[FOM] Counterfactuals in relative computability theory

[Matthias Jenny]

On Thu, Aug 18, 2016 at 5:23 PM Timothy Y. Chow <tchow at alum.mit.edu> wrote:

>
> I'm also having difficulty following your train of thought, so what
> follows is conjectural.
>
> It sounds like your thinking proceeds along roughly the following lines.
>
> 1. So-called "abstract objects" retain their identity across all possible
> worlds, at least if they are "fully precise."
>
> 2. If a connection can be established between a particular word, such as
> "algorithm," and an "abstract object," then the word also retains its
> identity across all possible worlds.
>
> 3. The connection in 2 is an ontological one.  Whether the connection is
> known by people in the actual world is irrelevant to the above
> considerations.
>

Thank you, this is helpful, as it allows me to discern where we're
misunderstanding each other.

Regarding 1: Actually, in S4 and stronger modal logics, *every* object
retains its identity across all possible worlds in the sense that it's a
theorem that, for all x,y, if x=y, then necessarily x=y.

Regarding 2: Two things. First, 'algorithm' is a predicate, so its semantic
value (i.e. the thing it "picks out"/expresses/refers to) is a property,
not an object. But even aside from that, I certainly don't subscribe to 2.
In fact, I'm not sure if I understand what you mean by it. Which of the
following two do you have in mind?

(a) If a connection can be established between a particular word, such as
"algorithm," and an "abstract object," then the word refers to the same
thing in all possible worlds.

(b) If a connection can be established between a particular word, such as
"algorithm," and an "abstract object," then the word has the same meaning
in all possible worlds.

I certainly don't subscribe to (a), and putting the point in terms of
objects instead of properties, 'Matthias Jenny's favorite natural number'
establishes a connection between a particular word and an abstract object,
but the referent is different in different possible worlds. (I take it you
to agree with this.)

I also don't subscribe to (b). Of course 'algorithm,' or really any word,
could've meant something slightly different, and it could even have meant
something radically different. But I don't think that this is relevant. The
following is often attributed to Abraham Lincoln:

Q: If you called a tail a leg, how many legs would a dog have?
A: Four, because calling a tail a leg doesn't mean it is one.

Similarly, even though 'algorithm' could've meant 'President of the United
States,' a possible world where it does is still a world where the set of
algorithms is the same as the set of algorithms in our world. At least
that's what I'm arguing.
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