought experiment? – virmaior Sep 22 '17 at 3:33
  • The discussion in the comments to the answer below clarify the example. This example asks if we have a truth here that cannot be described by modal logic. The more general question is if you have a contingent truth that is false in W1 and false in W2, and in general false in the other possible worlds, but true in W1+W2, and contingent even though those two worlds are not causally linked, then how does that affect our definition of necessary truth and falsehood? Or in simpler terms, formerly this necessary falsehood is contingently true in W1+W2, do we thereby have to modify our definitions. – Pythagorus Sep 23 '17 at 19:18
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It looks like you're not describing possible worlds as in a possible world semantic, but rather two parallel universes. The concepts are slightly different and serve different purposes.

Possible world semantic is a model for modal logic. It is a tool to analyse statements involving necessities, possibilities, or conditional and counterfactual statements. The aim is not to describe specific parallel universes that could interact or not. In modal logic, there are no proper names to refer to particular possible worlds, only modal operators such as "possibly" and "necessarily". So you cannot describe your setting using modal logic.

In possible world semantic, talking about causal relations between world is meaningless. There are only accessibility relations between worlds. So even saying that there's no causal relations (whatever the analysis of causation you provide) seems meaningless to me (even more so if you analyse causation in modal terms). Finally, assuming that W1 loses energy when W2 gains energy assumes that both worlds share a time referential, which does not go without saying.

In a sense there's no inconsistency since you describe possible worlds which are logically consistent, but in another sense, this seems to stem from a misunderstanding of what possible world semantic is about.

  • David Lewis "On the Plurality of Worlds", p.81 "Likewise across worlds. No trans-world causation, no trans-world causal continuity; no causal continuity, no survival, no travel." Obviously, talk of the lack of causality between possible worlds is not meaningless. – Pythagorus Sep 20 '17 at 1:37
  • The two worlds DO NOT share a time referential at all. They just act in concert when viewed from a semantic point of view where you can see analytically the true statements in W1, W2, and W1 + W2. There is no "when", but it is a fact of the two worlds that they pair in the way described, so when viewed semantically there is a W1 + W2 truth that looks like a regular transformer even though the two do NOT interact in any way. There is no misunderstanding, and perhaps no inconsistency, but something IS strange about it. – Pythagorus Sep 20 '17 at 1:44
  • Ok fair enough. Lewis' theory is a particular case because he sees possible worlds as real entities, akin to parallel universes (which almost nobody else assumes) and in that case it makes sense. Concerning time, then I don't understand your setting. Losing or gaining energy is a dynamical process. How can you say they lose and gain energy in the same proportions without referring to particular times? Do you mean W1 and W2 exist at all time and lose or gain the same quantity of energy at any time? – Quentin Ruyant Sep 20 '17 at 12:53
  • I mean that for some time and place in W1 and for some time and place in W2, where these times and places are otherwise arbitrary in their respective worlds, they act such that if they were together in a single world, they would act like a normal transformer. These two worlds do not have our physical laws. It is quite normal in W1 for energy and magnetic flux to disappear in an inductor, and it is quite normal in W2 for energy and magnetic flux to suddenly appear in an inductor. So, we have here an example of a W1+W2 truth that looks exactly like causal continuity but is not. – Pythagorus Sep 20 '17 at 17:41
  • Ok so there's some kind of mapping between the times of both worlds, such that W1 and W2 act in this way, but this mapping is arbitrary? Yes there's no inconsistency here. Now this state of affairs cannot be described with modal logic. – Quentin Ruyant Sep 20 '17 at 18:44

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