## Nencha on counterpart semantics

Informal talk about de re necessity is sometimes "weak" and sometimes "strong", in Kripke's terminology. When I say, 'Elizabeth II could not have failed to be the daughter of George VI', I mean – roughly – that Elizabeth is George's daughter at every world at which she exists. By contrast, when I say, 'Elizabeth II could not have failed to exist', I don't just mean that Elizabeth exists at every world at which she exists. My claim is that she exists at every world whatsoever. The former usage is "weak", the latter "strong".

When people give a semantics for the language of Quantified Modal Logic (QML), they typically treat the box as strong. '$$\Box Fx$$' is assumed to say that x is F at every accessible world, not just at every accessible world at which x exists.

There are good reasons for this choice. If the box is given a weak interpretation, the logic becomes odd. $$\Box(A \land B)$$ no longer entails $$\Box A$$. Also, many intuitive statements of weak necessity can't be expressed in the language of QML. We might want to say that Elizabeth is necessarily the daughter of George, but that George isn't necessarily the father of Elizabeth. You can't say this in the standard language of QML.

One of two mistakes Lewis made when he introduced his counterpart-theoretic interpretation of QML, in Lewis (1968), is to give the box a weak interpretation. He reads '$$\Box \phi(x)$$' as 'at every world, every counterpart of x satisfies $$\phi(x)$$'. Worlds where x has no counterpart are effectively ignored.

Lewis did consider a strong interpretation, suggested to him by Kaplan. On Kaplan's proposal, '$$\Box \phi(x)$$' means 'at every world, some counterpart of x is F'. As Lewis, points out, this has unacceptable consequence for the diamond, assuming the diamond is the dual of the box. '$$\Diamond \phi(x)$$' now means 'at some world, every counterpart of x satisfies $$\phi(x)$$'. If, say, Plato has no counterpart at a world in which there is nothing but empty spacetime, and 'x' picks out Plato, it follows that all sentences of the form $$\Diamond Fx$$ are true. But surely the mere fact that Plato could have failed to exist doesn't entail that he could have been a scrambled egg, and that he could have been a round square!

Nencha (2022) says that Lewis should simply have allowed for both readings. After all, ordinary modal discourse sometimes involves strong necessity and sometimes weak necessity. We should use Lewis's original interpretation when we formalise 'Plato couldn't have been a scrambled egg' as $$\Diamond Fx$$, and we should use Kaplan's interpretation when we formalise 'Plato could have failed to exist' as $$\Diamond \neg\exists y(y=x)$$. The former involves the dual of weak necessity, the latter the dual of strong necessity.

In fact, Nencha doesn't quite accept Kaplan's proposal for strong necessity. She says, plausibly enough, that $$\Box \phi(x)$$ should imply that all counterparts of x satisfy $$\phi(x)$$. On her proposal, the strong reading of '$$\Box \phi(x)$$' says 'x has a counterpart at every world and all counterparts of x at all worlds satisfy $$\phi(x)$$'. The dual '$$\Diamond \phi(x)$$' accordingly says 'either x has no counterpart at some world or some counterpart of x at some world satisfies $$\phi(x)$$'.

I think Lewis was right to reject Kaplan's proposal. And he would have been right to reject Nencha's. There is no natural reading of 'possible' on which the mere fact that Plato could have failed to exist entails that it is possible that he is a round square.

The problem also arises for 'necessary'. Consider '$$\Box \neg Fx$$'. On Nencha's account, this says that x has a counterpart at every world and that every counterpart of x at every world is not F. Assuming that Plato is not a necessary existent, 'it is necessary that Plato is not a round square' comes out as false. (It's also false on Kaplan's account.) This doesn't seem right.

Neither Kaplan's nor Nencha's interpretation of the box captures the ordinary strong reading of 'necessary'. Nencha's account looks plausible for simple, un-negated prejacents, but not for more complex formulas.

This brings us to Lewis's second mistake in Lewis (1968): his decision to interpret the language of QML by means of recursive translation rules. It would have been better to use the more standard technique of model theory that recursively defines 'true at a world in a model relative to an assignment'.

One can, of course, translate Lewis-style rules into a more standard model-theoretic semantics, as explained e.g. in Varzi (2020). But the converse isn't true. Many natural model-theoretic interpretations are difficult to spell out in terms of translation rules.

As it turns out, it is hard to give an adequate semantics of strong necessity using translation rules. Kaplan's won't do. Nencha's won't do either. Ramachandran and Forbes had a go at this in the 1980s and 1990s, but didn't find an acceptable answer – at least not without compromising other aspects of counterpart semantics.

If we use the model-theoretic approach, we can easily find a nice semantics for strong necessity, as I explain in Schwarz (2012).

What does it take for '$$\Box \phi(x)$$' to be true at a world w under some assignment g? On the strong reading of the box, we require that '$$\phi(x)$$' is true at all (w-accessible) worlds. We also need to shift the interpretation of 'x', so that when the point of evaluation is at some world v then 'x' picks out a counterpart at v of its original referent, g(x). What if there is no such counterpart at v? Then 'x' goes empty: the term picks out nothing relative to a world where g(x) doesn't have a counterpart.

We now need to say how to evaluate formulas with empty terms. A natural idea is that atomic formulas with empty terms are all false. On this approach, '$$\Box Fx$$' is equivalent to 'x has a counterpart at all worlds, and all counterparts of x at all worlds are F' (ignoring accessibility). But '$$\Box \neg Fx$$' is equivalent to 'no counterpart of x at any world is F'. Accordingly, '$$\Diamond Fx$$' is equivalent to 'some counterpart of x at some world is F', but '$$\Diamond \neg \exists y(y=x)$$' is equivalent to 'at some world, x doesn't have a counterpart'. This is all as it should be.

(There is, of course, another option: We could say that if an individual doesn't have a counterpart at some world, then it still has a counterpart in the world's "outer domain". This has some formal advantages, and it simplifies the problem of interpreting strong necessity, but one might say it is cheating.)

TLDR: Let's all stop doing counterpart semantics with translation rules!

Lewis, David. 1968. “Counterpart Theory and Quantified Modal Logic,” 15.
Nencha, Cristina. 2022. “On the Translation from Quantified Modal Logic to Counterpart Theory.” Synthese 200 (5): 383. doi.org/10.1007/s11229-022-03813-9.
Schwarz, Wolfgang. 2012. “How Things Are Elsewhere: Adventures in Counterpart Semantics.” In New Waves in Philosophical Logic, edited by G. Russell and G. Restall, 8–29. Basingstoke: Palgrave MacMillan.
Varzi, Achille C. 2020. “Counterpart Theories for Everyone.” Synthese 197 (11): 4691–4715.