Tuesday, July 27, 2004

Naturalism and the Status of Logical Laws

In a recent article, Victor Reppert writes, “If one accepts the laws of logic, as one must if one claims to have inferred one belief from another belief, then one must accept some nonphysical,nonspatial, and nontemporal reality, at least something along the lines of the Platonic forms.” (Philosphia Christi, 5:1 (2003), 19) This is part of Reppert’s case against naturalism since naturalism, as Reppert understands it, cannot countenance any nonspatial and nontemporal realities.

Although I have some objections to some of Reppert’s arguments, which I shall inflict upon him in due course, I am basically on his anti-naturalist side and propose to defend him against some of the objections raised by Richard Carrier in the latter’s lengthy critique of Reppert’s work. Carrier’s animadversions are in boldface. My comments are signalled by ‘BV.’ ‘[SNIP]’ indicates that I have omitted a portion of Carrier’s text.

For logical laws are just like physical laws, because physical laws describe the way the universe works, and logical laws describe the way reason works--or, to avoid the appearance of begging the question, logical laws describe the way a truth-finding machine works, in the very same way the laws of aerodynamics describe the way a flying machine works, or the laws of ballistics describe the way guns shoot their targets. The only difference between logical laws and physical laws is the fact that physical laws describe physics and logical laws describe logic. But that is a difference both trivial and obvious.

BV: There are a couple of reasons for thinking that the difference between physical and logical laws are not “trivial and obvious.” The first is that the laws of logic are necessarily true, whereas the laws of physics are not. The Law of Non-Contradiction (LNC), for example, does not merely happen to be true. For suppose ‘~(p & ~p)’ is contingently true. Then its negation, ‘(p & ~p),’ is possibly true, which is absurd. The negation of a law of nature, however, does not issue in a logical contradiction. Laws of nature are logically contingent.

The second reason for thinking that the difference between physical and logical laws is not “trivial and obvious” is that the laws of logic are prescriptive rather than descriptive: they prescribe how we ought to think if we want to arrive at truth in our reasoning. They do not “describe the way reason works” since reason often malfunctions. Carrier assimilates logic to psychology, thereby falling into the error of psychologism. At the very least, this is an issue that must be seriously addressed.

Reppert thinks there are other strange differences, but he doesn't think them through. For example, he says logical laws are unlike physical laws in that the former "pertain across possible worlds, including worlds with no physical objects whatsoever" (81; cf. 94). But that is again just the same trivial difference, that physical laws, not logical laws, describe physics.

BV: Carrier misses the point entirely. The laws of physics that hold in our world, do not hold in every logically possible world. But the laws of logic do hold in every logically possible world.
That’s the point. Perhaps Carrier can argue against it; but first he must see it. By the way, it is sloppy to say that physical laws describe physics. That’s just false taken literally. Physical laws describe nature or physical reality. Physics is the study of nature; as such, it is distinct from nature.

The reason logical laws pertain to "possible worlds" is the very fact that the sphere of possibility entailed by such a phrase is the sphere of imagination capable of being explored by the virtual model computations of our brain (the assembly and reassembly of the elements of experience),

BV: Not so. The sphere of logical possibility is not coextensive with the sphere of imagination. There are objects that are logically possible that cannot be imagined, Descartes’ chiliagon (one thousand-sided plane figure) for example.

For a world (or system or object within a world) to be "possible" is literally to be capable of simulation in our brain (or in any adjunct to our brain that extends its mental power--like a computer, hypothetical or real).

BV: This is false since impossible states of affairs can be simulated. Surely I can think impossible situations which I see to be impossible, as well as impossible situations which appear possible but which analysis shows to be impossible. What is possible and what is not cannot be tied to the representational powers of some such system as the human brain. Think about it. Could the possibilities and impossibilities pertaining to the brain itself depend on the representational powers of the brain? It is obvious that the possibility that human brains come into existence is prior to and independent of the representation powers of human brains. The world does not get its modal structure from the meat between our ears.

Since reason is a function performed on computed data, especially but not only that category of computed data defined by virtual models, obviously the rules of reason will by definition describe what is applicable to everything that can potentially be computed. It is no accident that Gödel's Theorem connected with work in computation: he proved that the realm of the possible excluded certain things precisely because those things could never be computed. The limits of computation literally are the limits of logic, because computation itself literally is logic (or "a" logic--as there are many types of computation, there are many logics).

BV: This is deeply confused. One of the things that Goedel showed is that there are undecidable propositions of arithmetic: propositions such that neither they nor their negations can be derived from the Principia Mathematica axiom set. Now if P is undecidable relative to that axiom set, then there is a sense in which P is not “computable.” But that is not to say that P is outside the “realm of the possible.” P is a truth of arithmetic. As such it is possibly true, and thus well within the realm of the possible.

It is false that “The limits of computation literally are the limits of logic” since P is logically possible but not computable. It is also false that “computation itself is literally logic....” Computation is a process while logic is a set of truths. Computation exists contingently; the truths of logic exist necessarily. Computation is going on in the CPU in front of me now, but not in other places such as my left foot. The truths of logic are not locatable. The move from “many types of computation” to “many types of logic” is a breath-taking non sequitur in the absence of a great deal of careful argumentative support.

Missing all this,

BV: Reppert “missed” all this? All what? He missed a farrago of nonsense, and did well to miss it.

Reppert goes on to declare that "if one accepts the laws of logic, as one must if one claims to have rationally inferred one belief from another" (emphasis added) "then one must accept some nonphysical, nonspatial and nontemporal reality" rather like Plato suggested (81). Note how close Reppert is to getting it--but just when you think he has it, he puts the cart before the horse, then observes that the whole caboose won't go, and from that concludes it can't go, without some wizard to cast a spell to levitate the cart so it can drag the horse along where it needs to go. If you think that is a silly way to respond to an inverted horse-and-cart, then you will agree Reppert's approach to logical laws is silly, too. The reason one must accept the laws of logic to rationally infer anything is the very same reason one must accept the laws of aerodynamics to fly. Surely Reppert would not conclude that we need some sort of supernatural powers and beings to explain why we need to follow the laws of aerodynamics to fly. The reason we need them is that it is physically impossible to fly any other way, and the only way flight is physically possible is exactly the way described by those laws. All you need for that to be true is a physical universe that is a certain way. Of Plato's hypothesis we have no need.

BV: Carrier is missing a very important difference between physical laws and logical laws, namely, that one can violate logical, but not physical, laws. There are logical inferences and illogical inferences. But there is no corresponding distinction between physical motions and unphysical motions. Every motion is physical, but not every inferring is logical. One cannot fly except in a way that satisfies the laws of aerodynamics, but one can reason in a way that fails to satisfy the laws of logic. Carrier ignores the NORMATIVITY of logic.
Then we came to discover how to precisely describe the operation of such a computer, when we discovered the laws of ballistics themselves (by observing and testing and so on), and defined them (using languages, like mathematics, specially adapted to such purposes). Later, with this knowledge in hand, we were able to build nearly flawless ballistics computers superior to our own. But even without that technology we could use our own general-purpose computer--the cerebral cortex--to run a nearly flawless ballistics computation by running a program called "mathematics."

BV: So mathematics is a program that the cerebral cortex runs. But this is bizarre. A program is a set of instructions that tells a computer to execute certain tasks, e.g. go to memory location A, take the quantity stored there, raise it to the second power, add it to the quantity stored in memory location B, then deposit the result of the computation in memory location C. This all presupposes the truths of mathematics. I see no clear sense in which the infinity of mathematical truths (many of them undiscovered) can be described as a program.

[SNIP: chunk of text deleted]]
He [Aristotle] goes on to explain that words have definite meanings assigned by human convention, and for that very reason words cannot also mean what they by definition deny (ibid.1006a-1007a). Thus, for Aristotle, logical laws derive necessarily and automatically from the existence of communication (defining terms and reasoning with others) and computation (reasoning with oneself). The moment you have those, in any possible universe, you will always have logical laws. It can never be any other way. This is exactly what I argue above and elsewhere. And since one does not need anything more than physics to have communication and computation, you do not need anything more to have logical laws.

BV: The argument is this:
1. Logical laws derive from communication and computation.
2. Physics (i.e. the physical world) is all that is needed for communication and computation.
3. Physics (the physical world) is all that is needed for logical laws.
I see no reason at all to accept premise (1). First of all, it is not clear what ‘derive’ means here. Does it mean that logical laws are empirical generalizations from the way people reason as a matter of fact? But if LNC were an empirical generalization, it might be falsfied by future experience – which is absurd. Logical laws do not derive from communication and reasoning: they are imposed on them. They are criteria of what is to count as true, and what is to count as real; as such, they cannot be said to derive from existing physical facts.

Extending the point to physical reality, Aristotle argues: Again, if all contradictory predications of the same subject at the same time are true, clearly all things will be one. For if it is equally possible either to affirm or deny anything of anything, the same thing will be a trireme and a wall and a man, which is what necessarily follows for those who hold the theory of Protagoras. For if anyone thinks that a man is not a trireme, he is clearly not a trireme [i.e. in their conception], but he also is a trireme if the contradictory statement is true. So the result is the dictum of Anaxagoras, "all things are mixed together," so that nothing truly exists. (Metaphysics 1007b) Aristotle goes on to explain that a posit asserts that something exists, while a negation asserts that it does not, so that to assert both is to declare, literally, nothing (ibid. 1007b-1008a). That is, a self-contradiction communicates nothing, and represents nothing even in the mind of one who wishes to declare it. Thus, it cannot correspond to anything real except the null set.

BV: This is sloppy. It confuses a self-contradictory utterance with a meaningless one. ‘I am both sitting and not sitting now’ is self-contradictory but not meaningless. If it were meaningless, it could not have a truth-value. Since every contradiction is false, every contradiction is meaningful.
As Reppert himself says, "Part of what it means to say anything is to imply that the contradictory is false" (82). Indeed, that is too wishy-washy washy: the fact is that what it means to say anything is literally at the same time to say (not imply, but assert) that the contradictory is false.

BV: This is false. Suppose a child asserts that all his toy fire trucks are red. He is not thereby asserting that it is not the case that some of his toy fire trucks are not red. In general, if one asserts that p, and p implies q, it does not follow that one asserts that q. Reppert had it right.