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In addition to the many books attempting to explain them, a book has been written detailing all the ways people’s thinking is influenced (or seduced) by Kurt Gödel’s Incompleteness theorems. I wonder why people are not similarly enchanted by his positive result around the same time, the completeness of the predicate calculus. For the uninitiated, completeness (or incompleteness) is a property of a formal system in logic. A complete system is one in which every true statement is a theorem: that means any truth can be deduced from the system’s axioms using only circumscribed rules of inference. Alternately, a complete system is one in which either a statement A or its negation ~A is provable: not both, or neither. As Gödel’s first incompleteness theorem shows, any logical system powerful enough to account for the properties of arithmetic fails to be complete because one can generate a statement which is true but not provable by the methods of ordinary logic. Arithmetic (number theory, set theory) is not a complete theory.

But for ordinary logic by itself, all truths are theorems. Without going into the proofs of this theorem, which are various and variously complicated, I invite the reader to consider what consequences this result might have for our thinking in other areas. One aspect of completeness which strikes me is the way that logical truth, far-ranging and complex, is linked to a very economical manner of proceeding with proofs. In a way, completeness is like a codification of first-order logic rather than a vindication: and this might hold out hope that reductive treatments in other areas (like mental phenomena) might prove adequate, in a strong sense, to their “target” phenomena: perhaps the reductions describe a certain area of the field to be reduced completely, rather than accounting for all phenomena very well. Perhaps we should even re-examine our philosophy of science, such that we don’t look for “approximations” to the truth, but rather explanations that “redefine” the phenomena such that a new order is revealed. If we have an example of such a redefinition, it must certainly be possible to pursue this in various other fields.

Another facet has to do with the systematic treatment of truth and interpretation used in modern logic, model theory. Another way to state the completeness theorem is to say that every consistent set of sentences in first-order logic has a model: that is, for any statements that don’t contradict each other we can assign an orderly interpretation that explains how they come out true, explaining the semantic relationships between statements by analyzing them into their constituents. Perhaps this has consequences for the philosophy of language: perhaps completeness is an argument for the Davidsonian view of language, that there is no coherent significance without systematicity of the sort captured by a Tarskian theory of truth.

Can you think of any more “far-out” consequences of completeness?

In mainstream philosophy, serious interest in consciousness has been rekindled for about a decade now. However, I find this current discussion, judged by the standards of debates which have been going a little longer, slightly one-sided. These consciousness studies so far have focused on the phenomenal aspects of consciousness — what it’s like to see the color red and what makes this different from knowing all about the color red, for example. But if we go back in history to classic philosophers, a different way of approaching consciousness appears, quite different in its preoccupations. Kant and Hegel, the great German Idealists, each talked a great deal about consciousness. But their concerns are very different from simple reflection on the sensorium and what kind of neural apparatus implements it. For Kant and Hegel, consciousness is distinguished by its presenting special objects of thought: although we entertain thoughts about the external world “independent of consciousness”, consciousness itself is an important source of categories for thinking in general.

I think one could, without too much violence, call this the logicist approach to consciousness, by analogy with the position of the same name in the philosophy of mathematics. Mathematical logicism holds that we achieve mathematical truth through the discovery of abstract mathematical objects, not because empirical data has quantitative regularities we abstract from, or because the human mind reasons to mathematical truths along subjectively intuitive lines, or through the application of empty formal schemata. Kant and Hegel both believe that consciousness contains elements which we need to reason about the
world as a whole: these elements are conceptually, not just experientially, important, and form the basis of much of our cogitating — without them, we would lack not only feeling but also important “objective” aspects of thought. If we conider the virtues of this logicism I suspect we will find, that the position of the idealists represents a superior supercession of the mechanical imagery that dominates contemporary psychology: just because the mind is “in” the head doesn’t mean that what we find in consciousness isn’t implicated in our thinking about things outside our heads.

On the other hand, one might object that the idealists’ talk of “consciousness” is really a poorly-conceived mapping of thought in general lacking the modern concept of intentionality. But the idea that intentional states do not require consciousness, since we have beliefs we are rarely if ever conscious of and the like, is really only one of the options available to us, and most appealing if we do not think deeply. Kant and Hegel would find the eliminative materialist’s insistence that we can have the truths of science without the operations of the mind nonsensical; and their reasons for this, that conceptual activity is at every stage confirmed by conscious unity, are hard to escape. So perhaps there are reasons to follow them in this viewpoint; it certainly seems the case that a healthy respect for recent ideas can hardly require us to treat the detailed analyses of the Phenomenology of Spirit, arguing as they do that consciousness and self-consciousness are limited but complex, as mere dross.