After he read my paper, Anthony told me something that I already knew — that the Discussion section was brief, and the Conclusion sudden. I had been both sick of working on the paper, and having trouble thinking about it in natural language. So those parts were… lacking. Anthony's gentle remarks increased my sense that they were inadequate.

I have expanded and reörganized the Discussion section (cannibalizing part of the Conclusion), and added to the Future Work section between it and the Conclusion. In that context, the Conclusion should seem less sudden, and I have added some thoughts to it (as well as having taken one from it).

Anthony also suggested that the paper could be made more accessible by discussion of the historical background of the problem, and of real-world examples. But, as I told him, I fear to alienate experts by such discussion; and I have since learned that I am already bumping-up against the size-limits for a submission to the journal of my choice.

Anyway, the latest version of my paper is at

The mathematical expressions that appear (tiled) in the background of this 'blog; are from a paper on which I have been working (off-and-on) over a long time. I'm far from perfectly happy with the present state of the paper, but I've finally put-together something like a complete draft of it, in PDF, at

There are some temporary issues with presentation of this draft:

• The layout needs to be fixed, by the insertion of page breaks, so that things such as section headings are pushed to a next page, rather than orphaned.
• The OpenOffice formula editor does not support use of the relational symbols ≻, ≽, ≿, or ⪰. (For now, I am representing strict preference with pref and weak preference with wpref; later, I will output the paper as LAΤΕΧ and then tweak the formulæ to give me ≻ for strict preference and one of the other three for weak preference.)

At this point, I welcome comments, from experts and from non-experts, both on the underlying content and on how I have expressed myself.

Up-Date (2009:04/08): Professor Gamst of UCSD caught an error in what had been formula (10). It was easily patched. I have up-dated the on-line version of the paper.

Much of my time of late has been going into my paper on operationalizing a model of preference in which strict preference and indifference don't provide a total ordering.

Quite a while ago, I reälized very precisely what sort of system the assumptions would have to imply; I mistakenly presumed that I would relatively quickly identify sufficient assumptions (beyond those already recognized). But, at this point, I have a sufficient assemblage, each member of which is, taken by itself, at least passably acceptable. Jointly, however, there's an issue of factoring.

The paper derives its results from three sets of propositions. The first and second sets seem perfectly fine to me, and I don't expect them to provoke much dispute. The third set are more ad hoc. For the purposes of the paper they function as axiomata, but some or all of them would more ideally be derived from deeper principles (the pursuit of which, however, would be mostly a distraction from my goals).

It's amongst this last set of propositions that the factoring problem exists. One of them used to play an important rôle; right now it's doing nothing but occupying space. I'd remove it, except that I suspect that, in conjunction with the very principle that seemed to make it superfluous, it renders redundant another principle which feels even more ad hoc.

At the same time, I am now wrestling with what sort of discussion to provide after presenting the theoremata. I just don't seem to be in much of a frame-of-mind to ruminate.