“Qualitative Theory and Chemical Explanation” by Weisberg is a bit frustrating to read. He is mistaken in his mathematical notation, and this causes a serious problem in trying to read what he is trying to say. He also makes a serious mathematical error regarding infinite sets, subsets, and equality.

On notation, on page 1075 in ‘instantiations’ (5) and (6) he means to write that the constant *k* can vary across a range of values from 0.9 to 1.1 or 0.99 and 1.01 in the respective instantiations. He says as much in the following sentences, but the problem his ‘instantiations’ do not say that. If one remembers their order of operations, placing a constant like 0.1 or 0.01 directly next to a parenthesis indicates multiplication to be carried out *before* one adds or subtracts the product from the newly introduced constant 1.0 or 1.01. In addition, while in certain scientific notation the plus/minus symbol might mean ‘within this given margin’, it more commonly refers to both adding or subtracting the precise amount after the symbol, after the product is taken, in mathematics. It took me half an hour of trying to find a technical definition in statistics of ‘precision’ that might explain his huge leaps in logic, before I realized it was just strange notation outside of the well established order of operations.

When writing of p-generality, Weisberg makes a serious error regarding set size. While it is true that the range of numbers from 0.99 to 1.01 is a subset of 0.9 to 1.1, it does not follow that the subset is smaller than the full set. The part Weisberg misses is that both are infinite sets. Set theory defines sets as having equal size if and only if there is a function that produces a one to one correspondence between the members of the two sets. If we take a member of the set from 0.99 to 1.01 to be *x* and 0.9 to 1.1 to be *y*, the function that provides a one to one matching is f(*x*) = 10(*x –* 0.9) or *y* = 10(*x* – 0.9). Take any element of the subset and place it into f(*x*) and you will get an element of the other set. It is a simple matter of algebra to derive the inverse of this function if one starts with a *y* to find *x*, the corresponding element of the subset. The inverse function is: *x* = 0.9 + *y*/10 or *x* = 0.9 + 0.1*y*.

Thus as the sizes of the sets of logically possible values of *k* have not changed, there has been no change in p-generality, while, by definition there is an increase in precision. A subset being defined as more precise than the set it comes from (p. 1075- indented part 2/3 of the way down). As p-generality remains constant- that is infinite- precision increases. To be charitable, his logic holds *only* if the sets involved are finite or of different orders of infinity. In his example, without more, the orders of infinity involved are the same.

The notation used here is standard, and it is understood that what is being multiplied is “1.0 +/- 0.1”, not merely the last element.

I think that as far as set size goes we can principle of charity a reasonable reading here. This is, after all, philosophy of science, not math. We are dealing with the finite abilities and tools available to scientists. Technically the equations each pick out an infinite set of models, but scientists may only be able to measure to an accuracy in the tens of thousandths. We would then understand the models picked out by the equation to vary by a ten thousandth, and not infinitesimally.

So, I left this comment Saturday… not sure how it didn’t get up.

Notation: this is pretty standard notation (we’re dealing with Philosophy of Science after all, they’ve done this once or twice). Any time a +/- is there I’d say it’s safe to assume that just the next number is the margin. If more is intended there will generally be parentheses.

Sets: while it’s technically true that both pick out an infinite number of sets as written (W drops the “smaller” in his 2009, perhaps because of this) I don’t think that it is important. That one is a subset of the other is all that’s necessary for it to be less precise. Further, I think we can principle of charity a reasonable reading. W is talking about model use in science. Models that make predictions that are indistinguishable by finite humans and their finite machines are, for all practical purposes, identical. The sets then, for all practical purposes, are finite.

Weisberg corrected his notation for the 2009 piece by using parentheses as I would have liked and altered his definitions to avoid the problem I noted.

When speaking of p-generality we are speaking of the possible range of values and his specific example included infinite sets. In the 2009 re-framing of the issue he defined away the problem by defining a smaller p-generality as that p-generality that is a proper subset of the greater p-generality, while leaving the possibility that the sets could have the same number of elements.

I did note the charitable reading, but as he defined away the problem in a later work, I suspect someone alerted him to the difficulty in the intervening years. I had not read the later work yet and wanted to save anyone else the difficulty I had with his notations in the earlier piece.