'Denote something as "X" ' vs 'denote something by "X" '

Question

This question "Denoted by" or just "denoted"? discusses whether we need to say "denoted by" or just "denoted".

I am wondering whether it is grammatically correct to say:

• "We denote the natural probability measure as P"

or whether we need to say

• "We denote the natural probability measure by P"

Or is it perhaps best to say:

• "We denote the natural probability measure P" ?

Answer 1

. A Google 2-grams search (for denoted as; denoted by) shows that both prepositions are used, but it seems that the latter has always been distinctly more popular, with 'denoted as' increasing fairly steadily from a low value since about 1930, to being about half as popular nowadays.

Examples given include:

• ...denoted by the symbol B ... [Introduction to Programming Languages - Page 113; Arvind Kumar Bansal · 2013]
• ... denoted by φ , ψ , χ , ω , ... , constant truth by T , and its negation by 1. [Belief Change - Page 79; Dov M. Gabbay, ‎Philippe Smets · 2012]

.......

• ... (denoted as q) , • the loss in market share as the monetary value of the decrease in market share that Median experiences (denoted as m) [Security Risk Models for Cyber Insurance - Page 116; David Rios Insua, ‎Caroline Baylon, ‎Jose Vila · 2020]

The 'as' option focuses on the representation, the 'by', perhaps a little more the representing. There is little difference in overall meaning.

Omitting the preposition is grammatical, but perhaps decreases ease of reading. F'x states in the partial duplicate that Google Scholar shows that again, while the form with preposition deletion is used, it is less popular than 'denoted by'.

As an aside, it is good to use 'we' (not the egotistical-sounding 'I'!). And the passive is thankfully only deemed necessary by a few nowadays.

Answer 2

I see nothing wrong with the passive, making the thing denoted the subject. Ngrams shows that it is used more frequently than "We denote", FWIW.

The natural probability measure is denoted by P.

You would opt for the "We denote" form when you are departing from standard practice, or referring to something new and thus get to give it a symbol, like naming a new comet.

If using the "We denote" form, use with:

We denote {whatever it is} with {symbol}.

(The paraphrase of that form is "We use {symbol} for {whatever it is}).

You could also make the symbol the subject:

P denotes the natural probability measure.

That form appears to beat out the others, though with Ngrams and truncated search terms, we can't be sure of that.

If you are writing an introductory textbook, you could use the adverb usually or conventionally when following usual practice or standard practice, respectively.