The following analysis shows how developments in epistemic logic can play a non-trivial role in cognitive neuroscience. These obtain a striking correspondence between two modes of identification, as distinguished in the epistemic context, and two cognitive systems distinguished by neuroscientific investigation of the visual system (the “where” and “what” systems). It is argued that this correspondence is not coincidental, and that it can play a clarificatory role at the actual working levels of neuroscientific theory.

Introduction

While most work in neuroscience is conducted at the cellular and sub-cellular level, brain research that catches the eye of philosophers is likely to come from a relatively recent interdisciplinary hybrid known as “cognitive neuroscience.” Explanations from cognitive neuroscience are of interest to philosophers since they offer the possibility of connecting brain and behavior through the specification of the information processing properties of parts and processes of the brain. However, despite the prominence of the information-processing approach in the brain and behavioral sciences, it is difficult to know exactly what cognitive neuroscientists mean by “information.” Historically, contexts in which this term has been given a precise definition include the so-called mathematical theory of communication, the theory of semantic information of Carnap and Bar-Hillel, and later the theories of informational complexity associated with Kolmogorov and Solomonoff. Most uses of the term “information” by cognitive scientists and neuroscientists conform to none of these three contexts.

Philosophers frequently complain of a lack of precision in scientific uses of the notion of information.

If Thomas Kuhn had not sworn to me a long time ago that he would never again use the p-word, I would have been tempted to introduce my viewpoint in this volume by saying that contemporary epistemology draws its inspiration from an incorrect paradigm that I am trying to overthrow. Or, since the individuation of paradigms is notoriously difficult, I might have said instead that our present-day theory of knowledge rests on a number of misguided and misguiding paradigms. One of them is in any case a defensive stance concerning the task of epistemology. This stance used to be expressed by speaking of contexts of discovery and contexts of justification. The former were thought of as being inaccessible to rational epistemological and logical analysis. For no rules can be given for genuine discoveries, it was alleged. Only contexts of justification can be subjects of epistemological theorizing. There cannot be any logic of discovery, as the sometime slogan epitomized this stance—or is it a paradigm? Admittedly, in the last few decades, sundry “friends of discovery” have cropped up in different parts of epistemology. (See, for example, Kleiner 1993.) However, the overwhelming bulk of serious systematic theorizing in epistemology pertains to the justification of the information we already have, not to the discovery of new knowledge. The recent theories of “belief revision”—that is, of how to modify our beliefs in view of new evidence—do not change this situation essentially, for they do not take into account how that new evidence has been obtained, nor do they tell us how still further evidence could be obtained.

In discussions of the ethics of science, the practice of omitting data, also referred to often as “data selection,” has played a significant role as an interesting test case of real or alleged scientific fraud. Babbage's classic taxonomy of scientific frauds distinguishes three kinds of such fraud—to wit, “forging,” “cooking,” and “trimming” of data. The meaning of these terms is obvious, with omitting data as a clear-cut case of cooking. In the literature dealing with dishonesty in science, several prominent scientists have been accused of omitting data, among them no lesser a figure than Isaac Newton, who has been charged with maintaining the impossibility of an achromatic lens while in possession of evidence suggesting the possibility of such a lens. (See Bechler 1975; Kohn 1988, 36–39.) The most thoroughly analyzed case is undoubtedly Robert A. Millikan's famous oil drop experiment, which helped him to earn a Nobel Prize. (See Franklin 1981 and 1986, ch. 5 and 229–232; Holton 1978; Broad and Wade 1982, 34–36.) As is well known, Millikan's experiments aimed principally at measuring the electric charge, e. They “depended on introducing droplets of liquid into the electric field and noting the strength of the field necessary to keep them suspended.” (Broad and Wade 1982, 34.) Contrary to his public pronouncements, Millikan excluded as many as 49 observations out of a total of 140.

One of the major current developments in cognitive psychology is what is usually referred to as the “theory of cognitive fallacies,” originated by Amos Tversky and Daniel Kahneman. The purported repercussions of their theory extend beyond psychology, however. A flavor of how seriously the fad of cognitive fallacies has been taken is perhaps conveyed by a quote from Piatelli-Palmerini (1994, xiii), who predicted “that sooner or later, Amos Tversky and Daniel Kahneman will win the Nobel Prize for economics.” His prediction was fulfilled in 2002.

The theory of cognitive fallacies is not merely a matter of bare facts of psychology. The phenomena (certain kinds of spontaneous cognitive judgments) that are the evidential basis of the theory derive their theoretical interests mainly from the fact that they are interpreted as representing fallacious—that is, irrational judgments on the part of the subject in question. Such an interpretation presupposes that we can independently establish what it means for a probability judgment to be rational. In the case of typical cognitive fallacies studied in the recent literature, this rationality is supposed to have been established by our usual probability calculus in its Bayesian use.

The fame of the cognitive fallacies notwithstanding, I will show in this chapter that at least one of them has been misdiagnosed by the theorists of cognitive fallacies. In reality, there need not be anything fallacious or otherwise irrational about the judgments that are supposed to exhibit this “fallacy.”

The Unreasonable Effectiveness of the a priori in Epistemology

Aristotle said that philosophy begins with the experience of wonder. But different phenomena are experienced as wondrous by different thinkers and to a different degree. The wonder that is the theme of this chapter seems to have struck some non-philosophers more keenly than most professional philosophers. Indeed, the most vivid formulation of the problem is probably in the title of Eugene Wigner's 1960 paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Historically speaking, Wigner's amazement is nevertheless little more than another form of the same reaction to the success of mathematics in science as early modern scientists' sense that the “book of the universe is written in mathematical symbols.” For a philosopher, this question is in any case a special case of the general problem of the role of our a priori knowledge, mostly codified in the truths of logic and mathematics, in the structure of our empirical knowledge.

This overall philosophical problem assumes different forms in different contexts. Wigner's puzzle has been taken up in the same form by relatively few philosophers, most extensively by Mark Steiner in his 1998 book, The Applicability of Mathematics as a Philosophical Problem. This problem can be given technical turns—for instance, by asking whether scientific laws can be expected to be computable (recursive), and even more specifically by asking which known laws are in fact computable. (See, e.g., Pour-El and Richards 1989.)

Knowledge and Decision-Making

Epistemology seems to enjoy an unexpectedly glamorous reputation in these days. A few years ago, William Safire wrote a popular novel called The Sleeper Spy. It depicts a distinctly post-Cold War world in which it is no longer easy to tell the good guys—including the good spies—from the bad ones. To emphasize this sea change, Safire tells us that his Russian protagonist has not been trained in the military or in the police, as he would have been in the old days, but as an epistemologist.

But is this with-it image deserved? Would the theory of knowledge that contemporary academic epistemologists cultivate be of any help to a sleeper spy? This question prompts a critical survey of the state of the art or, rather, the state of the theory of knowledge. I submit that the up-to-date image is not accurate and that most of the current epistemological literature deals with unproductive and antiquated questions. This failure is reflected in the concepts that are employed by contemporary epistemologists.

What are those concepts? It is usually thought and said that the most central concepts of epistemology are knowledge and belief. The prominence of these two notions is reflected in the existing literature on epistemology. A large chunk of it consists in discussions of how the concept of knowledge is to be defined or is not to be defined. Are those discussions on the target?

Right back in Chapter 2 we stated Turing's Thesis: a numerical (total) function is effectively computable by some algorithmic routine if and only if it is computable by a Turing machine. Of course, we initially gave almost no explanation of the Thesis. It was only very much later, in Chapter 31, that we developed the idea of a Turing machine and saw the roots of Turing's Thesis in his general analysis of the fundamental constituents of any computation.

Meanwhile, in Chapter 29, we had already introduced the idea of a µ-recursive function and noted the initial plausibility of Church's Thesis: a numerical (total) function is effectively computable by an algorithmic routine if and only if it is µ-recursive.

Then finally, in Chapter 32, we outlined the proof that a total function is Turing computable if and only if it is µ-recursive. Our two Theses are therefore equivalent.

Given that equivalence, we can now talk of

Crucially, this Thesis links what would otherwise be merely technical results about µ-recursiveness/Turing computability with intuitive claims about effective computability; and similarly it links claims about recursive decidability with intuitive claims about effective decidability. For example: it is a technical result that PA is not a recursively decidable theory. But what makes that theorem really significant is that – via the Thesis – we can conclude that there is no intuitively effective procedure for deciding what's a PA theorem.

Comparing incompleteness arguments

Our informal incompleteness theorems, Theorems 5.7 and 6.2, aren't the same as Gödel's own theorems. But they are cousins, and they seem quite terrific results to arrive at so very quickly.

Or are they? Everything depends, for a start, on whether the ideas of a ‘sufficiently expressive’ arithmetic language and a ‘sufficiently strong’ theory of arithmetic are in good order.

Now, as we've already briefly indicated in Section 2.2, there are a number of standard, well-understood, ways of formally refining the intuitive notion of decidability, ways that turn out to locate the same entirely definite and welldefined class of numerical properties and relations. In fact, these are the properties/relations whose application can be decided by a Turing machine. The specification ‘all decidable relations/properties of numbers’ can therefore be made perfectly clear. And hence the ideas of a ‘sufficiently expressive’ language (which expresses all decidable two-place numerical relations) and a ‘sufficiently strong’ theory (which captures all decidable properties of numbers) can also be made perfectly clear.

But by itself, that observation doesn't take us very far. For it leaves wide open the possibility that a language expressing all decidable relations or a theory that captures all decidable properties has to be very rich indeed. However, we announced right back in Section 1.2 that Gödel's own arguments rule out complete theories even of the truths of basic arithmetic.

This Interlude discusses a couple of further questions about incompleteness that might well have occurred to you as you have been reading through recent chapters. But, given that the chapters since the last Interlude have been quite densely packed, we should probably begin with a quick review of where we've been.

Taking stock

Here's some headline news which is worth highlighting again:

1. First, we showed that the restriction of the First Theorem to p.r. axiomatized theories is in fact no real restriction. Appealing to a version of Craig's Theorem, we saw that the incompleteness result applies equally to any consistent axiomatized theory which contains Q (or, indeed, is otherwise p.r. adequate). (Section 19.1)

2. But our pivotal new result was Theorem 20.4, the general Diagonalization Lemma: if T is a nice theory and ϕ(x) is any wff of its language with one free variable, then there is a ‘fixed point’ γ such that T ⊢ γ ↔ ϕ(⌜γ⌝). And further, if ϕ(x) is Π1, then it has a Π1 fixed point. (Section 20.5)

3. We then proved the rather easy Theorem 21.1: if γ is any fixed point for ¬ProvT(x), then, if T is nice, T ⊬ γ, and if T is also ω-consistent, then T ⊬ ¬γ. Since the Diagonalization Lemma tells us that there is a fixed point for ¬ProvT(x), that gives us the standard incompleteness theorem again.

4. […]

In this chapter, we use the really rather beautiful Diagonalization Lemma a number of times over. Here's a quick guide through the sections:

1. First, it is worth seeing how we can use the Lemma to prove Gödel's First Theorem again. We can think of the Theorem as in fact generated by putting together two separate strands of thought – (i) the general Diagonalization Lemma which tells us that the wff ¬Prov(x) in particular has a fixed point, plus (ii) reflections on the logical behaviour of Prov(x).

2. After a brief aside on the very idea of a Gödel sentence …

3. … we then use the Diagonalization Lemma again to derive the Gödel-Rosser Theorem which previously we left unproved.

4. Next, we show that no nice theory T can capture its own provability property ProvT.

5. Then we prove Tarski's key Theorem about the ‘indefinability’ of truth.

6. We note that this gives us what in a sense might be thought of as the master argument for incompleteness.

7. Finally, as a coda, we show some results that concern how long the proofs have to be.

This chapter introduces the notion of a µ-recursive function – which is a very natural extension of the idea of a primitive recursive function. Plausibly, the effectively computable functions are exactly the µ-recursive functions (and likewise, the effectively decidable properties are those with µ-recursive characteristic functions).

Minimization and µ-recursive functions

The primitive recursive functions are the functions which can be defined using composition and primitive recursion, starting from the successor, zero, and identity functions. These functions are computable. But they are not the only computable functions defined over the natural numbers (see Section 11.5 for the neat diagonal argument which proves the point). So the natural question to ask is: what other ways of defining new functions from old can we throw into the mix in order to get a broader class of computable numerical functions (hopefully, to get all of them)?

As explained in Section 11.4, p.r. functions can be calculated using bounded loops (as we enter each ‘for’ loop, we state in advance how many iterations are required). But as Section 3.6 already reminds us, we also count unbounded search procedures – implemented by ‘do until’ loops – as computational. So, the obvious first way of extending the class of p.r. functions is to allow functions to be defined by means of some sort of ‘do until’ procedure. We'll explain how to do this in four steps.

(a) Here's a simple example of a ‘do until’ loop in action. Suppose that G is a decidable numerical relation.

Let's finish by taking stock one last time. At the end of the last Interlude, we gave a road-map for the final part of the book. So we won't repeat the gist of that detailed local guide to recent chapters; instead, we'll stand further back and give a global overview. And let's concentrate on the relationship between our various proofs of incompleteness. Think of the book, then, as falling into three main parts:

(a) The first part (Chapters 1 to 7), after explaining various key concepts, proves two surprisingly easy incompleteness theorems. Theorem 5.7 tells us that if T is a sound axiomatized theory whose language is sufficiently expressive, then T can't be negation complete. And Theorem 6.2 tells us that we can weaken the soundness condition and require only consistency if we strengthen the other condition (from one about what T can express to one about what it can prove): if T is a consistent axiomatized theory which is sufficiently strong, then T again can't be negation complete.

Here the ideas of being sufficiently expressive/sufficiently strong are defined in terms of expressing/capturing enough effectively decidable numerical properties or relations. So the arguments for our two initial incompleteness theorems depend on a number of natural assumptions about the intuitive idea of effective decidability. And the interest of those theorems depends on the assumption that being sufficiently expressive/sufficiently strong is a plausible desideratum on formalized arithmetics.