In reply to Michael Gordon:
(sneak preview - Summary: if you want to exactly obtain four separate quantities, which are either independent or dependent on each other in a non-linear way, you need to have four numbers!)
Time to get anal... on which note some of the below is probably complete rubbish (it is late at night and I am working this through as I go!). I hope nobody actually bothers reading me making a simple concept very complicated...
We have the following information:
D (ifficulty) The overall difficulty of the route. We define this as the overall success/failure probability (see bottom) of the entire route using the best sequence and all available gear beta. This is basically the French grade.
C (rux move) The difficulty of the hardest move. We define this as the overall success/failure probability (see bottom) of the hardest move using the best sequence. This is basically the English tech grade.
O (nsightability) The onsightability of the route. We define this as the change in probability of success from that expressed by quantity 1 if the route is climbed 'onsight' i.e. without beta. This is not expressed individually by any grading system I know of.
X (danger) The 'danger' of the route. This can be calculated by taking each metre of the rule, multiplying the likelihood of a fall (normalised for the grade) by the consequences, and summing for the whole route. This is basically the American danger grade.
Note that I cannot and do not attempt to put numerical values or units on any of these but this is not required.
Now we examine the relationships (or lack of) between these three quantities) - independent here meaning that it is not affected by the quantity (it may affect that quantity).
The overall difficulty (D) clearly depends on the technical grade (C), but it does not depend on the onsightability or the danger.
The technical grade is independent of overall difficulty, onsightability or the danger.
The onsightability is independent of the overall difficulty by design. You could argue that it does depend on the ratio of the tech grade to the overall difficulty (e.g. a cruxy route being harder to onsight) but I will not assume this.
The danger grade is independent of the overall difficulty by design. It does depend on the tech grade since it depends on the details of the climb e.g. where and how hard the crux is. It may also be affected by the onsightability.
I have chosen these quantities because these are the ones we want to extract from the grade - overall chance of success, how hard the hardest move is, how easy it is to onsight and how dangerous it is.
System A: the British trad grade.
E: the adjectival grade is a function of the overall difficulty, the onsightability and the danger.
T: the tech grade is a pure quantity (not affected by the other quantities).
So E = E(D, O, X). But we also have D = D(T), O = O() and X = X(T), so E = E(D(T), O, X(T)). We know the relationship between D and T is non-linear; generally a tech grade sets a minimum difficulty but a long sustained pitch can be much harder than this minimum. We also know the relationship between X and T is non-linear, since it depends on the position of difficult parts and the availability of gear.
So if you have a given E and T (a complete British trad grade) you will always have ambiguity in D, O and X which you cannot simply resolve. How do we get any information at all? In practice a tech grade sets a likely range of difficulties, I think the onsightability is in practice not very important and at least at the lower grades the danger only changes the grade by one or two. So you can never tell if your VS 4b is very sustained 4b or just poorly protected, or if your HS 4c is just very easy to onsight with one hard move or super well protected (in practice a lot of the easy grades are probably used as shorthand i.e. VS 4b gets used for a 'poorly protected VS', regardless of whether there are 4b or 4c moves or how sustained it is). What you actually get is a region of probability; by assuming the difficulty is typical for the tech grade and ignoring the onsightability you can make an estimate of the danger grade but it is not exact. You can use further guidebook information (i.e. if it is 8m high it is probably not super-sustained) to further constrain you guess but you can never be exact.
System B: French grade and a danger grade.
D: the overall difficulty.
X: the danger grade.
Since these are both quantities we want, we obtain them perfectly (barring the usual grading debates), however we have lost all of the information of the hardest technical grade (although we know the range it might reasonably fall within) and the onsightability. We actually have a _better_ idea of the danger since we have the exact value rather than the inexact version we estimate from the British trad grade.
We can trivially change this by adding the tech grade, at which point we also know the tech grade (unsurprisingly). I was going to say that sport climbers have never asked for this but then I remembered that (hard) routes are often described with cruxes measured in boulder grades so actually this is sometimes included...
Summary: if you want to exactly obtain four separate quantities, which are either independent or dependent on each other in a non-linear way, you need to have four numbers!
Appendix:
When I say 'overall success/failure probability', what I _actually_ mean is that for an averaged climber with a 'skill level' matching the grade (e.g. a prototypical 'VS' climber) the chance of success is equal to that of other routes of the same grade - i.e. if our VS climber can climb 95% of all VS climbs, then there is a 95% chance of success of climbing this one as it too is a VS climb. I have explained this very badly but it isn't important!
Post edited at 22:18