Homage to Haskell B. Curry

Haskell B. Curry was a logician who lived more or less the same seven decades as Frederick B. Fitch. In fact, Curry and Fitch worked mostly on the same sorts of logic, the same philosophical issues, with more or less the same ideals to guide them. Since the main logic of their concern was combinatory logic, philosophers on the whole don’t know them well, though in computer science there is a technique still known as currying, named after him.

These are two of my heroes in logic, heroes who failed to reshape modern logic nearer to their hearts desire. While I knew Fitch, and even taught one semester in his place from a text he had written, I never knew Curry or even got to see him. But I return, more often now, to his conception of natural language, the language in which live, with increasing admiration. And lately I’ve gone back to something he is famous for among philosophers as well, Curry’s Paradox.

There is a popular version of this paradox, not Curry’s own, that displays the sentence:

(*) Either this sentence is not true, or else God exists.

If this sentence is not true then its first disjunct says what is the case, and so the sentence as a whole is true. So the sentence is true, which means that its first disjunct is not true, hence the second disjunct must be true: God exists. QED (It’s rather like Danny Daniels’ proof that God is not omniscient, that I talked about in my first post about Fitch.)

The drawback of this version is that there are already 1001 ways to counter paradoxes that rely on self-reference and negation. So, in this form, it is fun to play with, but doesn’t prove anything.

The original paradox involved neither self-reference nor negation. So none of the usual suspects provides any way out. And Curry used it to prove a tremendously important result (… well, I guess the result that God exists wouldn’t be so unimportant, if it worked …).

The young Curry and his contemporaries had an ideal for logic: to construct a carefully designed language in which we could reason and express ourselves just as we do in natural language, at least to the extent needed for science. Such a language could be taken as a model or representation of the natural language in which even the logican is working.

Curry laid down explicitly two minimal requirements for success:

1. In this representation, the elementary logic that we employ in natural language reasoning must be valid;
2. In this representation, we can name and talk about all the concepts that we can formulate and express in that language.

(The bit of elementary logic he specified is really minimal, it is taught in every first logic course in universities today). He called these the requirements of deductive completeness and combinatorial completeness.

In the formulation he gave to it, the paradox became a proof that this ideal is unreachable. Nothing we could possibly construct can meet both those requirements. There is an absolute limit to how much of our natural language that can be captured, modeled, represented, in a formal construction. Our language, the language we work in, cannot be grasped as a whole.

I seem to remember a Greek myth where a god is amazed by what is so different about mortals: “They are always reaching for what they cannot hold”. Curry was reaching for what he could not hold, but came to understand that he was.

In his later writings, taking issue with Tarski and other writers about the relation of logic to language,Curry reflected on the inevitable limits to representation that he had found.

Curry distinguished between what he called our language in use, and any language that can be constructed or adequately represented by such a construction. Our language in use is the language within which we construct our theories, fictions, models, w e are within it, so to speak, but its horizon is ever receding as we move.

I would sum up his view, in my own way, like this:

Our natural language is beyond the very possibility of an adequate representation, as a whole.

Curry was not an existentialist, and yet he could have gone one step further. We are the being endowed with language. It is in this language, impossible to capture, in which we live and move and have our being, it is what singles out our way of being, what we are. So what we are is beyond the very possibility of adequate representation.

Perhaps we are only spume upon the waves. Still we have this logically provable claim to transcendence ….

NOTE.

Like others have done, I have tried to construct languages that can satisfy Curry’s requirements to some interesting extent, in unexpected ways, though obviously not entirely. My current effort I am calling “Curry’s Paradox Peacefully Accommodated” (in progress) and it relies on forms of inference that cannot be captured in tautologies.