A common formulation in mathematics is “Let A. Then B“, especially when A consists of more than one hypothesis.
E.g. “Let a and b be real numbers, and suppose x2+ax+b =0 has two distinct real solutions. Then a2 > 4b.”
The formulation “Let A. Then B” was recently reported to me as considered by some ‘not grammatically correct’ and furthermore one that ‘a native English speaker would not use,’ the formulation considered a corruption of the standard conditional “If A, then B.” Preferred in the above example would be: “Let a and b be real numbers. If x2+ax+b =0 has two distinct real solutions, then a2 > 4b.”
However, I and, according to anecdotal evidence gleaned from a random selection of mathematical texts, many other native English speakers would demur. The more marked division of scene-setting and hypothesis-making from the conclusion drawn achieved by the full stop and initial “Then” provides more syllogistic impulsion, giving greater weight to the conclusion. Also, if the hypotheses are lengthy to state, or there is no clear candidate for a key hypothesis, the use of
“Let [hypotheses]. Then [conclusion]”
can be clearer than the formulation
“Let [hypotheses]. If [key hypothesis], then [conclusion].”
To which part of speech “then” belongs in a conditional sentence of the form “If [protasis], then [apodosis]” is unclear, especially as “then” may be dropped without noticeably changing the meaning of the sentence (except when the word “then” carries stress in order to emphasize the apodosis, or to signal that it is the stated protasis that will in fact bring about the apodosis, not some more likely to be supposed condition). A sentence of the form
If [protasis], then [apodosis]
can also be written
[Apodosis] if [protasis].
Introducing as it does an independent clause, but not coordinating it with another independent clause, “then” is most commonly classified here as an adverb.
The Shorter OED lists four adverbial senses of “then”:
- At that time, at the time defined by a relative or other clause, at the time spoken or referred to;
- At the moment immediately following an action, next in order of time; in the next place esp. in a series or narrative;
- In that case, in those circumstances, when that happens;
- That being so, on that account, therefore, as may be inferred.
The first two are temporal, the last two consequential, and closest to the sense of “then” in a conditional sentence “If …, then…”. As a conjunctive adverb, “Then” may thus signal temporal sequence or consequence (empirical or logical). The latter sense perhaps justifies the initial use of “Then” to introduce consequences of hypotheses that have been listed previously. (It also fits the narrative sense of ‘following on’.)
Things are less contentious in a definition of the form
Let [declaration of domain]. Then [definition of term with scope the given domain],
e.g. Let X have property P. Then elements of X are said to be … if ….
where the use of “then” is redundant. The clause introduced by “let” performs the function of introducing notation and the set-up for, rather than a condition for, what follows. The connection is weaker than that of hypothesis-conclusion, and one can simply write:
Let X have property P. Elements of X are said to be … if …
Moreover, the “said to be” can be dropped if the term being defined is marked in another way (e.g. by italicization). For example, “A natural number p is prime if …” rather than “Let p be a natural number. Then p is said to be prime if …”
In sum, “then” as a conjunctive adverb (modifying the whole clause to which it is attached) can be understood in its adverbial sense of “as a consequence” or “it follows that” as well as in its temporal or sequential sense.