The Universe of Discourse

Gantō's axe does have computational content

A while back I was thinking about this theorem of intuitionistic logic:

$$((P\to Q)\land ((P\to \bot) \to Q)) \to {(Q\to\bot)\to \bot} \tag{$\color{darkred}{\heartsuit}$}$$

Since it's intuitionistically valid, its proof can be converted into a program with the corresponding type. But the function that you get seemed to be computationally vacuous, since it could only be called with arguments that were impossible to obtain, or else with arguments that render the whole thing silly. For the former, you can make it work in general if you have at hand a function of type !!(P\to\bot)\to Q!! — but how could you? And for the latter, you can make it work if !!Q=\mathtt{int}!!, in which case you don't really need the program at all since !!(Q\to\bot)\to\bot!! is trivially true.

Several people replied to me about this, but the most interesting response I got was from Simon Tatham, who observed that !!\color{darkred}{\heartsuit}!! is still intuitionistically valid if you replace !!\bot!! with arbitrary !!R!!:

$$((P\to Q)\land ((P\to R) \to Q)) \to {(Q\to R)\to R} \tag{$\color{purple}{\spadesuit}$}$$

The proof is essentially the same:

1. Suppose you have: !!f : P\to Q!! and !!g:(P\to R) \to Q!!
2. then if you also have !!h: Q\to R!!, you can compose it with !!f!! to get !!h\circ f : P\to R!!
3. which you can feed to !!g!! to get !!g(h\circ f) : Q!!
4. and then feed that to !!h!! to get !!h(g(h\circ f)) : R!!.

But unlike !!\color{darkred}{\heartsuit}!!, this is computationally interesting.

M. Tatham gives this example: Take !!R!! to be !!\mathtt{bool}!!. We can understand a function !!X \to R!! as a subset of !!X!!, represented by its characteristic function. Say that !!Q!! is the type of subsets of !!P!!, perhaps in some representation that is more concrete than characteristic functions. If you'd like a more specific example, take !!P!! to be the natural numbers, and !!Q!! to be (recursive) sets of natural numbers as represented by (possibly infinite) lists in strictly increasing order.

Then:

• !!f!! is some arbitrary function that assigns, to each element of !!P!!, a related subset of !!P!!. Say for example that !!f(p)!! is the semispatulated closure of !!p!!.

• !!g!! could be the isomorphism that converts a subset of !!P!!, represented as a characteristic function, into one represented as a !!Q!!.

• !!h : Q\to \mathtt{bool}!! is a single subset of !!Q!!, represented as a characteristic function on !!Q!!. Say, !!h(q)!! is true if and only the set !!q\subset P!! has the Cosell property.

With these interpretations, !!(h\circ f)(p) : P\to \mathtt{bool}!! tells you whether !!p!!'s semispatulated closure has the Cosell property, and !!g(h\circ f)!! is the element of !!Q!! that represents, !!Q!!-style, the set of all such !!p!!.

Then !!h(g(h\circ f)) : R!! computes whether this set itself also has the Cosell property: true if so, false if not.

This is certainly nontrivial, and if the made-up properties were replaced by real ones, it could even be interesting.

One could also replace !!R=\mathtt{bool}!! with some other set, and then instead of a characteristic function, !!X\to R!! would be some sort of valuation function.

I had asked:

Is the whole thing just trivial because there is no interesting way to instantiate data objects with the right types? Or is there some real computational content here?

I think M. Tatham's example shows that there is, and that the apparent vacuity of the theorem !!\color{darkred}{\heartsuit}!! arose only because !!R!! had been replaced with the empty set.

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