Користувач:Галактион/Modus ponens

Подстраница "Користувач:Галактион/Modus ponens" создана для того, чтобы перенести информацию из раздела "Обговорення" статьи "Modus ponens". Галактион 14:48, 5 березня 2010 (UTC)

Я не владею Украинским языком, поэтому дополню статью "Modus ponens" некоторыми сведениями на Русском и Английском языках. В первую очередь указанные сведения предназначены для людей, которые интересуются математикой и её приложениями.

Начальные сведения (Нулевой уровень)

Предварительные замечания

0. Сведения подразделяются на знания (факты) и предположения (гипотезы).

Если математическому предложению предшествует верификатор "|–", то это предложение каталогизируется как элемент множества знаний (факт).

Если математическому предложению предшествует верификатор "h |–", то это предложение каталогизируется как элемент множества предположений (гипотеза).

1. Правилами логического вывода называются правила получения [новых] знаний из [известных] знаний.

Каждое правило вывода можно записать в виде:

$~\vdash {\mathfrak {a}}_{0},\quad \vdash {\mathfrak {a}}_{1},\quad ...,\quad \vdash {\mathfrak {a}}_{n}\quad \Rrightarrow \quad \vdash {\mathfrak {b}}$,

где

1) предложения $~{\mathfrak {a}}_{0},\ {\mathfrak {a}}_{1},...,{\mathfrak {a}}_{n}$ с предшествующими им верификаторами $~\vdash$ – это известные факты,

2) предложение $~{\mathfrak {b}}$ с предшествующим ему верификатором $~\vdash$ – это новый (получаемый) факт.

3) символ $~\Rrightarrow$ – это аналог Английского слова "therefore", а также Русского словосочетания "стало быть" или Русского слова "поэтому".

2. Modus ponens – это правило вывода, которое позволяет получать из двух [известных] фактов один [новый] факт.

Примечание

Существуют и другие правила вывода, которые позволяют получать из двух [известных] фактов один [новый] факт.

Формулировки "Modus ponens"

$~\vdash {\mathfrak {a}},\quad \vdash \ {\mathfrak {a\to b}}\quad \Rrightarrow \quad \vdash {\mathfrak {b}}$

$~\vdash {\mathfrak {a}},\quad \vdash \ {\mathfrak {a\to b}}\quad \therefore \quad \ \vdash {\mathfrak {b}}$

$~\vdash {\mathfrak {a}},\quad \vdash \ {\mathfrak {a\to b}}\quad \mathrm {Therefore} \quad \vdash {\mathfrak {b}}$

Examples

$~\vdash \ 0<1,\quad \vdash \ 0<1\ \to \ 0+1<1+1\quad \Rrightarrow \quad \vdash \ 0+1<1+1$

$~\vdash \ 0<1,\quad \vdash \ 0<1\ \to \ 0+1<1+1\quad \therefore \quad \ \vdash \ 0+1<1+1$

$~\vdash \ 0<1,\quad \vdash \ 0<1\ \to \ 0+1<1+1\quad \mathrm {Therefore} \quad \vdash \ 0+1<1+1$

Менее формальные формулировки "Modus ponens" и пояснения к ним.

Формулировка

It is known that $~{\mathfrak {a}}.$ [Besides] it is known that "$~{\mathfrak {a}}.$" implies "$~{\mathfrak {b}}.$". $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Примеры

It is known that I am a man. Besides, it is known that "I am a man." implies "I am mortal.". $~\Rrightarrow$ It is known that I am mortal.

Примечание

Предложение "It is known that "I am a man." implies "I am mortal."." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложение "It is known that I am a man." и предложение "It is known that I am mortal." - это предложения Английского предметного языка с верификаторами "It is known that...".

Данный пример показывает, что из верного предложения Английского предметного языка и верного предложения Английского метаязыка выводимо верное предложение Английского предметного языка.

It is known that 0 < 1. Besides, it is known that "0 < 1" implies "0 + 1 < 1 + 1". $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

Примечание

Предложение "It is known that "0 < 1" implies "0 + 1 < 1 + 1"." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложение "It is known that 0 < 1." и предложение "It is known that 0 + 1 < 1 + 1." - это предложения предметного математического языка с Английскими верификаторами "It is known that ..."

Данный пример показывает, что из верного предложения математического предметного языка и верного предложения Английского метаязыка выводимо верное предложение математического предметного языка.

It is known that 2 Na _(s) + 2 H₂O _(l) → 2 NaOH _(aq) + H₂ ↑. Besides, it is known that "2 Na _(s) + 2 H₂O _(l) → 2 NaOH _(aq) + H₂ ↑" implies "2 Na + 2 H₂O = 2 NaOH + H₂". Therefore, it is known that 2 Na + 2 H₂O = 2 NaOH + H₂.

Примечание

Предложение "It is known that "2 Na _(s) + 2 H₂O _(l) → 2 NaOH _(aq) + H₂ ↑" implies "2 Na + 2 H₂O = 2 NaOH + H₂"." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложения "It is known that 2 Na _(s) + 2 H₂O _(l) → 2 NaOH _(aq) + H₂ ↑." и "It is known that 2 Na + 2 H₂O = 2 NaOH + H₂." - это предметные предложения на языке моделирования химических процессов с верификаторами "It is known that ...".

Данный пример показывает, что из верного предметного предложения на языке моделирования химических процессов и верного предложения Английского метаязыка выводимо верное предметное предложение на языке моделирования химических процессов.

Формулировка

It is known that $~{\mathfrak {a}}.$ [Besides,] It is known that if $~{\mathfrak {a}},$ [then] $~{\mathfrak {b}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Примеры

It is known that I am a man. Besides, it is known that if I am a man, [then] I am mortal. $~\Rrightarrow$ It is known that I am mortal.

Примечание

Предложение "It is known that I am a man.", предложение "It is known that if I am a man, [then] I am mortal." и предложение "It is known that I am mortal." - это предложения Английского предметного языка с верификаторами "It is known that...".

It is known that 0 < 1. Besides, it is known that if 0 < 1, [then] 0 + 1 < 1 + 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

Формулировка

It is known that $~{\mathfrak {a}}.$ [Besides,] it is known $~{\mathfrak {b}}$ if $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Примеры

It is known that I am a man. Besides, it is known that I am mortal if I am a man. $~\Rrightarrow$ It is known that I am mortal.

Примечание

Предложение "It is known that I am a man.", предложение "It is known that I am mortal if I am a man." и предложение "It is known that I am mortal." - это предложения Английского предметного языка с верификаторами "It is known that...".

It is known that 0 < 1. Besides, it is known that 0 + 1 < 1 + 1 if 0 < 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash {\mathfrak {a}}\quad \Rrightarrow \quad \vdash {\mathfrak {b}}$

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash {\mathfrak {a}}\quad \therefore \quad \ \vdash {\mathfrak {b}}$

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash {\mathfrak {a}}\quad \mathrm {Therefore} \quad \vdash {\mathfrak {b}}$

Examples

$~\vdash \ 0<1\ \to \ 0+1<1+1,\quad \vdash \ 0<1\quad \Rrightarrow \quad \vdash \ 0+1<1+1$

$~\vdash \ 0<1\ \to \ 0+1<1+1,\quad \vdash \ 0<1\quad \therefore \quad \ \vdash \ 0+1<1+1$

$~\vdash \ 0<1\ \to \ 0+1<1+1,\quad \vdash \ 0<1\quad \mathrm {Therefore} \quad \vdash \ 0+1<1+1$

It is known that "$~{\mathfrak {a}}.$" implies "$~{\mathfrak {b}}.$". [Besides] it is known that $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that "I am a man." implies "I am mortal." Besides, it is known that I am a man. $~\Rrightarrow$ It is known that I am mortal.

It is known that "0 < 1" implies "0 + 1 < 1 + 1". Besides, it is known that "0 < 1". $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

It is known that if $~{\mathfrak {a}},$ [then] $~{\mathfrak {b}}.$ [Besides,] it is known that $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that if I am a man, [then] I am mortal. Besides, it is known that I am man. $~\Rrightarrow$ It is known that I am mortal.

It is known that if 0 < 1, [then] 0 + 1 < 1 + 1. Besides, it is known that 0 < 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

It is known that $~{\mathfrak {b}}$ if $~{\mathfrak {a}}.$ [Besides,] it is known that $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I am mortal if I am a man. Besides, it is known that I am a man. $~\Rrightarrow$ It is known I am mortal.

It is known that 0 + 1 < 1 + 1 if 0 < 1. Besides, it is known that 0 < 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

Другие формулировки "Modus ponens"

$~\vdash \ {\mathfrak {a}},\quad {\mathfrak {a}}\Rightarrow {\mathfrak {b}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

Example

$~\vdash \ 0<1,\quad 0<1\Rightarrow 0+1<1+1\quad \Rrightarrow \quad \vdash \ 0+1<1+1$

It is known that $~{\mathfrak {a}}.$ [Besides] "$~{\mathfrak {a}}.$" is known to imply "$~{\mathfrak {b}}.$". $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I am a man. Besides, "I am a man." is known to imply "I am mortal.". $~\Rrightarrow$ It is known that I am mortal.

It is known that 0 < 1. Besides, "0 < 1" is known to imply "0 + 1 < 1 + 1". $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

It is known that $~{\mathfrak {a}}.$ [Besides,] if $~{\mathfrak {a}},$ then $~{\mathfrak {b}}.\ \Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I am a man. Besides, if I am a man, then I am mortal. $~\Rrightarrow$ It is known that I am mortal.

It is known that 0 < 1. Besides, if 0 < 1, then 0 + 1 < 1 + 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

$~{\mathfrak {a}}\Rightarrow {\mathfrak {b}},\quad \vdash \ {\mathfrak {a}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

Example

$~0<1\Rightarrow 0+1<1+1,\quad \vdash \ 0<1\quad \Rrightarrow \quad \vdash \ 0+1<1+1$

"$~{\mathfrak {a}}.$" is known to imply "$~{\mathfrak {b}}.$". Moreover $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

"I am a man." is known to imply "I am mortal.". Moreover I am a man. $~\Rrightarrow$ It is known that I am mortal.

"0 < 1" is known to imply "0 + 1 < 1 + 1". Moreover 0 < 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

If $~{\mathfrak {a}},$ then $~{\mathfrak {b}}.$ Moreover $~{\mathfrak {a}}.\ \Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

If I am a man, then I am mortal. Moreover I am a man. $~\Rrightarrow$ It is known that I am mortal.

If 0 < 1, then 0 + 1 < 1 + 1. Moreover 0 < 1. $~\Rrightarrow$ It is known that 0 + 1 < 1 + 1.

Дополнение к правилу вывода "Modus ponens"

$~\vdash \ {\mathfrak {a}},\quad \vdash \ {\mathfrak {a}}\leftrightarrow {\mathfrak {b}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

It is known that $~{\mathfrak {a}}.$ [Besides,] it is known that "$~{\mathfrak {a}}.$" means "$~{\mathfrak {b}}.$". $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I eat varenyky. Besides, it is known that [the sentence] "I eat varenyky." means [the sentence] "Varenyky is eaten by me.". $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that 0 < 1. Besides, it is known that [the sentence] "0 < 1" means [the sentence] "1 > 0". $~\Rrightarrow$ It is known that 1 > 0.

It is known that $~{\mathfrak {a}}.$ [Besides,] it is known that $~{\mathfrak {a}}$ if and only if $~{\mathfrak {b}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I eat varenyky. Besides, it is known that I eat varenyky if and only if varenyky is eaten by me. $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that 0 < 1. Besides, it is known that 0 < 1 if and only if 1 > 0. $~\Rrightarrow$ It is known that 1 > 0.

$~\vdash \ {\mathfrak {a}},\quad \vdash \ {\mathfrak {b\leftrightarrow a}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

It is known that $~{\mathfrak {a}}.$ [Besides,] it is known that $~{\mathfrak {b}}$ just in case $~{\mathfrak {a}}.\ \Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I eat varenyky. Besides, it is known that varenyky is eaten by me just in case I eat varenyky. $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that 0 < 1. Besides, it is known that 1 > 0 just in case 0 < 1. $~\Rrightarrow$ It is known that 1 > 0.

$~\vdash \ {\mathfrak {a}}\leftrightarrow {\mathfrak {b}},\quad \vdash \ {\mathfrak {a}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

It is known that "$~{\mathfrak {a}}.$" means "$~{\mathfrak {b}}.$". [Besides,] it is known that $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that "I eat varenyky." means "Varenyky is eaten by me.". Besides, it is known that I eat varenyky. $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that "0 < 1" means "1 > 0". Besides, it is known that 0 < 1. $~\Rrightarrow$ It is known that 1 > 0.

It is known that $~{\mathfrak {a}}$ if and only if $~{\mathfrak {b}}.$ [Besides,] it is known that $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I eat varenyky if and only if varenyky is eaten by me. Besides, it is known that I eat varenyky. $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that 0 < 1 if and only if 1 > 0. Besides, it is known that 0 < 1. $~\Rrightarrow$ It is known that 1 > 0.

$~\vdash \ {\mathfrak {a}},\quad {\mathfrak {a}}\Leftrightarrow {\mathfrak {b}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

It is known that $~{\mathfrak {a}}.$ [Besides,] "$~{\mathfrak {a}}.$" is known to mean "$~{\mathfrak {b}}.$". $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

It is known that I eat varenyky. Besides, "I eat varenyky." is known to mean "Varenyky is eaten by me.". $~\Rrightarrow$ It is known that varenyky is eaten by me.

It is known that 0 < 1. Besides, "0 < 1" is known to mean "1 > 0". $~\Rrightarrow$ It is known that 1 > 0.

$~{\mathfrak {a}}\Leftrightarrow {\mathfrak {b}},\quad \vdash \ {\mathfrak {a}}\quad \Rrightarrow \quad \vdash \ {\mathfrak {b}}$

"$~{\mathfrak {a}}.$" is known to mean "$~{\mathfrak {b}}.$". Moreover $~{\mathfrak {a}}.$ $~\Rrightarrow$ It is known that $~{\mathfrak {b}}.$

Examples

"I eat varenyky." is known to mean "Varenyky is eaten by me.". Moreover I eat varenyky. $~\Rrightarrow$ It is known that varenyky is eaten by me.

"0 < 1" is known to mean "1 > 0". Moreover 0 < 1. $~\Rrightarrow$ It is known that 1 > 0.

Первый уровень

Предварительные замечания

0. Математические предложения, каталогизируемые как элементы множества знаний, подразделяются на аксиомы, определения и теоремы.

Если математическому предложению предшествует верификатор $~a\vdash$, тогда это предложение каталогизируется как аксиома.

Если математическому предложению предшествует верификатор $~d\vdash$, тогда это предложение каталогизируется как определение.

Если математическому предложению предшествует верификатор $~t\vdash$, тогда это предложение каталогизируется как теорема.

1. Правилами логического вывода называются правила получения теорем из известных знаний (аксиом, определений или теорем).

Каждое правило логического вывода можно записать в виде:

$~\vdash {\mathfrak {A}}_{0},\quad \vdash {\mathfrak {A}}_{1},\quad ...,\quad \vdash {\mathfrak {A}}_{n}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$,

где

предложения $~{\mathfrak {A}}_{0},\ {\mathfrak {A}}_{1},\ ...,\ {\mathfrak {A}}_{n}$ с верификаторами $~\vdash$ - это известные элементы множества знаний,

предложение $~{\mathfrak {B}}$ с верификатором $~t\vdash$ - это [новая = получаемая] теорема,

символ $~\Rrightarrow$ - это аналог Английского слова "Therefore".

2. Modus ponens - это правило логического вывода, которое позволяет получать из двух известных элементов множества знаний одну теорему.

Указанное правило вывода можно записать в виде:

$~\vdash {\mathfrak {A}},\quad \vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

Examples

$~\vdash \ 0<1,\quad \vdash \ 0<1\to 0+1<1+1\quad \Rrightarrow \quad t\vdash \ 0+1<1+1$

It is known that I am a man. Besides, it is known that if I am a man, [then] I am mortal. Therefore, it is proved that I am mortal.

Также указанное правило вывода можно записать в виде:

$~\vdash {\mathfrak {A\to B}},\quad \vdash {\mathfrak {A}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

Examples

$~\vdash \ 0<1\to 0+1<1+1,\quad \vdash \ 0<1\quad \Rrightarrow \quad t\vdash \ 0+1<1+1$

It is known that if I am a man, [then] I am mortal. Besides, it is known that I am a man. Therefore, it is proved that I am mortal.

Формулировки Modus ponens

Вывод теоремы $~{\mathfrak {B}}$ из верного предложения $~{\mathfrak {A}}$ и аксиомы $~{\mathfrak {A\to B}}$

$~\vdash {\mathfrak {A}},\quad a\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is known that $~{\mathfrak {A}}.$ [Besides,] it is obvious that $~{\mathfrak {A\to B}}.$. Therefore, it is proved that $~{\mathfrak {B}}.$.

Вывод теоремы $~{\mathfrak {B}}$ из аксиомы $~{\mathfrak {A}}$ и аксиомы $~{\mathfrak {A\to B}}$

$~a\vdash {\mathfrak {A}},\quad a\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is obvious that $~{\mathfrak {A}}.$ [Besides,] it is obvious that $~{\mathfrak {A\to B}}.$. Therefore, it is proved that $~{\mathfrak {B}}.$

Выводы теоремы $~{\mathfrak {B}}$ из определения $~{\mathfrak {A}}$ и аксиомы $~{\mathfrak {A\to B}}$

$~d\vdash {\mathfrak {A}},\quad a\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is defined that $~{\mathfrak {A}}.$ [Besides,] it is obvious that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из теоремы $~{\mathfrak {A}}$ и аксиомы $~{\mathfrak {A\to B}}$

$~t\vdash {\mathfrak {A}},\quad a\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is proved that $~{\mathfrak {A}}.$ [Besides,] it is obvious that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из верного предложения $~{\mathfrak {A}}$ и теоремы $~{\mathfrak {A\to B}}$

$~\vdash {\mathfrak {A}},\quad t\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is known that $~{\mathfrak {A}}.$ [Besides,] it is proved that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из аксиомы $~{\mathfrak {A}}$ и теоремы $~{\mathfrak {A\to B}}$

$~a\vdash {\mathfrak {A}},\quad t\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is obvious that $~{\mathfrak {A}}.$ [Besided,] it is proved that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из определения $~{\mathfrak {A}}$ и теоремы $~{\mathfrak {A\to B}}$

$~d\vdash {\mathfrak {A}},\quad t\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is defined that $~{\mathfrak {A}}.$ [Besides,] it is proved that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из теоремы $~{\mathfrak {A}}$ и теоремы $~{\mathfrak {A\to B}}$

$~t\vdash {\mathfrak {A}},\quad t\vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is proved that $~{\mathfrak {A}}.$ [Besides,] it is proved that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из аксиомы $~{\mathfrak {A}}$ и верной импликации $~{\mathfrak {A\to B}}$

$~a\vdash {\mathfrak {A}},\quad \vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is obvious that $~{\mathfrak {A}}.$ [Besides,] it is known that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из определения $~{\mathfrak {A}}$ и верной импликации $~{\mathfrak {A\to B}}$

$~d\vdash {\mathfrak {A}},\quad \vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is defined that $~{\mathfrak {A}}.$ [Besides,] it is known that $~{\mathfrak {A\to B}}.$ Therefore it is proved that $~{\mathfrak {B}}.$

Вывод теоремы $~{\mathfrak {B}}$ из теоремы $~{\mathfrak {A}}$ и верной импликации $~{\mathfrak {A\to B}}$

$~t\vdash {\mathfrak {A}},\quad \vdash {\mathfrak {A\to B}}\quad \Rrightarrow \quad t\vdash {\mathfrak {B}}$

It is proved that $~{\mathfrak {A}}.$ [Besides,] it is known that $~{\mathfrak {A\to B}}.$ Therefore, it is proved that $~{\mathfrak {B}}.$

См. также

Користувач:Галактион/Modus tollens

Галактион 14:55, 18 серпня 2009 (UTC)