Clarify logical connective of "greater than either of"
Opened this issue · 2 comments
In the introductory section describing the properties of the right-angled triangle ABC, point 3 currently reads (page 6):
“The angle ABC is greater than either of the angles BAC or BCA.”
This phrasing is logically ambiguous. In modern usage, “either … or …” can be read as exclusive or inclusive disjunction, implying that the angle ABC is greater than at least one of the other two angles — which is trivially true and weaker than Euclid’s actual intent.
Expected meaning:
From the geometric context, Euclid means that the right angle is greater than each of the other two acute angles. Therefore, the logical connective should be AND, not OR.
Similarly for point 4.
The role of this example is purely illustrative, it doesn't follow from any preceding proofs, and it's not entirely clear what these properties are supposed to be based on. In general I try not to change things which are not incorrect in this book, and the introduction, including this particular section of it, was taken directly from Byrne's original as is.
I agree that AND is a better description of these properties of the right-angled triangle, though. What do you think if I add a margin note next to these points, suggesting to read them as something like this?
The angle ABC is greater than the angles BAC and the angle BCA.
The angle BCA and the angle CAB are less than the angle ABC.
I appreciate sticking to Byrne's original (and in general I admire the whole work that you have done 🙌). The “or” in the original text may be a linguistic artifact, not a logical one.
I like the idea of the margin note, which could avoid confusion in this very early example, but of course you have to decide if such a margin note is justified, with respect to the project goal of reproducing Byrne's original.