An excellent reply. "=" is used in different ways in these examples, so one can argue that such equations as these do not use "=" in the way that it is being used by logicians doing modal logic. If this is so then Kripke apparently overreaches in claiming the necessity of such equivalences. Following this line of thought, the relation between mean kinetic energy and temperature is not one of metaphysical identity.
The implication for Kripke is a weakening of the apparatus he uses to argue for a posteriori necessity and doubt cast on his argument against the identity of mind and brain.
One possible reply is that there
is something in common to equations, since in each case they use the "=" to state that the
value on the left will be the same as the
value on the right; that it is this
value that is rigidly designated, not the items in the equation. So in E=mc2, what is rigidly designated is that the value of E is the very same as the value of mc², and so on for each example. This would be to agree with you that in physics "=" does not always assert an ontological identity, but that if the use of "=" is to make any sense, it must assert the identity of the two values it equates.
On this account, the identity here is not between ontological entities but between their values within the structure put together by doing science. This would considerably weaken the applicability of Kripke's system.
A second possibility is that the historical use of "=" back to Russell's attempt to ground arithmetic in logic,
does show that the "=" in physics is the same as the "=" in logic. They are both uses of the identity relation set out in
Begriffsschrift, and that
all arithmetic equations are grounded in that logical interpretation. It's just what "=" means. So the mean kinetic energy
just is
.
This is a pretty dogmatic response, stating that the reason we can write such equations at all is that their effectiveness is dependent on or justified by the logic of identity, that accepting your argument would be tantamount to claiming that identity signs in physics are ambiguous and equivocal. Pretty harsh. My response to
suffered from something like this, and perhaps Tim might say something similar. Are physical equations really that precise?
A third approach might be to take on board what has been said, and go back to the basics to reassess how our modal logics work.
In propositional logic, one can substitute any proposition for any other provided that they have the same truth value. In predicate logic, one can substitute any individual variable for another provided that they designate the same individual. In modal logic, this fails: while two propositions may both be true in a given world, it does not follow that they are true in every possible world. Truth is evaluated not only by what individual is designated, but also by which world the evaluation takes place in.
However, at the level of possible worlds semantics, modal logic is extensional: formulas are assigned sets of possible worlds as their extensions, and modal operators like necessity (□) are interpreted as quantifying over those sets. That is, □A is true at a world w if A is true in every world accessible from w. Because substitution of formulas with identical extensions preserves truth across all worlds in the model, the possible worlds interpretation is extensional.
☐p is true in
w iff p is true in every world that is accessible from
w.
We also have that in S5 and elsewhere that it is valid that a=b→□(a=b). It is the consequences of this simple theorem that Kripke is teasing out. The salient bit is that we may find out only a posteriori that a is indeed equal to b. This is what leads to the conclusion that so many find objectionable, that there are necessities that are discovered by looking around at how things are rather than understood a priori.
In the simplest case, that a=b means that a and b are the very same individual. And becasue of the extensionality involved, with some standard considerations we have it that if some expression is true for a, it will also be true for b. There is no obvious reason that this analysis can't be somewhat extended, including to kinds. So if we find, as seems to be the case, that every animal that has a heart also has kidneys, then extensionally, the set of animals with hearts and the set of animals with kidneys are the very same set, and we can substitute "animals with hearts" for "animals with kidneys" while preserving truth.
So if, whenever we pick out an animal that has a heart, we also thereby pick out an animal that has a kidney, then necessarily, if an animal has a heart then it has kidneys. If they are extensionally equivalent in every possible world, then necessarily, if an animal has a heart is has kidneys.
We might do this if, for instance, we were to insist that if we were to come across some animal that appeared to have a kidney but no heart, what appears to be a kidney is not a kidney, but has been misidentified.
The third response, then is to note that Kripke's move treats identity statements as extensional, and not in the intensional fashion seen in Malcolm. These rigid designators refer to the same entity or set in all possible worlds. Substitution of such identical entities is permitted at a modal semantic level, so when we find that a=b a posteriori, we might stipulate this as a metaphysical necessity, and reject counter instances as errors of identification.
What we have here is an at least apparent conflict between two quite different approaches. Folk might be tempted to suppose, somewhat simplistically, that either one or the other must be true, and the other must be false. Is it the case that we must either adopt the extensional approach and Kripke, or the intensional approach of Malcolm? Or are they talking
past each other.
It might be interesting to look at Malcolm's approach through the lens of one of the formal intuitionist logics. Perhaps relevance logic would be informative.
This post has taken a few hours to put together, so thanks for the challenge. I hope you find it as interesting as I do.