Partial functions and free logic

How much should a mathematical logician care about free logic? Worries about empty domains or empty names aren’t going to give the mathematician much pause. But there is a more interesting case.

The standard semantic story treats function expressions of a FOL language as denoting total functions — for any object of the domain as input, the function yields a value in the domain as output. Mathematically, however, we often work with partial functions: that’s particularly the case in computability theory, where the notion of a partial recursive function is pivotal. Partial recursive functions, recall, are defined by allowing the application of a minimization or least search operator, which is basically a definite description operator which may fail to return a value. So, it might well seem that in order to reason about computable functions we will need a logic which can accommodate partial functions and definite descriptions that fail to refer, and this means we will need a free logic.

Or at least, this is a claim often made by proponents of free logic. And the claim is vigorously pressed e.g. by Oliver and Smiley in Ch. 11 of their Plural Logic (as they set up the singular logic on which they are going to build their plural logic). Yet O&S give no examples at all of places where mathematical reasoners doing recursive function theory actually use arguments that need to be regimented by changing our standard logic. And if we turn to mainstream theoretical treatments of partial recursive functions in books on computability — including those by philosophically minded authors like Enderton, Epstein & Carnielli or Boolos &Jeffrey — we find not a word about needing to revise our standard logic and adopt a free logic. So what’s going on here?

I think we have to distinguish two quite different claims:

  1. Suppose we want to revise the usual first-order language of arithmetic to allow partial recursive functions, and then construct a formal theory in which we can e.g. do computations of the values of the partial recursive functions (when they have one) in the way we can do simpler formal computations as derivations inside \mathsf{PA} (or inside \mathsf{PRA}, formal Primitive Recursive Arithmetic). Then this formal theory with its partial functions will need to be equipped with a free logic to allow for reference failures.
  2. When, it comes to proving general results about partial recursive functions in our usual informal mathematical style, we need to deploy reasoning which presumes a free logic.

Now, (1) may be true. But mathematicians in fact seem to have very little interest in that formalization project (though some computer scientists have written around the topic, though what I have read has been pretty unclear). What they care about is the general theory of computability.

And there seems no good reason for supposing (2) is true. Work through a mathematical text on the general theory of computability, and you’ll see that some care is taken to handle cases where a function has no output. For example, we introduce the notation f(x){\downarrow} to indicate that f indeed has an output for input x; and we introduce the notation f(x) \approx g(x) to indicate that either (i) both f(x){\downarrow} and g(x){\downarrow} and f(x) = g(x) or (ii) neither f(x) nor g(x) is defined. And then our theorems are framed using this sort of notation to ensure that the mathematical propositions which are stated and proved or disproved are straightforwardly true or false (and aren’t threatened with e.g. truth-valueness because of possibly empty terms). In sum, reflection on the arguments actually deployed by Enderton etc. suggests that the silence of those authors on the question of revising our logic is in fact entirely appropriate. Theorists of computability don’t need a free logic.

Big Red Logic Books: end-of-year report

As I’ve put it before, self-publishing seemed exactly appropriate for the Big Red Logic Books. They are aimed at students, so why not make them available as widely as can be? — free to download as PDFs, for those happy to work from their screens, and at minimal cost for the significant number who prefer to work from a physical copy.

So how did things go over 2021? The headline stats are these:

PDF downloadsPaperback sales
Intro Formal Logic10270905
Intro Gödel’s Theorems6529757
Gödel Without Tears2482831

The absolute download stats are very difficult to interpret, because if you open a PDF in your browser on different days, I assume that this counts as a new download — and I can’t begin to guess the typical number of downloads per individual reader. But the relative month-by-month figures will more significant: and for IFL and IGT these remain very stable, while those for GWT have increased quite a bit over the year. As for paperback sales, month-by-month, these remain very steady, and the figures are very acceptable. Modified rapture, then!

(Aside: There is hardback version of IFL and GWT available for libraries, and I’ve been paid for some sales to distributors. But how many hardbacks have actually been sold to real buyers I don’t know — only a few dozen, probably. I rather doubt that I will again go through the palaver of arranging hardback versions of any future books.)

I don’t know what general morals can be drawn from my experiences with these three books. As I’ve also put it before, every book is what it is and not another book, and every author’s situation is what it is. But open-access PDF plus very inexpensive but reasonably well produced paperback is obviously a fairly ideal model for getting stuff out there. I’d be delighted if more people followed the model. But I suppose you can only do this if you no longer need the reputational brownie points of publication by a university press (and if you have a good enough eye to use LaTeX or whatever to produce decent typography!). Maybe it is a model for the idle retired among us, who want to finish that book they’ve being meaning to write …

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