In Mathematics, it suffices that we find a counter-example to prove to ourselves or to any number of people that something is not true. If we find a large number of examples that validate our claim, we still must make use of allowed processes of generalization, say, to prove to others that we have a theorem, that is, a rule or a way to generate assertions.
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We should definitely have a similar thing in language, for it is very easy to come up with long lists of examples that validate claims that are not true.
Pieces of software that intend to translate/interpret texts in place of human beings probably contain theorems in them, is it not?
Have you ever stopped to think about how the systems analysts have succeeded in coming up with their systems, like what reasoning makes the system find a perfect match, amongst so many that are possible, for a certain word, from the original text, in the English language?
One of the so many possible ways to do that has to be using theorems that deal with each and every one of the pieces of any given word from the original text individually, is it not?
After thinking about the couple (English; Portuguese) for a time, we came up with quite a few possible theorems. Proving that some of them are not true is a really hard task, like finding a single counter-example, but proving that some of them are one hundred per cent true is almost impossible.
Consider, as possible theorem, the following: Whatever ends in ão in Portuguese will have an English language equivalent ending in ion.
We have built this claim after writing down the following matches: (constellation; constelação), (prohibition; proibição), (castration; castração), (constipation; constipação), (lotion; loção); (notion; noção), (portion; porção), (condition; condição), (preparation; preparação), (continuation; continuação), (manipulation; manipulação), (prostration; prostração), and etc.
We did not immediately notice, therefore, that we also have: (no; não); (pike; sapatão); (heart; coração), (solitude; solidão), (ledger; razão), (meal; refeição); (coat of arms; brasão), (pagan; pagão), (transport; condução), (pressure; pressão), (printing; impressão), (captain; capitão), (saint; são), (card; cartão), (sex drive; tesão), (reason; razão), and etc.
What about this one: Whatever ends in ia in Portuguese will have an English equivalent ending in y?
The initial list was: (lethargy; letargia), (metallurgy; metalurgia), (iridology; iridologia), (geology; geologia), (geography; geografia), (musicology; musicologia), (theosophy; teosofia), (philosophy; filosofia), (anthroposophy ; antroposofia), (allergy ; alergia), (mastectomy; mastectomia), (sexology; sexologia), (ontology; ontologia), (allegory; alegoria), and etc.
Again, we also have: (apologia; apologia), (claustrophobia; claustrofobia), (haemorrhage; hemorragia), (merchandise; mercadoria), (aunt; tia), (stationery shop; papelaria), (bookshop; livraria), (rectorship; reitoria), (directorship; diretoria), (bitch/slut; vadia), (neighbourhood; periferia), (mania; mania), and etc.
So coming up with theorems to generate automated tools to translate from one language into another may be a very tricky business, is it not?
It is still worth trying to come up with rules and testing hypotheses, however (even if it is for the sake of learning the exceptions to our possible rules).