The Tie No Sweetening Breaks

I.The First Known Thing — the tie that a small improvement fails to break

philosophy · value theory, comparability, and the small-improvement argument

It is ordinarily assumed that two things compared on some value stand in one of three relations: one is better, one is worse, or the two are equally good. The assumption is rarely argued for, because it is inherited from the behaviour of numbers, which can only be greater than, less than, or equal to one another. Against it stands an argument of remarkable economy, usually called the small-improvement argument, which has become the leading argument that the three relations are not enough.

the argument in four lines

Take two options between which no comparison seems to hold — the familiar examples are two careers, or two cities to live in. Neither appears better than the other. Now sweeten one of them very slightly: the same career with a little more money, the same city with a shorter commute. The sweetened option is plainly better than its unsweetened self. But it does not thereby become better than the rival. And that is the squeeze. If the original pair had been equally good, then improving one of them, however slightly, would have made it better than the other, because equality is a knife-edge and the smallest weight tips it. The pair survives the sweetening, so the pair was not equal; and it was not better or worse by hypothesis. Something else is going on.

what the something else is taken to be

On the strongest reading the pair is simply incomparable: no positive value relation holds. Ruth Chang’s proposal, set out in 2002, is less drastic and more useful. She argues for parity as a fourth basic value relation, available once one stops modelling evaluative comparison on the real numbers: two items may be on a par without either being better and without being equal. On this account the sweetened option remains on a par with the rival because parity, unlike equality, has width, and a small improvement moves an item within that width rather than out of it. Chang insists on a further condition that matters more here than it usually does in the literature: there is no comparison simpliciter. Every comparison is relative to a covering consideration, a specification of what the comparison is for. Change the covering consideration and the relation may change with it.

the rival diagnosis, and why the dispute will not end

The argument has a determined opposition, and the opposition is the reason this specimen exists. One reply treats the apparent gap as vagueness: it is neither true nor false that one option is better, and the trichotomy survives as an idealisation. A sharper reply is epistemicist. On epistemicism, vagueness is ignorance rather than indeterminacy, so there is a determinate fact about which option is better and we simply cannot reach it. Flanigan and Halstead argued in 2018 that if epistemicism is true then options in small-improvement cases are comparable after all, and the whole argument collapses into a report of what we do not know. Here is the stalemate. A relational gap and an epistemic gap produce exactly the same first-person evidence — the sweetening fails to move the verdict either way — and the discipline’s examples offer no procedure for telling them apart, because nobody can turn up the precision on a career. The cases are built out of intuitions, and an intuition has no replicate count.

II.The Second Known Thing — the scorecard that must rank two compounds

drug discovery · lead optimisation and multi-parameter trade-offs

A lead series is a family of related molecules, and optimising it means improving potency, selectivity, solubility, permeability, metabolic stability, and a list of safety liabilities at once, in a setting where almost every improvement to one costs something on another. Because the properties are measured in unrelated units, the universal device is a composite score. Each property is passed through a desirability function that maps its value onto a scale from nought to one, the resulting numbers are weighted and combined, and the series is ranked on the total. The apparatus is borrowed from industrial quality engineering and has been in medicinal chemistry for two decades.

two of the field’s own instruments, and the way their exchange rates were set

Two published scores show the practice clearly. The CNS multiparameter optimisation score of Wager and colleagues, from 2010, takes six calculated properties — molecular weight, logP, logD at pH 7.4, topological polar surface area, hydrogen-bond donors, and the pK_a of the most basic centre — scores each from nought to one, and adds them without weights to a total out of six, with four treated by convention as the mark of a favourable profile. The quantitative estimate of drug-likeness of Bickerton and colleagues, from 2012, uses eight properties and fits each desirability function to the observed distribution of that property across 771 orally dosed approved drugs, choosing asymmetric double sigmoidal shapes because the distributions are asymmetric.

the admission inside the second one

The weighting of that second score deserves its own paragraph, because the field has already recorded the awkward finding and then carried on. Having no external source for the exchange rates, the authors reasoned that the best weights are those that maximise information content, enumerated 390,625 weight combinations, and selected by Shannon entropy. They then reported that the maximally weighted, the mean-weighted and the unweighted versions perform comparably. The exchange rates between eight incommensurable properties were therefore set by a criterion internal to the arithmetic, and the index works about as well with them as without them. Equal weighting, as in the CNS score, is the same admission made by omission: it is not a measurement of how potency trades against polarity, it is a declaration that nobody knows.

what the field does about ties, which is already a great deal

None of this is naive, and the next two practices must be stated at full strength because the candidate has to survive them. The first is probabilistic scoring, developed by Segall and colleagues: the uncertainty in each measured or predicted property is propagated through the desirability functions into an uncertainty on the total, which yields a probability that one compound is better than another and, in their own worked example, the finding that the top five or six compounds cannot be confidently distinguished from the highest ranked on the available data. Their advice is to keep such compounds rather than reject them. The second is sensitivity analysis on the scoring profile itself, which asks how far the weights and desirability shapes can be moved before the top-ranked compound changes. The field, in short, knows that its ties are often artefacts of error, and knows that its weights are soft. What it does not have is a way of telling those two things apart in any particular pair.

III.The Bridge

no paper on the small-improvement argument cites a paper on compound ranking; both halls arrive at a pair that will not order, and only one of them can turn up the precision

The halls do not meet. The literature of parity and comparability is analytic value theory; the literature of composite desirability is medicinal chemistry and cheminformatics. Neither cites the other and no review spans them. Four neighbours have to be set aside, and the first two are close. Probabilistic scoring, above, treats indistinguishability as entirely a consequence of data uncertainty and takes the scoring profile as given; it answers how confident the ranking is, never whether a ranking exists to be confident about. Profile sensitivity analysis varies the weights, but it asks a robustness question about one chosen compound rather than describing a structure over the series, and it has no account of why the weights have the range they have. Third, multi-criteria decision analysis in health economics does use the word incommensurability, but rhetorically, as a caution against aggregation, and it issues no measurement. Fourth, and furthest, is Pareto analysis, which declines to order compounds on the Pareto front at all; that is a refusal to compare rather than a relation between the pair, it is computed from the properties alone, and it is silent about pairs that are not on the front.

X: a tie has two kinds, and they are told apart by what added precision does to them rather than by how either one feels

Call it parity under a bounded set of exchange rates. Where items are ordered by a composite of components measured in unrelated units, the order is a function of the exchange rates between those components, and that set of rates is bounded not by evidence but by what the practice is prepared to defend; so a pair whose preference reverses somewhere inside the defensible set is not ordered by any amount of data, and stands to the other pairs of the series in a relation that is neither better, worse, nor equal. Three things follow and they are not one thing said three times. The first is a diagnostic, and it is the small-improvement argument with a unit attached. Sweeten one member of a pair by a measured increment and ask what it takes to flip the preference. If the flip threshold is smaller than the measurement error, the tie is epistemic and replicates will break it; if the threshold is well outside the error yet the preference reverses inside the admissible set of profiles, no replicates will break it, because what is unsettled is the exchange rate and not the value. The two cases are separated by a single observable: whether the flip threshold shrinks as the replicate count rises, at the rate error shrinks, or stays flat. The second is that this is the dial the philosophers do not have. A relational gap and an epistemic gap present the same evidence in a thought experiment because a thought experiment cannot be run again with more care; an assay can, and the pair that remains unordered as precision improves is the first instance of a value gap that is not an artefact of ignorance about the facts, though it may still be an artefact of refusing to fix the rates. The third runs back the other way and is an obligation rather than a gift: if there is no comparison without a covering consideration, then a score reported as a number without a declared admissible set of exchange rates has not said what it is comparing for, and the question of which compound is better has not been asked yet. What the two halls together make available is not a flag on one choice but a partition of the series into classes its own committed values cannot order.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The philosopher has no reason to ask it. There is no assay in the examples, no organisation whose commitments bound the exchange rates, and no interest in a partition of forty molecules. The chemist has no reason either, because an unordered pair arrives at the bench as two overlapping error bars, which is a nuisance to be reduced rather than a structure to be measured, and because the weights are held as a necessary convention nobody expects to be informative. Held together, one question becomes available, and it is answerable with data that programmes already generate.

the parity-class hypothesis

For a congeneric lead series with replicated measurements on each property, elicit from the programme’s chemists not a set of weights but an admissible set: the ranges of weights and the families of desirability shapes each would defend as reasonable for this project. For every pair of compounds, compute the sweetening needed to flip the composite preference, expressed in units of that property’s own replicate standard deviation, and recompute it as replicates are added. Call a pair unordered when its preference reverses somewhere inside the admissible set. The hypothesis is that the series partitions into far fewer classes of mutually unordered compounds than it has members; that the flip threshold for pairs inside a class is flat in the replicate count while the threshold for pairs across classes falls as the root of it; that class membership survives replacing a readout with an independent one and re-eliciting the admissible set from different chemists; and that the programme’s own recorded decisions — re-rankings between meetings, late reversals, compounds carried in parallel — concentrate on pairs inside a single class.

why it would matter

Because the two kinds of tie have different remedies and the field currently applies one remedy to both. An epistemic tie is bought off with replicates, and buying it off is money well spent. A tie of the other kind consumes replicates without end, and its only honest resolutions are to declare a new covering consideration in advance — a criterion outside the present profile, named before the data arrive rather than after — or to carry both compounds and stop pretending the choice was made on the merits. A programme that could count its parity classes would know what share of its deliberation is spent on comparisons its own values do not make, which is a number no scorecard reports and every project has.

V.How It Could Be Proven — or Broken

a candidate must be breakable, or it is merely poetry

An intercognix candidate earns nothing until it names its own test. This one needs no new chemistry and no new assay, only replicates that were taken anyway, a scoring profile that most projects already maintain, and one addition the field does not currently make: eliciting the profile as a range the chemists will defend rather than as a point they will tolerate.

the honest status

Note the narrowness. The claim concerns a partition of one lead series under one elicited admissible set, and it says nothing whatever about which compound should be chosen inside a class, which is the decision that actually has to be made. Nor does it settle the philosophical dispute it borrows from: epistemicism could be true of values in general, and a flip threshold flat in the replicate count could still be the signature of chemists declining to commit rather than of a gap in the values themselves, which is why the elicitation has to be run twice on different people before the flatness is believed. What transfers is narrower and more modest than either hall’s ambitions. A tie has two kinds. They look identical from the inside and are told apart from the outside, by what added precision does to them. And a discipline that treats every tie as a shortage of data has been buying replicates against a quantity that does not respond to them.