Reducing logic

Cooper attempts to derive logic, and ultimately mathematics from evolutionary theory. He has a "ladder of reducibility" (p20):



He notes

The term 'implies' beside the upward arrows means theory-implies. It is intended in the sense of the Nagel model or one of its generalizations, in which one theory implies another if the terms of the second can be defined within the first and its propositions derived from the first. [p21]

The interesting point here is not that Cooper thinks that logic can be derived using Nagel reducibility from evolutionary theory, for if we evolved then it follows that a proper description of evolution will necessarily include a description of our evolved logic. It is, rather, that he thinks logic derives from evolutionary theory rather than from evolution. And only from an aspect of evolutionary theory at that - the aspect of life history strategy.

Why is this problematic? I have a number of concerns - for a start, there are no strategies in evolution itself, but only in the ways in which we describe life histories. As Michael Ghiselin once said, "We must not take the term "strategy" too literally. The only real strategists here are evolutionary theorists. Carrots do not ponder differential reproductive success." (The economy of Nature and the evolution of sex, University of California Press, 1974, p41)

So why the interest to derive logic from evolutionary theory, rather than from evolution, if the latter move is blocked by the divide of abstraction from reality? If logic is just a special case of evolutionary theory, then all we have done is provide a system E from which we can derive the system L. Surely the interesting thing would be to define logic as an outcome of evolution itself? And that would get us back to Wigner's and Hamming's claims about maths - that we evolved it to suit our cognitive needs.

More interesting to me, anyway, would be showing how logic derives from our evolutionary history and nature as primates, hominids and so forth, and evolutionary psychologists do attempt this. But that account founders a little on a lack of data. If you are going to make an etiological account, you have to be able to decide from the multitude of ways we might have evolved.

If Cooper is right about the formal reduction relations between each of these domains and logics, then it follows that the "implies" relation means that the next higher level is a subset of the one below it. For the formal outcomes of the next higher, or superordinate, level are contained fully in the one below (subordinate). So this table becomes a Venn diagram:



[Side comment - Venn did not invent these, but he introduced the shading of intersections (which we now do not really use). Euler is the originator of Venn diagrams, at least according to Kneale and Kneale.]

For all the interest this account has, it boils down to this: we can take a particular restricted population-genetics model, and from it we can extract various kinds of logics - decision theory (a kind of game theory), inductive logic, and deductive logic. Is this the same as saying that logic evolved?

No, it is not. For we may be able to find an extensive class of models, some of them unrelated to evolution, from which logic can be derived. At best, Cooper has shown that logic is consistent with evolutionary dynamics, and that we may have abstracted logic out of our evolutionary understanding of that dynamics.

But we did not evolve to understand evolution or evolutionary theory. We evolved to understand the necessities of life - mating, food gathering, disease and predator avoidance. And there is no requirement that the mechanisms of our cognitive systems are isomorphic with the actual evolutionary dynamics that created them. For example, we are born essentialists (see Susan Gelman's The Essential Child for a review of the evidence for this), but nature, so far as we can tell from scientific investigation, is rarely essentialistic. Yet essentialism is an adaptive mindset - it allows us to do things without worrying about the tails of the distributions of properties. We get more than a few false positives and negatives, sure (that is what statistics are all about), but that is a small price to pay for being able to immediately size up a situation.

Here's a possibility: It is not that we evolved as organisms that allows us to abstract evolutionary dynamics to formal logic and maths, but that we evolved as scientists - in short, the much more selectionist cultural evolution of science has a population dynamic similar to the "life-history strategy logic" that Cooper bases all this on. Maths is just an abstract version of the way we think when we do science.

This is more plausible to me - Aristotle, from whom much formal logic flows, was attempting to do science in the broad sense, when he worked through the implications of his "naive intuitions" in the Posterior Analytics. He generalised (something humans did evolve to do well) from the cultural movement of rational thinking he was heir to, including the Pythagorean mathematical mystery religion, to a context-free system of rules. And this matches very well Cooper's account.

Next, we shall look at Cooper's strategy tree approach. It's really rather elegant. But that will complete our look at the details of the book. After that I shall ruminate on reason and logic and evolution. But no chocolate...

References:

Gelman, Susan A. 2003. The essential child: origins of essentialism in everyday thought, Oxford series in cognitive development. Oxford; New York: Oxford University Press.

Kneale, William Calvert, and Martha Kneale. 1962. The development of logic. Oxford UK: Clarendon Press.