“For all” and “there exists”

Question

The two most common mathematical quantifiers are "for all" (∀) and "there exists" (∃).

I wondered how to render them in Latin. Here is my proposal:

• for all x: pro omnis x
• for all x,y: pro omnibus x, y
• there exists x such that y: x ut y existet/est

Answer 1

First, there is no universal "Latin". Each era, or even each author, has their own style of Latin.

According to Wikipedia, quantifiers ("for all" and "there exists") first appeared in the 4^th century BC. One of the notable works on this matter is De Interpretatione, composed by Aristotle, and translated into Latin in the 6^th century by Anicius Manlius Torquatus Severinus Boethius.

Una autem est affirmatio et negatio quae unum de uno significat, uel cum sit uniuersale uniuersaliter uel non similiter, ut 'omnis homo albus est', 'non est omnis homo albus', 'est homo albus', 'non est homo albus', 'nullus homo albus est', 'est quidam homo albus', si 'album' unum significat. Sin uero duobus unum nomen est positum ex quibus non est unum, non est una affirmatio; ut, si quis ponat nomen 'tunica' homini et equo, 'est tunica alba' haec non est una affirmatio nec negatio una; nihil enim hoc differt dicere quam 'est equus et homo albus', hoc autem nihil differt quam dicere 'est equus albus' et 'est homo albus'. Si ergo hae multa significant et sunt plures, manifestum est quoniam et prima multa uel nihil significat (neque enim est aliquis homo equus); quare nec in his necesse est hanc quidem contradictionem ueram esse, illam uero falsam. (Capitulum VIII)

The affirmation and negation are one, which indicate one thing of one, either of an universal, being taken universally, or in like manner if it is not, as "every man is white," "not every man is white," "man is white," "man is not white," "no man is white," "some man is white," if that which is white signifies one thing. But it one name be given to two things, from which one thing does not arise, there is not one affirmation nor one negation; as if any one gave the name "garment" to a "horse," and to "a man;" that "the garment is white," this will not be one affirmation, nor one negation, since it in no respect differs from saying "man" and "horse" are "white," and this is equivalent to "man is white," and "horse is white." If therefore these signify many things, and are many, it is evident that the first enunciation either signifies many things or nothing, for "some man is not a horse," wherefore neither in these is it necessary that one should be a true, but the other a false contradiction. (Chapter 8, translation by Octavius Freire Owen in the 19^th century)

We collect the affirmations and negations in Latin, along with their English translations:

• Omnis homo albus est = Every man is white.
• Non est omnis homo albus = Not every man is white.
• Est homo albus = Man is white.
• Non est homo albus = Man is not white.
• Nullus homo albus est = No man is white.
• Est quidam homo albus = Some man is white.

However, logic was treated as just its own field in mathematics. It was not until the modern era that mathematics was seen to be built on logic. For example, Gauss would write:

Productum e duobus numeris positivis numero primo dato minoribus per hunc primum dividi nequit. (Theorema 13, Disquisitiones arithmeticae)

The prouct of two positive numbers smaller than a given prime number is not divisible by that prime number. (My translation)

Whereas in the modern terminology, influenced by logic, one would write "For any prime number p, for any positive numbers a and b each smaller than p, their product ab is not divisible by p."

From the example, one can see that Gauss would either write nothing specific (as in the case of a and b), or would write "datus" for a "given" number (as in the case of p).

He then went on to write:

Si quis neget, supponamus dari numeros b, c, d, etc. omnes < p, ita ut ab ≡ 0; ac ≡ 0; ad ≡ 0 etc. (mod. p).

If anyone denies (this), let us suppose numbers b, c, d, etc. be given, each < p, so that ab ≡ 0; ac ≡ 0; ad ≡ 0 etc. (mod. p). (My translation)

Now let's examine what Euler wrote:

§. 28. Omnis numerus primus, qui unitate excedit multiplum quaternarii, est summa duorum quadratorum. (Propositio V, E228 -- De numeris, qui sunt aggregata duorum quadratorum)

§. 28. Every prime number, which exceeds a multple of four by one, is a sum of two squares.

Here we can see used the quantifier "omnis", but where one would usually write "there exist numbers x and y such that p = x² + y²", Euler just wrote "est summa duourum quadratorum".

Searching through more works of Euler and Gauss, it seems that they just used some variant of "can" (queo, possum, possibilis, etc.) where we would write "exist". Of course, I am happy to be proven wrong.

For example, Gauss's paper proving the Fundamental Theorem of Algebra is entitled "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse", and Euler wrote:

§. 40. Qui numerus duobus pluribusque diversis modis in duo quadrata resolvi potest, ille non est primus, sed ex duobus ad minimum factoribus compositus. (Propositio VII, E228)

§. 40. Any number that can be resolved into two squares in two or more different ways, it is not prime, but composed of at least two factors. (My translation)