ackmanben wrote:Consider:
-2 times -4 equals 8
The sentence, "Bob does not have no apples" means bob has apples.
What do you think? Possible mathematical properties/patterns in language?
In standard English that is indeed the case, and negation follows the definition it operates by in formal logic. That is,
¬¬p ≡ p.
However, this is not the case in - probably - a majority of languages, and this includes many dialects of English. Now, to English-speakers that also happen to think rationally, this might sound dumb, but it's not really. The benefits of permitting multiple negations not to cancel out include:
- more redundancy, and thus more resistant to background noise
- less taxing on the mind to keep track of the parity of the negations this far
- the rough number of negations can, in some languages, provide a kind of idea of how intensive the feelings of the speaker are regarding the negation
The first point is kind of important - most speakers of language don't realize how much of the stuff we think we hear actually are interpolations our mind does for us, there's a huge percentage, really, that is nothing but really good guesses. If there's some extra noise, or we're fatigued or anything, we might get just too little information to reliably reconstruct the utterance in our minds. An extra negation thrown in will give us a bit more information, and is therefore helpful.
You can still, of course, think it's illogical, but it's not really: it's as if we're using a different operator, and let's arbitrarily pick the symbol
¤ to express it. Instead of the "classical" negation this new one has a truth table like this:
¤true = false
¤false = false.
Logic permits such functions, there's nothing inherently wrong about them, although you will not find any big treatment of them in books on logics, as there's not much interesting in ways of formal logical trickery you can do with them, unlike, say, the scheffer stroke or the pierce arrow.
The claim that double negation is illogical when used to express negatives really just betrays some ignorance about logic, or at least a mistaken assumption about language (i.e. that the words "not/no-/don't/..." encode the usual negation and not something else.
The list of languages that permit - and even
enforce double negation includes Afrikaans, Russian (and pretty much all the Slavonic languages), Portuguese, Ancient Greek, Hungarian, ...
Я здесь никого не знаю. /ya n
je znayo nitʃevo zdes
j/ I don't know anyone here (literally, "I don't know noone here". You can extend such a sentence with loads of negatives.
Apparently, in Chinese, double negatives do cancel out - but triple negatives also cancel out to the same truth value! (That probably goes hand in hand with my other point in the list above - once one negation's been cancelled out, the Chinese have figured it's best not to keep track of the number beyond that point, just let the truth-value of the sentence sit still at "true").
In English, if you make an utterance like "Well, technically speaking I don't know
no one that can help us ...", it's often somewhat marked - e.g. the intonation is somewhat off, the phrasing and context would help realize this does mean there probably is someone you think can help, etc. It's interesting how, say, someone speaking a dialect where double negation is permitted or even mandatory easily can change intonation so that we know when to parse the double negation as classical negation or as, well, let's call the other one 'utter negation' or something, just so we have a name for it.
In Russian, a similar effect would be be achieved by different means of making it clear you do know someone that could help. It might also be the case in Russian, not sure on this, that negation on nouns and pronouns uses the utter negation-operator, and negation on verbs uses classical negation, ot quite sure on this.
Russian does some other interesting things with regard to negation: objects of negative verbs (except for quirky case objects) are always in the genitive, whereas for non-negatives, there's more variation.
Another interesting thing to look at is Jerspersen's Cycle:
http://en.wikipedia.org/wiki/Jespersen%27s_cycle(Oh, apparently English "not" is diachronically a reduced form of nawiht - "no thing", and the negation took the form "I ne saugh nawiht", literally "I no saw nothing" ... wonder to what extent double negation in modern dialects is a retention of this rather than an innovation?)
Ultimately, and this is my last edit of this post, I promise, it's not that surprising we'd find mathsy things in language: language is built up by relations between constituent phrases and words and morphemes, and it's a thing pretty much every human ever deals with on a daily basis. Meanwhile, maths is the study of how systems of rules interact. Grammar is a system of rules or patterns, although it has some rather complex properties*.
So it's quite natural that mathematicians would take language as a starting point for developing some mathematical concepts, even useful ones. Now, it turns out there's a lot of different approaches that are of some relevance to linguistics: you have informatics and the ideas of how much information there actually is encoded in a string of symbols, you have game theory and neural networks and whatnot, you have algebras on strings of symbols and the whole field of formal languages. As it turns out, formal languages appear on at least two levels in natural languages, and here I'll contradict myself and mention three such levels:
First, you have what's known as phonotax, how sequences of sounds are permitted to combine. In English, strength is a permissible word, so the string of sounds transcribed as /stɹɛŋθ/ is permissible, however, /ŋɛθɹst/ is not permissible. In Georgian, თკვენ - /tkven/ is quite permissible. What sequences are permissible in a given human language is actually fairly easy to model using deterministic finite automata as long as we understand the actual restrictions involved.
Second, and this is not present in all languages, you get morphology, which tells how words combine with affixes to form new forms. I think this is doable with deterministic finite automata for most languages, although some small modifications of the definition of the automata would make the rules more concise. Some languages might have more complex rules which require pushdown automata or even turing-machines though.
As the third one, we get how words combine to form sentences, and here, the symbols we use are all words that pass the tests of phonotax and morphology. Here, I have heard the claim that some Australian aboriginal languages have grammars so complicated that turing-machines with finite tapes would be necessary to model them, but several languages by and large manage with something as simple as deterministic finite automata.
All this, of course, is maths because a mathematician has sat down and - intentionally or not - formalized some field of maths in a way amenable to use with language. Likewise, turns out we've formalized some fields of maths in ways that make them amenable to use with gambling. Or physics. Or economics. Or counting apples in baskets.
* For one, it's distributed - no human alive knows all there is to English. For another, it's malleable, flexible and changing. So, the maths we need to describe it, in case we aren't just going to come up with a very naive and simplistic mathematical model of language, would include provisions for changed the ruleset in ways that don't apply everywhere simultaneously - e.g. double negation might exist in one register but not in another register as spoken by the same speaker, and different speakers may adopt or lose double negation due to education, changes in sociolinguistic attitudes, etc, etc over time.)
Thanks to LoneWolfEburg for the correction of my Russian sample, btw.