I was helped immensely in my understanding of the structure of Kripke's argument by a critical response to Kripke by Scott Soames. In an intricate bit of philosophical analysis, Soames shows that Kripke is equivocating between two different meanings of "determine." I think Ross is making a similar, but more basic, mistake, as I will explain.
What does it mean for a set of facts (F) to determine another set of facts (G)? This is the fundamental issue of determinacy. In order to be clear about Ross's argument, we need to know what he thinks F is, what he thinks G is, and what he means by "determine." For Soames, the problem lies in the last of these. For Ross, it lies in the other two.
The first thing we have to get clear is that Ross is not talking about the indeterminateness of meaning, Feser's claims notwithstanding. If he were, he would have to discuss the meaning of "meaning", as Kripke (?) and Soames do. Also, because Kripke's argument leads to skepticism about whether humans ever mean anything, as well as machines, Ross would owe us an account of how he can avoid the skeptical conclusion about humans while affirming skepticism about machines. But he does none of this. Indeed, as one commenter noted at the beginning of the discussion, Ross never mentions "meaning" in the article. Furthermore, neither Ross nor the naturalist thinks an adding machine means anything when it performs an operation, so if meaning were the issue, the entire discussion about the adding machine would be beside the point.
So the short reply to Ross' use of Kripke is that Ross has divorced the quaddition argument from the Kripkean context. The result is that quaddition becomes simply another version of the problem of limited data. And we have already seen that the problem of limited data helps Ross not at all. If you are convinced of this point, you can skip the rest of this too-long post. What follows simply expands and explains this point.
Ross never discusses meaning - his discussion is entirely about whether the machine is executing a function, and whether the machine's future outputs are determinate. Let's look again at the way he begins his argument:
Whatever the discriminable features of a physical process may be, there will always be a pair of incompatible predicates, each as empirically adequate as the other, to name a function the exhibited data or process "satisfies." That condition holds for any finite actual "outputs," no matter how many. That is a feature of physical process itself, of change. There is nothing about a physical process, or any repetitions of it, to block it from being a case of incompossible* forms ("functions"), if it could be a case of any pure form at all. That is because the differentiating point, the point where the behavioral outputs diverge to manifest different functions, can lie beyond the actual, even if the actual should be infinite; e.g., it could lie in what the thing would have done, had things been otherwise in certain ways. For instance, if the function is x(*)y = (x + y, if y < 1040 years, = x + y + 1, otherwise), the differentiating output would lie beyond the conjectured life of the universe.And later on:
Secondly, opposed functions that are infinite (that is, are a "conversion" of an infinity of inputs into an infinity of outputs) can have finite sequences, as large as you like, of coincident outputs; they can even have subsequences that are infinitely long and not different (e.g., functions that operate "the same" on even numbers but differently on odd numbers). So for a machine process to be fully determinate, every output for a function would have to occur. For an infinite function, that is impossible. The machine cannot physically do everything it actually does and also do everything it might have done.And from Thought and World
If the machine is not really adding in the single case, no matter how many acutal outputs seem "right," there might eventually be nonsums.
I interpret Ross to be saying that what is not determined - his G - is what function the system is computing. Further, on the basis of the preceding quotes, I take it that he cashes out this G in terms of
a) what the system might output at a future time, and
b) what the system would have output for inputs it might have had, but did not have.
Let's consider a system with two inputs, x and y. Then the question is, "Is the output z determined for every possible x and y?"
Now we come to central question: determined by what? What is Ross's set F - the facts that (fail to) determine the output, z?
Since we are considering a purely physical system, a prime candidate for F would be the set
F1: All the physical facts about the system.
In this case, the question being asked becomes "Is the output z determined by the physical facts about the system, for all possible inputs x and y?" But this is nothing more nor less than the question of physical determinism. In a deterministic world, the set F1 certainly does determine the possible outputs, z, even for cases that the machine hasn't actually computed. Setting aside issues of quantum indeterminacy (which Ross never mentions), it seems that all outputs are determined by F1.
But F1 is not what Ross has in mind. He never attempts any discussion of physical determinism. Instead, he seems to have in mind something like
F2: The physical facts that are known about the system at some time T.
I take this from his talk of the "discriminable features of a physical process" in the first quote above and from his talk of "empirical adequacy," though I have to say that Ross is extremely vague about this.
If this is Ross's argument - if he is saying "What the system might have done, or will do, is not determined by the physical facts that we know about the system" - then we should simply reply, "So what?" As we saw already with the problem of limited data, there is no way to argue from an epistemological lack to a metaphysical conclusion.
There is another possibility: suppose that, instead of the physical facts that are known about the system, what Ross really means is
F3: All the physical facts that can be known about the system.
But now we have to be careful. What does is mean to say something "can be known"? Does this mean all the physical facts that can in principle be known about the system? Then F3 is the same as F1 - all the physical facts about the system can in principle be known (barring quantum uncertainty). In that case, there is no reason to think the outputs are undetermined. However, in actual fact we can never know all the physical facts about the system, no matter what set of observations we make. Thus, any given set of observations, no matter how detailed, is consistent with incompatible functions. Does this justify Ross's conclusion? No, because in that case, there is again only an epistemological lack.
Let me explain the last remark using the example of a computer. The computer seems at first to be a simple counterexample to Ross's claim that the outputs z are not determined by the physical features of the system. For I can look at the program the computer is running (it is encoded physically somewhere in the computer's memory) and see what the function is: I can deduce, for example, what the output would have been for some inputs x and y that the computer has not actually calculated. But (as Feser points out) a wire might burn out or a transistor go bad inside the computer, so that the actual output is not what the computer program would lead us to believe. This is true, but it doesn't really answer the objection. For suppose I insist on a more detailed physical description of the computer: one so detailed that the failure of the wire/transistor is predictable by this description and so is accounted for. Then we see that the indeterminacy was only apparent: the output is in fact determined by this more detailed set of facts. (In this case the computer would be executing something like quaddition, rather than addition.) If we dig deep enough, we will always find some set of physical facts that do determine the output.
We can now see why Ross is wrong to say that "for a machine process to be fully determinate, every output for a function would have to occur." A sufficiently detailed set of physical facts about the system determines not only what outputs will occur at a future time, but also what outputs would have occurred for other inputs that were not actually submitted to the machine.
I have gone into this at length because I think it shows both why Ross's argument is so appealing and why it is wrong. For any given set of physical facts about the system, there are infinitely many inequivalent functions compatible with the behavior of the system. But the set of possible observations is not fixed: if we find a particular set of physical facts leaves the outcome undetermined, we can always ask for a more detailed description of the system. For a sufficiently detailed set of physical facts, we find the outcome is determined by those facts. What makes the argument seem reasonable is the slide from a given set of facts, to any possible set of facts, to all physical facts.
- Ross's argument is not about whether meanings are determined by the physical facts about the system, but whether a functional form is determined.
- Whether a functional form is determined is cashed out in terms of whether future or counterfactual outputs are determined.
- Ross is unclear about what F it is that fails to determine the functional form.
- If we take F to be the set of all physical facts about the system, then all outputs are determined, and Ross's argument fails.
- If we take F to be the set of known physical facts, then outputs are indeed undetermined, but this is only an epistemological issue. For a sufficiently detailed set of physical facts, the output is determined.
- Thus, all three of Ross's main arguments: quaddition, grue, and the problem of limited data, only point to a lack of knowledge about the system.
- These epistemological concerns are not enough to draw the conclusion that the system is "physically and logically" indeterminate.
None of this addresses Professor Feser's point about the meaning of a physical process, which I will have to address (I hope!) another time.
* I take Ross to mean "inequivalent" rather than "incompossible" here and throughout - see Richard's remarks in the previous post.