Some may feel this is not appropriate for the mathematics stack exchange, but it is a question in logic, and I feel it is entirely a good fit. The following argument has been put forth by the philosopher Alvin Plantinga in favor of mind-body dualism. I cannot see any logical error, but also have a hard time believing dualism has been demonstrated by a purely logical argument. If someone could please point out what is logically wrong with this, I would be appreciative.
The argument:
- If A and B are identical then any statement of A is true of B and vice versa.
- I can imagine existing without my body, for example in the body of a bird. I cannot imagine my body existing without my body.
- By (2) we showed the existence of a statement that is true of (me) but not true of my (body).
- By (1) my body and me are not identical.
End argument.
- is true by definition. The first part of 2. is true by my experience -- I can imagine waking up in a bird's body, can you? The second part of 2. is true by my experience, and if I'm not mistaken it must be true of everybody's experience since one cannot imagine something logically invalid -- my body is my body by definition and cannot exist and not exist at the same time (P and $\not$ P cannot both be true). 3. and 4. are simply logical conclusions.
Has Plantinga rigorously proven an age old philosophical position defended by Descartes hundreds of years ago?
Update: Maybe this is better suited for the philosophy stack exchange? Could I please have it moved? Thank-you.
I think this is barely relevant to this site, but ultimately that it is relevant, and the reason is that it is a good motivataion for modal logic.
Specifically, I would argue that $(1)$ is not true, at least the way that we might want it to be - for instance, "the eveningstar" and "the morningstar" are both Venus, but I can imagine that the eveningstar would not rise first in the morning, whereas I certainly can't imagine the morningstar not rising first in the morning (because that's how it's defined). Does this show that the eveningstar and the morningstar are different?
We model this sort of thing via possible world semantics, or Kripke frames. See https://en.wikipedia.org/wiki/Kripke_semantics. Briefly speaking, a Kripke frame (usually pictured as a directed graph) consists of a family of worlds (vertices) together with an "accessibility relation" (arrows). Each usual propositional sentence is true or false at each given world (worlds can disagree), but the Kripke frame structure lets us form modal statements as well, using "possible" and "necessary." Specifically, say that a world $w$ in a frame $K$ satisfies "it is necessary that $p$ holds" if, in every world accessible from $w$, $p$ is true. Similarly, "it is possible that $p$" is interpreted as "it is not necessary that $\neg p$."
For this example, the accessibility relation we're interested in is "$w'$ is an imaginable world, from the point of view of $w$." (It might be more natural to replace "world" by "state of affairs.")
In this context, (1) splits into two pieces:
If $A=B$, then any purely propositional statement holds of $A$ iff it holds of $B$.
If $A=B$, then any statement at all (including modalities) holds of $A$ iff it holds of $B$.
These two princples are very different, as the morningstar-eveningstar exapmle shows! We can have two formulas $\alpha$ and $\beta$, and worlds $w$ and $w'$, such that relative to $w$ $\alpha$ and $\beta$ define the same object, but relative to $w'$ they don't.
(NOTE: the question, "What graph properties should a Kripke frame satisfy in order to reflect such-and-such a logical situation?" is a great one, but not relevant to this question; for now, it's enough to realize that there can be more than one world, at all.)
So that said, I've actually skipped over a really important point!
I introduced worlds in Kripke frames as knowing about propositional statements. But we’re talking about objects, so we’re living in the world of predicate logic.
Specifically, we're trying to understand a statement of the form $(*)$ "The thing defined as '$A$' equals the thing defined as '$B$'." Intuitively, I want to say that there might be a world $w$ at which "$(*)$" is true but "necessarily $(*)$" is false. But this involves interpreting names, or definitions, across different worlds! How does that work?
It turns out this is a really huge deal - there’s lots of ways of potentially axiomatizing how Kripke frames “ought” to work in first-order logic, but there isn’t an obviously nicest one (see https://math.berkeley.edu/~buehler/First-Order%20Modal%20Logic.pdf, especially around 1.3). So the modal logic that’s being butchered (:P) by Plantinga’s argument is actually even more complicated than the run-of-the-mill propositional modal logic we know and love.
The mistake Plantinga’s making, though, does have a purely propositional analogue: it is not the case that “$p\iff q$” is the same as “It is necessary that $p\iff q$.” This is essentially the mistake being made: swapping identity for necessary identity.