What part of «no» do you not understand?
cliché I have already said «no,» and that is my final answer, so stop trying to elicit a different response from me. A: «Come on, Mom, just let me borrow the car tonight!» B: «Billy, what part of ‘no’ do you not understand?» I’ve rejected their application three times already. What part of «no» do they not understand?
See also: not, of, part, what
Farlex Dictionary of Idioms. © 2022 Farlex, Inc, all rights reserved.
- What part of «no» don’t you understand?
- what part of no don’t you understand?
- isnt
- it isn’t worth it
- it’s not worth it
- rock the house
- without (even) batting an eye
- without (even) batting an eyelash
- without (even) blinking an eye
- without batting an eye
ILLUSTRATION BY ELLEN SURREY
Not long ago, I walked into a friend’s kitchen and found her opening one of those evil, impossible-to-breach plastic blister packages with a can opener. This worked, and struck me as brilliant, but I mention it only to illustrate a characteristic that I admire in our species: given almost any entity, we will find a way to use it for something other than its intended purpose. We commandeer cafeteria trays to go sledding, “The Power Broker” to prop open the door, the Internet to look at kittens. We do this with words as well—time was, spam was just Spam—but, lately, we have gone in for a particularly dramatic appropriation. In certain situations, it seems, we have started using “no” to mean “yes.”
Here’s Lena Dunham demonstrating this development, during a conversation with the comedian Marc Maron on his podcast “WTF.” The two are talking about people who reflexively disparage modern art:
MARON: They can look at any painting and go, “Eh.” They can look at a Rothko and go, “Hey, three colors.” And then you want to hit them.
DUNHAM: No, totally.
Dunham is twenty-eight years old, but the “No, totally!” phenomenon is not limited to her generation. It’s not even limited to “No, totally.” I first started noticing it when a fiftysomething acquaintance responded to a question I asked by saying, “Yup! No, very definitely.” That sent me looking for other examples, which turn out to be almost nonexistent in written English but increasingly abundant in speech. In 2001, the journalist Bernard Kalb told the White House correspondent Dana Milbank that it was the job of reporters to thoroughly investigate political candidates, to which Milbank responded, “Oh, no, yes, I agree with you there.” In 2012, Anderson Cooper, talking with the CNN senior political analyst Gloria Borger, referred to Newt Gingrich as “the guy who has come back from the dead multiple times.” Borger’s reply veered toward Molly Bloom terrain: “Yes, no, exactly, exactly, exactly.”
“No, totally.” “No, definitely.” “No, exactly.” “No, yes.” These curious uses turn “no” into a kind of contranym: a word that can function as its own opposite. Out of the million-odd words in the English language, perhaps a hundred have this property. You can seed a field, in which case you are adding seeds, or seed a grape, in which case you are subtracting them. You can be in a fix but find a fix for it. You can alight from a horse to observe a butterfly alighting on a flower.
Such words—also called auto-antonyms, antagonyms, Janus words, and antiologies—can arise for different reasons. Some are just a special kind of homonym; what appears to be one word with two opposite meanings is really two different words with identical spellings and pronunciations. Thus “clip,” meaning “to attach together,” comes from the Anglo-Saxon clyppan, while “clip,” meaning “to cut off,” comes from the Old Norse klippa. Other contranyms arise when nouns becomes verbs. Sometime around 1200 A.D., dust turned into a verb and, as dust will do, went every which way: “to dust” can mean to remove dust, as from a bookshelf, or to add something dusty, as flour to a cake pan or snow to the streets of Brooklyn. Alternatively, a contranym can reverse meanings when it is used as a different part of speech. As a noun, “custom” refers to a behavior that is common to many people. As an adjective, it refers to something designed for just one person.
Occasionally, however, a contranym arises through a process called amelioration, whereby a normally negative word develops a secondary, positive meaning. This phenomenon is particularly common in slang: “bad” becomes good, “wicked” becomes awesome, and “sick” and “ill” become wonderful. (They have been ameliorated: made better.) The use of “no” to mean “yes” appears to be an example of amelioration, but with one important distinction: “no” can’t mean “yes” on its own. Consider a slightly abridged version of Lena Dunham’s conversation about art appreciation:
MARON: And then you want to hit them.
DUNHAM: No.
Take away the “totally” and Dunham appears to be rejecting anti-philistine violence. By contrast, you can take away the “no” without doing any evident semantic damage at all. A perfectly fine response to “Then you want to hit them” is “Totally”—or, for that matter, “Yes, totally,” or just “Yes.” In fact, every instance of “No, totally” and its kindred phrases can be replaced with “Yes,” without any disruption of grammar or meaning. So why do we sometimes use “no” instead?
At first blush, “no” does not appear to be the kind of word whose meaning you can monkey with. For one thing, there is its length. At just two letters and one syllable, it lacks the pliable properties of longer words. You can’t stuff stuff inside it. (You can say “unfreakingbelievable,” but you cannot say “nfreakingo.”) You can’t mangle it, à la “misunderestimate” or (the finest example I’ve heard lately) “haphazardous.” On the contrary, it is so simple and self-contained that it is a holophrasm, a word that can serve as a complete sentence. (Holophrasms aren’t common in English, but any verb in command form can be holophrastic—“Go,” “Help,” “Run”—and babies just learning to talk use single words to express complex ideas all the time, albeit without regard to grammar: “Ball,” “Up,” “Want.”) Moreover, the word has the apparent fixity and clarity of a logical operator: like “if,” “then,” “and,” “or,” and “not,” “no” seems designed to be unambiguous. When we ask, in the face of excessive pestering, “What part of ‘no’ don’t you understand?,” what we mean is: “Unless you are a complete cretin, there is no part of ‘no’ that you could possibly misunderstand.”
Well, perhaps you would care to join me for a while in the land of complete cretinhood. For instance, answer me this, if you can: What part of speech is “no”? I thought it over for a while and concluded that it must be an interjection, even though it fails the Mad Libs test. (“The burglar bumped into the dresser and exclaimed, ‘___, my toe!’ ” The last time someone filled in a blank like that with “no” was never.) At a generous estimate, I was only one-sixth correct—but, in my defense, “no” resists all ready grammatical categorization. It is not an interjection, except when it is. (“Oh, no, I missed the train.”) It is not a noun, except when it is. (“The nos have it.”) It is not an adjective, except when it is. (“I have no idea what you’re talking about.”) It is not an adverb, except when it is. (“I’m no clearer on this than I was when I began.”) Some linguists grant it the separate part-of-speech status of “sentence word,” because, as I noted, it can serve as a stand-alone sentence. Others consider it a particle—even though, as a rule, the point of particles is precisely that they can’t stand alone; they exist to affect the meaning of other words.
In addition to this grammatical ambiguity, “no” also sometimes suffers from semantic ambiguity—which is odd, considering that we regard it as absolute. But consider the question “You aren’t a fan of cilantro?” The answer “No” is confusing, since it can mean either “No, it tastes like dish soap” or “No, I adore it.” Some languages avoid this type of indeterminacy. In Japanese, for instance, hai and iie, although generally translated as “yes” and “no,” actually mean something closer to “That’s correct” and “That’s incorrect.” This eliminates the grey area. “You’re not a fan of cilantro?” “That’s incorrect,” you are a fan. In English, by contrast, we must resort to elaboration: “No, I like it fine, I just don’t want any on my pancakes.”
Until the end of the sixteenth century or thereabouts, English had a tidier solution to this problem: we had two words for “no,” which we used in distinct ways. Those two words formed half of what’s called a four-form system of negation and affirmation. If you speak French (or, in a statistical unlikelihood, Norwegian, Swedish, Danish, or Icelandic), you are familiar with a three-form system: in French, n__on can negate anything, oui is used only in response to positively phrased questions or statements, while si is used to contradict questions or statements phrased in the negative. In Franglish:
Would you like to have dinner with me on Friday?
Oui, I’d like that very much.
You don’t like the cilantro pesto I made?
Si, it’s delicious!
Back when English was a four-form system, it, too, had a si—a word used specifically to contradict negative statements. That word was “yes.” To affirm positive statements, you used “yea”:
Shoot, there aren’t any open pubs in Canterbury at this hour.
Yes, there are.
Is Chaucer drunk?
Yea, and passed out on the table.
Similarly, “nay” was used to respond to positive statements or questions, while “no” was reserved for contradicting anything phrased in the negative:
Is the Tabard open?
Nay, it closed at midnight.
Isn’t Chaucer meeting us here?
No, he went home to bed.
Over time, the distinction withered, “yea” and “nay” became obsolete, and “yes” and “no”—the words that started out as special cases, for responding exclusively to negatives—came to hold their current status. Or, as the case may be, statuses.
What does all this have to do with the strange case of “No, totally”? The linguists I spoke with thought that this use of “no” might be a response to an implicit or explicit negative in the preceding statement: the type of “no” we used back when we also had “nay.” In modern English, you need to use something to clear up the cilantro-style confusion—so why not “totally” or its ilk? Here’s ABC News’s Joy Behar talking to the comedian Ricky Gervais about how girls, unlike boys, are not encouraged to make fools of themselves in public:
BEHAR: Well, they don’t get rewarded for acting stupid.
GERVAIS: No, exactly, yes.
Because Behar’s statement is negative, either “yes” or “no” on its own would be a confusing response. Gervais chooses “no,” then has to add “exactly, yes” to indicate that he doesn’t mean “No, Joy, you’re wrong.” You could argue that there’s also a negative, this one implicit, in the exchange between Mark Maron and Lena Dunham. By that logic, Maron is really saying, “You want to hit them [because these guys don’t know anything about art],” and Dunham’s reply means, “No, they don’t, I totally agree.”
In suggesting this negation theory of “No, totally,” linguists are borrowing from their far more developed explanation of a seemingly similar expression: “Yeah, no.” The “no” in that phrase is generally thought to retain its customary negative function. I’m a little dubious about whether that’s the whole story, but it doesn’t matter, because I’m going to ignore “Yeah, no” here. For one thing, those who are interested can refer to what the Oxford English Dictionary lexicographer Jesse Sheidlower called “the extensive ‘Yeah, no’ literature.” (An excellent place to start is this three-part analysis by the University of Pennsylvania linguistics professor Mark Liberman.) For another, the comparison only gets us just so far—because in many examples of “No, totally!” there doesn’t seem to be any negation whatsoever. Consider:
LYDIA: That book is constructed so brilliantly. It’s like a locked-room mystery.
IVAN: No, totally.
In this case and many others like it, “No, totally” appears to be all affirmation—a surprised and happy seconding. A rough translation might be, “Wow, that’s just how I feel!”
We’ve been using “no” to express surprise, including happy surprise, for a very long time. You hear that use in “No way!” You hear it (or heard it) in the early-aughts slang “Oh no you di’int!”And you hear it in “The Adventures of Tom Sawyer,” after Tom asks Huck Finn what you can do with a dead cat. Why, you can use it to cure warts, Huck replies. “No! Is that so?” Tom exclaims. Here, “no” is again serving as an interjection, akin to the “damn” in the phrase, “Damn, that’s smart”—“damn” being another normally negative word that can sometimes swap polarity and become positive. With both “damn” and “no,” the slimmest hint of the negative might linger, in the form of envious admiration. But, for the most part, this enthusiastic “no” has very little negative meaning, or really much semantic content at all. It is more like verbal punctuation—like the initial, upside-down exclamation mark in Spanish that alerts you to impending excitement: ¡Totally!
EXP: 08888
HITS: {{lyricTraining.hits}}888
STREAKS: {{lyricTraining.streaks}}888
TIME: {{lyricTraining.timeout / 1000}}88
Complete the lyrics by typing the missing letters.
Each perfect answer brings you 20 points
Hints are displaying when you get stuck
You have 60 seconds to complete the challenge.
Streaks will give you bonus in exp and time.
You can skip the sentence to get the next one.
[Ctrl] Replay the sentence.
[Enter] Skip the sentence
[Space] Pause / Play
{{word.word}}
{{cue.translated_text}}
x{{playblackSpeed.toFixed(2)}}
Loading ..
Not Found
{{annotation.word}} ({{annotation.pos_simplify}})
/{{annotation.phonetic}}/
{{annotation.translation}}
{{annotation.name}}
{{annotation.sub_type}}
Back
Games & Quizzes
Training Mode — Typing
Fill the gaps to the Lyric — Best method
Training Mode — Picking
Pick the correct word to fill in the gap
{{game.name}}
{{game.description}}
Requires {{game.required_annotions}} vocabulary annotations
Congratulations! You gained {{totalPoint}} XP
You may need to watch a part of the video to unlock quizzes
Don’t forget to Sign In to save your points
Repeat
Skip
({{game.question_index}}/{{game.questions.length}})
{{part}}
Repeat
Skip
({{game.question_index + 1}}/{{game.questions.length}})
Make correct pairs from two columns
{{lyricTraining.finished && lyricTraining.timed_out ? ‘Timed Out’ : ‘Challenge Accomplished’}}
| PERFECT HITS | {{lyricTraining.stats.PERFECT_HITS}} | +{{lyricTraining.stats.PERFECT_HITS * 20}} |
| HITS | {{lyricTraining.stats.HITS}} | +{{lyricTraining.stats.HITS * 10}} |
| LONGEST STREAK | {{lyricTraining.stats.LONGEST_STREAKS}} | +{{lyricTraining.stats.LONGEST_STREAKS * 10}} |
| REMAINING TIME | {{lyricTraining.timeout / 1000}}s | +{{lyricTraining.timeout / 1000}} |
| TOTAL | +{{lyricTraining.exp}} |
We couldn’t find definitions for the word you were looking for.
Or maybe the current language is not supported
{{simpleDefinition.pos}}
{{simpleDefinition.def}}
Sentences & parts of speech:
-
{{tag.normal}} {{tag.pos_name}}
|
{{a.word || a.name}} /{{a.phonetic}}/ {{a.type || a.sub_type}} {{a.translation}} |
{{a.definition}} |
The example sentences of WHAT PART OF «NO» DO YOU NOT UNDERSTAND in videos (15 in total of 10000)
opening verb, gerund or present participle
the determiner
mind noun, singular or mass
to to
all determiner
possibility noun, singular or mass
and coordinating conjunction
then adverb
deciding verb, gerund or present participle
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
all determiner
possibility noun, singular or mass
we personal pronoun
choose verb, non-3rd person singular present
to to
make verb, base form
things noun, plural
clearer verb, non-3rd person singular present
we personal pronoun
should modal
instead adverb
use verb, base form
parenthesis noun, singular or mass
to to
decide verb, base form
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
expression noun, singular or mass
you personal pronoun
asked verb, past tense
for preposition or subordinating conjunction
so adverb
little adjective
did verb, past tense
n’t adverb
you personal pronoun
read verb, non-3rd person singular present
these determiner
passages noun, plural
do verb, non-3rd person singular present
you personal pronoun
not adverb
understand verb, non-3rd person singular present
i personal pronoun
‘m verb, non-3rd person singular present
all determiner
—
powerful adjective
—
why wh-adverb
do verb, non-3rd person singular present
you personal pronoun
not adverb
understand verb, non-3rd person singular present
that preposition or subordinating conjunction
the determiner
need noun, singular or mass
of preposition or subordinating conjunction
my possessive pronoun
people noun, plural
for preposition or subordinating conjunction
their possessive pronoun
controller noun, singular or mass
is verb, 3rd person singular present
greater adjective, comparative
than preposition or subordinating conjunction
your possessive pronoun
need noun, singular or mass
for preposition or subordinating conjunction
your possessive pronoun
friend noun, singular or mass
?
when wh-adverb
you personal pronoun
add verb, non-3rd person singular present
music noun, singular or mass
,
you personal pronoun
can modal
adjust verb, base form
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
song noun, singular or mass
you personal pronoun
want verb, non-3rd person singular present
to to
play verb, base form
so adverb
desperately adverb
to to
make verb, base form
me personal pronoun
like preposition or subordinating conjunction
you personal pronoun
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
that determiner
is verb, 3rd person singular present
fair adjective
tori proper noun, singular
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
that determiner
is verb, 3rd person singular present
fair adjective
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
no determiner
no determiner
noo proper noun, singular
noo proper noun, singular
noo proper noun, singular
!
the determiner
comments noun, plural
below preposition or subordinating conjunction
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
world noun, singular or mass
you’re proper noun, singular
from preposition or subordinating conjunction
a determiner
traditional adjective
dish noun, singular or mass
and coordinating conjunction
you personal pronoun
never adverb
the determiner
only adverb
thing noun, singular or mass
,
depending verb, gerund or present participle
on preposition or subordinating conjunction
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
world noun, singular or mass
you personal pronoun
live verb, non-3rd person singular present
in preposition or subordinating conjunction
,
your possessive pronoun
bicycle noun, singular or mass
brakes noun, plural
but coordinating conjunction
also adverb
specifically adverb
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
projects verb, 3rd person singular present
you personal pronoun
‘d modal
be verb, base form
more adverb, comparative
interested adjective
in preposition or subordinating conjunction
working verb, gerund or present participle
on preposition or subordinating conjunction
—
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
,
no determiner
.
so adverb
,
exactly adverb
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
brain noun, singular or mass
is verb, 3rd person singular present
this determiner
device noun, singular or mass
reading noun, singular or mass
and coordinating conjunction
how wh-adverb
does verb, 3rd person singular present
it personal pronoun
work verb, non-3rd person singular present
?
um proper noun, singular
what wh-pronoun
was verb, past tense
that preposition or subordinating conjunction
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
this determiner
what wh-pronoun
is verb, 3rd person singular present
that preposition or subordinating conjunction
i personal pronoun
love verb, non-3rd person singular present
these determiner
okay adjective
that wh-determiner
have verb, non-3rd person singular present
to to
tell verb, base form
you personal pronoun
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
the determiner
the determiner
time verb, base form
it personal pronoun
is verb, 3rd person singular present
on preposition or subordinating conjunction
that determiner
time noun, singular or mass
like preposition or subordinating conjunction
what wh-determiner
part noun, singular or mass
of preposition or subordinating conjunction
your possessive pronoun
past adjective
it personal pronoun
‘s verb, 3rd person singular present
just adverb
i personal pronoun
mean verb, non-3rd person singular present
everywhere adverb
from preposition or subordinating conjunction
touring verb, gerund or present participle
to to
just adverb
How to use «what part of «no» do you not understand» in a sentence?
-
The great secret of true success, of true happiness, is this: the man or woman who asks for no return, the perfectly unselfish person, is the most successful.
-
There is no real and true joy of if that joy is not imbued with love. Love cannot exist without joy.
-Torkom Saraydarian-
-
Love expects no reward. Love knows no fear. Love Divine gives — does not demand. Love thinks no evil; imputes no motive. To Love is to share and serve.
-Sivananda-
-
I just enjoy working with really wonderful actors and amazing creative people and I hope to keep doing that, no matter where.
-Stana Katic-
-
No one is saved by good works (Ephesians 2:8-9), but every Christian has been saved for good works (verse 10).
-David Jeremiah-
-
There is for a free man no occupation more worthy and delightful than to contemplate the beauteous works of nature and honor the infinite wisdom and goodness of God.
-John Ray-
-
Oh, I’ve a love, a true, true love, who waits upon yon shore… and if my love won’t be my love, then I will live no more.
-Libba Bray-
-
The French are funny, sex is funny, and comedies are funny, yet no French sex comedies are funny.
-Matt Groening-
Special thanks to The Free Dictionary
and azquotes
for providing us with the information used in this web page
На основании Вашего запроса эти примеры могут содержать грубую лексику.
На основании Вашего запроса эти примеры могут содержать разговорную лексику.
разве вы не понимаете
ты не понял
вы не поняли
ты не поняла
тебе не понятна
вам непонятна
тебе непонятна
вам непонятно
непонятного
разве вы не видите
Неужели вы не понимаете
Разве ты не понимаешь
Неужели ты не понимаешь
тебе непонятно
Почему вы не понимаете
Предложения
How those of you who have given your widow’s last mite to this Ministry, do you not understand?
Как те из вас, кто отдал вашу последнюю лепту вдовы этому Служительству, разве вы не понимаете?
What about «no working with family» do you not understand?
Что на счет «не работать со своей семьей» ты не понял?
Honey, what part of the fact that she has class tonight do you not understand?
Милый, какую часть из того факта, что у нее сегодня занятия, ты не понял?
Which of those two words do you not understand?
Is that your answer or do you not understand the question?
Alfred What do you not understand?
What part of «off the books» do you not understand?
What do you not understand about me saying,
What part of «sit down» do you not understand?
What part of «swoop» do you not understand?
Birkhoff, what part of «do not disturb» do you not understand?
Биркофф, что именно из слов «Не беспокоить» ты не понял?
Justin, do you not understand?
What part of stop do you not understand?
Mother, do you not understand?
What part of no do you not understand?
Why therefore do you not understand my word?
How do you not understand this…
What, do you not understand?
Which pieces do you not understand?
What part of scientific method do you not understand?
Интересно, какие же аспекты Вашего научного метода я не понимаю?
Предложения, которые содержат do you not understand
Результатов: 302. Точных совпадений: 302. Затраченное время: 340 мс
Documents
Корпоративные решения
Спряжение
Синонимы
Корректор
Справка и о нас
Индекс слова: 1-300, 301-600, 601-900
Индекс выражения: 1-400, 401-800, 801-1200
Индекс фразы: 1-400, 401-800, 801-1200
-
#1
What part of the chapter don’t you understand?What part of the chapter do you not understand?
Any nuance between them?
Last edited by a moderator: Dec 15, 2014
-
#2
What part of the chapter don’t you understand?
This is the ordinary question I’d expect to hear.
What part of the chapter do you not understand?
The speaker seems to be emphasizing «not» for some reason.
Last edited: Dec 15, 2014
-
#3
Thank you very much, i got it
-
#5
Actually it could equally be the «you» which is being emphasized.
Student A : I don’t understand x
Student B : I don’t understand y
Teacher (to Student C) : What part of the chapter do you not understand?
There may or may not be a difference between the two forms. It all depends on whether the speaker uses intonation/emphasis.
Русский
English
Český
Deutsch
Español
عربى
Български
বাংলা
Dansk
Ελληνικά
Suomi
Français
עִברִית
हिंदी
Hrvatski
Magyar
Bahasa indonesia
Italiano
日本語
한국어
മലയാളം
मराठी
Bahasa malay
Nederlands
Norsk
Polski
Português
Română
Slovenský
Slovenščina
Српски
Svenska
தமிழ்
తెలుగు
ไทย
Tagalog
Turkce
Українська
اردو
Tiếng việt
中文
Примеры использования
Do you not understand
в предложениях и их переводы
можешь приехать ко мне, прошу, прямо сейчас» ты не понял?
Биркофф, что именно из слов»
Не
беспокоить» ты не понял?
Результатов: 75,
Время: 0.0232
Пословный перевод
Фразы по алфавиту
Поиск в Английско-Русском словаре по букве
Лучшие запросы из словаря
Английский
—
Русский
Русский
—
Английский
How do I understand the Hindley-Milner rules?
Hindley-Milner is a set of rules in the form of sequent calculus (not natural deduction) that demonstrates that we can deduce the (most general) type of a program from the construction of the program without explicit type declarations.
The symbols and notation
First, let’s explain the symbols, and discuss operator precedence
-
𝑥 is an identifier (informally, a variable name).
-
: means is a type of (informally, an instance of, or «is-a»).
-
𝜎 (sigma) is an expression that is either a variable or function.
-
thus 𝑥:𝜎 is read «𝑥 is-a 𝜎«
-
∈ means «is an element of»
-
𝚪 (Gamma) is an environment.
-
⊦ (the assertion sign) means asserts (or proves, but contextually «asserts» reads better.)
-
𝚪 ⊦ 𝑥 : 𝜎 is thus read «𝚪 asserts that 𝑥, is-a 𝜎«
-
𝑒 is an actual instance (element) of type 𝜎.
-
𝜏 (tau) is a type: either basic, variable (𝛼), functional 𝜏→𝜏’, or product 𝜏×𝜏’ (product is not used here)
-
𝜏→𝜏’ is a functional type where 𝜏 and 𝜏’ are potentially different types.
-
𝜆𝑥.𝑒 means 𝜆 (lambda) is an anonymous function that takes an argument, 𝑥, and returns an expression, 𝑒.
-
let 𝑥 = 𝑒₀ in 𝑒₁ means in expression, 𝑒₁, substitute 𝑒₀ wherever 𝑥 appears.
-
⊑ means the prior element is a subtype (informally — subclass) of the latter element.
-
𝛼 is a type variable.
-
∀ 𝛼.𝜎 is a type, ∀ (for all) argument variables, 𝛼, returning 𝜎 expression
-
∉ free(𝚪) means not an element of the free type variables of 𝚪 defined in the outer context. (Bound variables are substitutable.)
Everything above the line is the premise, everything below is the conclusion (Per Martin-Löf)
Precedence, by example
I have taken some of the more complex examples from the rules and inserted redundant parentheses that show precedence:
-
𝑥 : 𝜎 ∈ 𝚪 could be written (𝑥 : 𝜎) ∈ 𝚪
-
𝚪 ⊦ 𝑥 : 𝜎 could be written 𝚪 ⊦ (𝑥 : 𝜎)
-
𝚪 ⊦ let 𝑥 = 𝑒₀ in 𝑒₁ : 𝜏
is equivalently
𝚪 ⊦ ((let (𝑥 = 𝑒₀) in 𝑒₁) : 𝜏) -
𝚪 ⊦ 𝜆𝑥.𝑒 : 𝜏→𝜏’ is equivalently 𝚪 ⊦ ((𝜆𝑥.𝑒) : (𝜏→𝜏’))
Then, large spaces separating assertion statements and other preconditions indicates a set of such preconditions, and finally the horizontal line separating premise from conclusion brings up the end of the precedence order.
The Rules
What follows here are English interpretations of the rules, each followed by a loose restatement and an explanation.
Variable
Given 𝑥 is a type of 𝜎 (sigma), an element of 𝚪 (Gamma),
conclude 𝚪 asserts 𝑥 is a 𝜎.
Put another way, in 𝚪, we know 𝑥 is of type 𝜎 because 𝑥 is of type 𝜎 in 𝚪.
This is basically a tautology. An identifier name is a variable or a function.
Function Application
Given 𝚪 asserts 𝑒₀ is a functional type and 𝚪 asserts 𝑒₁ is a 𝜏
conclude 𝚪 asserts applying function 𝑒₀ to 𝑒₁ is a type 𝜏’
To restate the rule, we know that function application returns type 𝜏’ because the function has type 𝜏→𝜏’ and gets an argument of type 𝜏.
This means that if we know that a function returns a type, and we apply it to an argument, the result will be an instance of the type we know it returns.
Function Abstraction
Given 𝚪 and 𝑥 of type 𝜏 asserts 𝑒 is a type, 𝜏’
conclude 𝚪 asserts an anonymous function, 𝜆 of 𝑥 returning expression, 𝑒 is of type 𝜏→𝜏’.
Again, when we see a function that takes 𝑥 and returns an expression 𝑒, we know it’s of type 𝜏→𝜏’ because 𝑥 (a 𝜏) asserts that 𝑒 is a 𝜏’.
If we know 𝑥 is of type 𝜏 and thus an expression 𝑒 is of type 𝜏’, then a function of 𝑥 returning expression 𝑒 is of type 𝜏→𝜏’.
Let variable declaration
Given 𝚪 asserts 𝑒₀, of type 𝜎, and 𝚪 and 𝑥, of type 𝜎, asserts 𝑒₁ of type 𝜏
conclude 𝚪 assertslet𝑥=𝑒₀in𝑒₁ of type 𝜏
Loosely, 𝑥 is bound to 𝑒₀ in 𝑒₁ (a 𝜏) because 𝑒₀ is a 𝜎, and 𝑥 is a 𝜎 that asserts 𝑒₁ is a 𝜏.
This means if we have an expression 𝑒₀ that is a 𝜎 (being a variable or a function), and some name, 𝑥, also a 𝜎, and an expression 𝑒₁ of type 𝜏, then we can substitute 𝑒₀ for 𝑥 wherever it appears inside of 𝑒₁.
Instantiation
Given 𝚪 asserts 𝑒 of type 𝜎’ and 𝜎’ is a subtype of 𝜎
conclude 𝚪 asserts 𝑒 is of type 𝜎
An expression, 𝑒 is of parent type 𝜎 because the expression 𝑒 is subtype 𝜎’, and 𝜎 is the parent type of 𝜎’.
If an instance is of a type that is a subtype of another type, then it is also an instance of that super-type — the more general type.
Generalization
Given 𝚪 asserts 𝑒 is a 𝜎 and 𝛼 is not an element of the free variables of 𝚪,
conclude 𝚪 asserts 𝑒, type for all argument expressions 𝛼 returning a 𝜎 expression
So in general, 𝑒 is typed 𝜎 for all argument variables (𝛼) returning 𝜎, because we know that 𝑒 is a 𝜎 and 𝛼 is not a free variable.
This means we can generalize a program to accept all types for arguments not already bound in the containing scope (variables that are not non-local). These bound variables are substitutable.
Putting it all together
Given certain assumptions (such as no free/undefined variables, a known environment, ) we know the types of:
- atomic elements of our programs (Variable),
- values returned by functions (Function Application),
- functional constructs (Function Abstraction),
- let bindings (Let Variable Declarations),
- parent types of instances (Instantiation), and
- all expressions (Generalization).
Conclusion
These rules combined allow us to prove the most general type of an asserted program, without requiring type annotations.