Co-partial Functions: a Step Too Far?

Mathematicians interested in computability theory have to deal with partial functions. Those interested in the theory of complex variables talk of multi-valued functions. Even in elementary arithmetic we use functional expressions which take plurals or lists: we talk, e.g., of ‘the sum of the first ten numbers’ and ‘the sum of 4, 7, 11, 19’. A logic that is apt for regimenting the reasoning of mathematicians without undue distortion will therefore have to go beyond our standard first-order logic which treats all functions as total, single-valued, and mapping a determinate number of things to some value. We’ll need to acknowledge partial functions by allowing functional expressions with no referent, and this will require adopting a free logic. Further, the logic will have to be expanded to deal with plural terms, to cope with the outputs of many-valued functions, and with the inputs of even the humdrum ‘sum’ function. Or so the revisionist story goes – see Oliver and Smiley (2006a). Conservatives will try to resist the challenge. They will note that if we inspect standard texts on computable functions, we won’t in fact find ineliminable use of empty terms. Likewise, modern texts on complex variables tend to offer ways of parsing away superficial talk of many-valued functions. Meanwhile, common-or-garden talk of sums of pluralities invites regimentation in terms of the repeated application of the basic operation of adding one thing to another. Mathematicians, say the conservatives, can stick to standard logic after all. How to settle the debate between conservatives and revisionists is a tough question, and not the topic here (it isn’t even clear what the rules of the game are). But both sides should agree that the revisionism will look the more attractive the neater and smoother the non-standard logic that is offered: the aims and claims of logical regimentation, whatever exactly they are, are surely ill-served by unnecessary complexity. And the point of this note is to raise a question about the complexity of Oliver and Smiley’s own preferred revisionist logic. For their treatment of functions not only deals with partial functions (in a free-logical framework) and allows plural inputs and plural outputs, but there’s another wrinkle. It also accommodates what our authors call co-partial functions, i.e. functions that map nothing to something. But do we need to buy this additional complication? Or is allowing for co-partial functions a step too far down the revisionist road? The question has some independent interest.