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I am interested in translating the word "then" in logical statements like this: "If a number is prime, then it is squarefree." Or maybe better: "If x is even and x+y is odd, then y is odd." In common language one would not use a "then" at all, but in mathematics (including logic) it is very common to emphasize the consequence with a "then". This logical "then" is not temporal but causal, and I have never seen it in ancient sources at all. I have not found this sense of "then" in Latin dictionaries.

What would be a good translation of this "then"? Was there a fixed phrase in the mathematical Latin of, say, 18th and 19th century? My intuition is to translate "if–then" with si–deinde. I want something that works within a sentence, so that the "then" does not start a stand-alone sentence, but just a consequence clause.

It seems that Gauss does not use a "then" at all in this example. It is possible that the logical "then" was introduced after Latin went out of fashion in mathematics.

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    Si ... deinde looks perfectly all right to me. I would certainly use 'then' in conversation, though not on every occasion. Perhaps igitur would suit? Or possibly ipso facto? — a [prime number is] ipso facto [squarefree]. – Tom Cotton Mar 13 '17 at 11:08
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    It seems that Newton uses (from time to time) dein and ergo. If anyone wants to write the answer, be my guest: I can't do more research for some hours yet. – Rafael Mar 13 '17 at 12:53
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    Adelard of Bath (1080-1152) has a parenthetical 'necesse est' ( once with igitur,) to indicate certainty. For a confident opinion, he finishes the clause with 'non dubitas.' – Hugh Mar 13 '17 at 14:39
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Option 1: sequitur, ut

Browsing L&S I came to the entry on the verb sequor, meaning II.B.4, that reads:

In logical conclusions, to follow, ensue; with subject-clause,

especially with ut. And it cites a pretty clean example from Cicero:

Si hoc enuntiatum: "Veniet in Tusculanum Hortensius" verum non est, sequitur, ut falsum sit.” (Cic. Fat. 28)

Several other examples from classical times may be found with this construction.

Option 2: ergo

Ergo seems to me like the most natural choice (one word, no comma needed.)

L&S, lists at least two examples of meanings where ergo introduces the consequence in conditional clauses.

  • Consequently, therefore (...) with si, cum, quia, etc.: “ergo ego nisi peperissem, Roma non oppugnaretur,” Liv. 2.40.8; "Ergo quia sum tangere ausus, haud causificor quin eam ego habeam potissimum." Pl. Aul. 4.10.25
  • With imperatives and future, then: “ergo, si sapis, mussitabis,” Pl. Mil. 2.5.66.

More explicitly, in modern scientific literature, Newton uses ergo (e.g. in his famous Philosophiae Naturalis Principia Mathematica, sect. I.)

Anyway, Newton himself i) builds many conditional clauses starting with si and using a comma as the only link to the consequent, and ii) uses ergo alone to introduce stand-alone sentences as consequence of the reasoning in the previous ones.

Option 3: dein and other phrasal forms

Newton also uses dein (see L&S entry for dein) in the same context, and apparently more smoothly.

It is also worth mentioning what Tom and Hugh noted: deinde (long for dein) and additional phrases like necesse est. In fact, I also found Newton using dico quod as the link in the Principia (Sect. III. Prop. XIV. Theor. VI.)

  • I'm still not sure whether it does the trick, it seems that may also be used to introduce a consequent in a separate sentence. – Rafael Mar 21 '17 at 18:52
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    Newton's use of dein is the kind of thing I was after, and I accept this answer because of that. I consider ergo to be more "therefore", and ex A sequitur, ut B more "it follows from A that B". Although these are logically equivalent to "then" in some sense, they would not be used in the exact same way. I know I might be stubborn, but I look at it from the perspective of what I would like to express as a mathematician: "if A, then B". I did not expect the Romans to use modern style mathematical language, but it is interesting to see how close they got. – Joonas Ilmavirta Mar 29 '17 at 21:16
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I think it is worth mentioning in this context that a modern notation like

pq

which we read in English either as p implies q or if p then q, can be read in Latin as ex p, (sequitur) q with an optional sequitur, along the lines of ex falso, (sequitur) quodlibet. In most examples of si p sequitur q, p is a proposition; when it is not, ex + Abl. can be used. It is a spatial metaphor, of course, but it works much better than its English counterparts from and out of.

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