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.
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:
- ,
- где
- 1) предложения с предшествующими им верификаторами – это известные факты,
- 2) предложение с предшествующим ему верификатором – это новый (получаемый) факт.
- 3) символ – 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.
Examples
- Менее формальные формулировки "Modus ponens" и пояснения к ним.
- Формулировка
- It is known that [Besides] it is known that "" implies "". It is known that
- Примеры
- It is known that I am a man. Besides, it is known that "I am a man." implies "I am mortal.". 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". 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 [Besides,] It is known that if [then] It is known that
- Примеры
- It is known that I am a man. Besides, it is known that if I am a man, [then] I am mortal. 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. It is known that 0 + 1 < 1 + 1.
- Формулировка
- It is known that [Besides,] it is known if It is known that
- Примеры
- It is known that I am a man. Besides, it is known that I am mortal if I am a man. 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. It is known that 0 + 1 < 1 + 1.
Examples
- It is known that "" implies "". [Besides] it is known that It is known that
- Examples
- It is known that "I am a man." implies "I am mortal." Besides, it is known that I am a man. 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". It is known that 0 + 1 < 1 + 1.
- It is known that if [then] [Besides,] it is known that It is known that
- Examples
- It is known that if I am a man, [then] I am mortal. Besides, it is known that I am man. 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. It is known that 0 + 1 < 1 + 1.
- It is known that if [Besides,] it is known that It is known that
- Examples
- It is known that I am mortal if I am a man. Besides, it is known that I am a man. It is known I am mortal.
- It is known that 0 + 1 < 1 + 1 if 0 < 1. Besides, it is known that 0 < 1. It is known that 0 + 1 < 1 + 1.
Example
- It is known that [Besides] "" is known to imply "". It is known that
- Examples
- It is known that I am a man. Besides, "I am a man." is known to imply "I am mortal.". It is known that I am mortal.
- It is known that 0 < 1. Besides, "0 < 1" is known to imply "0 + 1 < 1 + 1". It is known that 0 + 1 < 1 + 1.
- It is known that [Besides,] if then It is known that
- Examples
- It is known that I am a man. Besides, if I am a man, then I am mortal. It is known that I am mortal.
- It is known that 0 < 1. Besides, if 0 < 1, then 0 + 1 < 1 + 1. It is known that 0 + 1 < 1 + 1.
Example
- "" is known to imply "". Moreover It is known that
- Examples
- "I am a man." is known to imply "I am mortal.". Moreover I am a man. It is known that I am mortal.
- "0 < 1" is known to imply "0 + 1 < 1 + 1". Moreover 0 < 1. It is known that 0 + 1 < 1 + 1.
- If then Moreover It is known that
- Examples
- If I am a man, then I am mortal. Moreover I am a man. It is known that I am mortal.
- If 0 < 1, then 0 + 1 < 1 + 1. Moreover 0 < 1. It is known that 0 + 1 < 1 + 1.
- It is known that [Besides,] it is known that "" means "". It is known that
- 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.". 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". It is known that 1 > 0.
- It is known that [Besides,] it is known that if and only if It is known that
- 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. 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. It is known that 1 > 0.
- It is known that [Besides,] it is known that just in case It is known that
- Examples
- It is known that I eat varenyky. Besides, it is known that varenyky is eaten by me just in case I eat varenyky. 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. It is known that 1 > 0.
- It is known that "" means "". [Besides,] it is known that It is known that
- Examples
- It is known that "I eat varenyky." means "Varenyky is eaten by me.". Besides, it is known that I eat varenyky. 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. It is known that 1 > 0.
- It is known that if and only if [Besides,] it is known that It is known that
- 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. 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. It is known that 1 > 0.
- It is known that [Besides,] "" is known to mean "". It is known that
- Examples
- It is known that I eat varenyky. Besides, "I eat varenyky." is known to mean "Varenyky is eaten by me.". It is known that varenyky is eaten by me.
- It is known that 0 < 1. Besides, "0 < 1" is known to mean "1 > 0". It is known that 1 > 0.
- "" is known to mean "". Moreover It is known that
- Examples
- "I eat varenyky." is known to mean "Varenyky is eaten by me.". Moreover I eat varenyky. It is known that varenyky is eaten by me.
- "0 < 1" is known to mean "1 > 0". Moreover 0 < 1. It is known that 1 > 0.
0. Математические предложения, каталогизируемые как элементы множества знаний, подразделяются на аксиомы, определения и теоремы.
- Если математическому предложению предшествует верификатор , тогда это предложение каталогизируется как аксиома.
- Если математическому предложению предшествует верификатор , тогда это предложение каталогизируется как определение.
- Если математическому предложению предшествует верификатор , 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:
- ,
- где
- предложения с верификаторами - это известные элементы множества знаний,
- предложение с верификатором - это [новая = получаемая] теорема,
- символ - 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:
- Examples
- 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:
- Examples
- 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.
Вывод теоремы из верного предложения и аксиомы
- It is known that [Besides,] it is obvious that . Therefore, it is proved that .
- Вывод теоремы из аксиомы и аксиомы
- It is obvious that [Besides,] it is obvious that . Therefore, it is proved that
- Выводы теоремы из определения и аксиомы
- It is defined that [Besides,] it is obvious that Therefore, it is proved that
- Вывод теоремы из теоремы и аксиомы
- It is proved that [Besides,] it is obvious that Therefore, it is proved that
Вывод теоремы из верного предложения и теоремы
- It is known that [Besides,] it is proved that Therefore, it is proved that
- Вывод теоремы из аксиомы и теоремы
- It is obvious that [Besided,] it is proved that Therefore, it is proved that
- Вывод теоремы из определения и теоремы
- It is defined that [Besides,] it is proved that Therefore, it is proved that
- Вывод теоремы из теоремы и теоремы
- It is proved that [Besides,] it is proved that Therefore, it is proved that
Вывод теоремы из аксиомы и верной импликации
- It is obvious that [Besides,] it is known that Therefore, it is proved that
Вывод теоремы из определения и верной импликации
- It is defined that [Besides,] it is known that Therefore it is proved that
Вывод теоремы из теоремы и верной импликации
- It is proved that [Besides,] it is known that Therefore, it is proved that
Користувач:Галактион/Modus tollens
Галактион 14:55, 18 серпня 2009 (UTC)