No one has ever mentioned to me MSCD chapter 13 (Numbers and Formulas) or asked me any questions about it, so evidently the subject isn’t high on anyone’s list of concerns. But I have a soft spot for that chapter anyway—it discusses the fiendishly subtle forms of ambiguity that can arise in expressing formulas in prose and mathematical equations, and it shows you how to avoid that ambiguity.
I was disappointed not to be able to cite any caselaw in that chapter, but thanks to reader Mark Anderson, I won’t have that problem in the third edition of MSCD. Mark pointed me to his firm’s analysis of a recent English case in which the following language was at issue:
23.4% of the price achieved for each Residential Unit in excess of the Minimum Guaranteed Residential Unit Value less the Costs and Incentives.
The problem is that this formula expresses two alternative meanings. Here’s how I’d express them:
An amount equal to (1) 23.4% of the price achieved for each Residential Unit in excess of the Minimum Guaranteed Residential Unit Value minus (2) the Costs and Incentives.
An amount equal to 23.4% of the result of (1) the price achieved for each Residential Unit in excess of the Minimum Guaranteed Residential Unit Value minus (2) the Costs and Incentives.
The idea is to spot this kind of ambiguity and instead articulate whichever of the alternative meanings you intend.
6 thoughts on “An Instance of Formula Ambiguity”
One way to deal with potential ambiguities in formulas is to always include an example following the formula.
Thus, any potential ambiguity could be dealt with by including something like the following after the sentence at issue in the case: “So, for example, assume the the Minimum Guaranteed Residential Unit Value is $1M; the price achieved for each Residential Unit is $1.5M; and the Costs and Incentives are $50,000. Then the amount referred to is calculated by subtracting the Costs and Incentives from the Minimum Guaranteed Residential Unit Value ($1M – $50,000 = $950,000) and calculating the excess of the price achieved over this figure ($1.5M – $950,000 = $550,000) and taking 23.4% of the result (23.4% of $550,000 = $128,700).
If you do as a general practice, then it becomes less critical to parse every formula for potential ambiguities.
Kazu: Agreed, but I suspect that once anyone is attuned to this issue, they’d find it simplest just to make sure that the formula isn’t ambiguous. Of course, if a formula is sufficiently complex you’d want to include an example regardless. Ken
I have to say that when you wade through this judgement, even with its examples, it’s still not that easy following Lord Hoffman’s reasoning! That must be why I’m not a Law Lord! LOL
I was just about to comment on the fact that Mark Anderson (in the example given on his webpage) doesn’t use square brackets to ‘nestle’ the parentheses. After checking Wikepedia it appears that usage of square brackets ‘…is not technically correct, but is quite common.’
I’m sure I was taught differently at school all those years ago!
I referred to MSCD chapter 13 twice this year because I was indeed concerned with putting some formulas into words.
Perhaps it’s a good comment on the clarity of your instructions that you haven’t been asked a lot of questions about chapter 13.
Re: 9/30/2009 Mathematical Formulas
In my experience, many lawyers choose the profession because it is one of the few that does not require the study of science or mathematics. This was a significant, if not material, consideration for me. [Unfortunately, as things worked out, I spent virtually all of my time in legal practice, and much of my time outside of practice, working on heavily quantitative matters.]
As for the meaning of, “23.4% of the price achieved for each Residential Unit in excess of the Minimum Guaranteed Residential Unit Value less the Costs and Incentives,” settled principles of order of operations in mathematics require the following result:
1. Overall, as a matter of pure mathematics, this language means 0.234 * RU – MGRUV – C + I.
2. Since multiplication is performed before addition and subtraction, the first step would be to multiply 0.234 * RU (“0.234RU”).
3. The addition and subtraction would then be performed left to right, so the result would be 0.234RU – MGRUV – C + I.
I am NOT suggesting that this language is clear. It is not. But, application of accepted mathematics principles seems to lead to the first meaning you suggest [although I’ve looked at your alternatives numerous times and still am not sure], provided that RU > MGRUV. Your second alternative would be mathematically summarized as 0.234 * (RU – MGRUV – C + I) [I think. You describe the second alternative in 2 steps, but it doesn’t matter; principles of mathematics hold that operations are performed left to right. To be clear, these principles also require performing all of the operations in parentheses before doing the multiplication.]
Note: Assuming Meaning 1 is the intended one, even if RU > MGRUV, the result could still be negative depending on the sum of C + I. I doubt the parties contemplated the consequences of a negative result.
1. While I agree with the post suggesting that an example can be helpful, this will only be true if the example is consistent with the wording.
2. The use of “in excess of,” “more than,” “lower than,” etc. should be avoided unless it is clear that you mean “greater than” or “less than” in the sense of inequality rather than mere sum or difference.
3. Similarly, use “less,” “decreased by,” “difference,” “plus,” “increased by,” “sum” to indicate addition and subtraction.
4. Use definitions and multiple sentences.
Using the language quoted in the original post for sample terms without intending to suggest that the following is even close to what the drafters intended:
“Net Price” means, for each Residential Unit (presumably defined elsewhere), the Residential Unit amount received by X less the Minimum Guaranteed Residential Unit Value (presumably defined elsewhere) less Costs & Incentives (a combined term presumably defined elsewhere). Then, in a substantive provision on payment, Provided that Net Price is greater than $0, X shall pay to Y 23.4% of the Net Price.
[Note the use of the ampersand in the combined – and, hopefully, defined) term. This avoids the use of “and” or “plus” before “Incentives,” which would require yet another definition! Further, separating Costs from Incentives may complicate consideration of a negative result. (What if the result after deducting Costs is positive, but further deduction of Incentives makes the net result negative?) My preference would be to incorporate everything into “Costs.”]
[Note also the use of “amount received.” This is to clarify that payment is not due until X actually receives the money. (I’m guessing from the wording used that this was some kind of real estate deal where closing can be delayed for a significant period after price is agreed, or the deal may not close at all.)]
BTW: I arbitrarily use square brackets in ordinary text, basically to avoid having to use the Shift key more than necessary. I express no opinion on the proper use of brackets in mathematics formulas.