# "If and only if"

In mathematical literature "if and only if" (sometimes abbreviated as "iff"1) is a relatively common phrase. Saying "A if and only if B" means that A and B are equivalent logical statements. This is equivalent to but less clumsy than saying "A if B and A only if B". It is convenient to have conjunctions to say "A is implied by B" (if), "A implies B" (only if) and "A is equivalent to B" (if and only if).

I assume that this conjunction or phrase appears in Latin, too. To my surprise, I found no trace of it in this text of Gauss. When, if ever, did this phrase appear in Latin mathematical literature? And more importantly, what is this phrase in Latin? Of course, I could translate it as si et solum si, but I would like to know what was actually used.

Partial answers and non-mathematical contexts are also welcome if you have any insight to share.

1 The English abbreviation is "iff" (< if), the Finnish one is "joss" (< jos), the Italian one is "sse" (< se), the Spanish one is "ssi" (< si), the Swedish one is "omm" (< om), and so forth. By analogy, I would abbreviate the thing in Latin as ssi. But this question is about the fuller and more formal version.

• I've been looking around, and I can't seem to find any such usage either. The more I think about it, though, I wouldn't be surprised to find that there wasn't really a set expression. By the time mathematics started to become set-theoretically rigorous, Latin wasn't really being used any more, and while it was, mathematics was mostly concerned with polynomials and things you would do with them when you got your hands on them, which doesn't really lend itself to 'if and only if' theorems. By no means am I trying to suggest that these guys didn't understand logic, but since it wasn't (cont.) Mar 2, 2017 at 4:34
• (cont.) the case that they had occasion to say 'if and only if' all the time, it wouldn't necessarily have coalesced into a set phrase like it has in modern mathematics. Mar 2, 2017 at 4:34
• Vicipaedia uses si et tantum si consistently, along with other modern practitioners. @MarJohnson is right: ancient mathematicians understood logic perfectly (actually, they invented it) but had no set phrase for “iff”, either in Greek, Arabic or Latin. In modern mathematics, “if and only if” appeared in the 19th Century. Give me some time to source this. Mar 5, 2017 at 1:18
• Interestingly, German uses another phrasing, instead of the archaic dann und nur dann, wenn we say genau dann, wenn (maybe translatable as "exactly if") Mar 16, 2017 at 17:31

I think the phrase neque aliter would fit. as in si, neque aliter fieri potest. . . .

Neque aliter has plenty of precedents, easily found by googling, including some classical (e.g. in Cic. pro Sest. 97), though not in the exact sense you are looking for. I'm fairly sure that I came across it some years ago while translating KF Gauss's disquisition on capillary action, but it would take rather a long time to find the precise reference.

Here is an example from his Arithmeticorum Liber Secundus, (Propositio 26a, 171 Propositae cuiuspiam quantitatis radicem cubicam extrahere) which isn't exactly what you are asking for, but may give you the confidence to use it :

Itaque deinceps: nam maior numerus distinctius partes exprimit, quia numerosior; neque aliter geometrico puncto accedere licet propter incommensurabilitatem quaesitae radicis in nullum numerum cadentis.

Maybe an alternative is "si et modo si". I constructed it like a literal translation of "if and only if", so not sure if it makes much sense. Yet, I found two sources using it:

• A mathematical page:

Nota 3. Duo essentiae quaecumque elementi x necessarie aequivalentes sunt. "φ essentia elementi x (φ Ess.x) est si et modo si omni proprietati ψ elementi x, et necessarie y, quod si y habeat proprietatem φ, tum proprietatem ψ habet".

Quamquam hoc enuntiatum, "casa rubra est", verum est, si et modo si casa illa vere rubra est, e contra hoc enuntiatum, "furari iniustum est", non ideo tantum falsum est, quod furari iniustum est.

I cannot really defend it further than this.