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| Truth function ： ウィキペディア英語版 |
Material conditional.'' IMHO by no means the material conditional may not be referred to or abbreviated as the adjective "truth functional", omitting a noun like "conditional" or "implication". Seems more like an internal link spamming rather than an appropriate dab hatnote. --Incnis Mrsi -->In mathematical logic, a truth function is a function from a set of truth values to truth values. Classically the domain and range of a truth function are , but they may have any number of truth values, including an infinity of these.A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true ''if, and only if'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence:"''Mary believes that Al Gore was President of the USA on April 20, 2000''"is true while:"''Mary believes that the moon is made of green cheese''"is false. In both cases, each component sentence (i.e. "''Al Gore was president of the USA on April 20, 2000''" and "''the moon is made of green cheese''") is false, but each compound sentence formed by prefixing the phrase "''Mary believes that''" differs in truth-value. That is, the truth-value of a sentence of the form "''Mary believes that...''" is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply ''operator'' since it is unary) is non-truth-functional.The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulas may be interpreted as either true or false.== Table of binary truth functions ==In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.
In mathematical logic, a truth function is a function from a set of truth values to truth values. Classically the domain and range of a truth function are , but they may have any number of truth values, including an infinity of these.
A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true ''if, and only if'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.
Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence
:"''Mary believes that Al Gore was President of the USA on April 20, 2000''"
is true while
:"''Mary believes that the moon is made of green cheese''"
is false. In both cases, each component sentence (i.e. "''Al Gore was president of the USA on April 20, 2000''" and "''the moon is made of green cheese''") is false, but each compound sentence formed by prefixing the phrase "''Mary believes that''" differs in truth-value. That is, the truth-value of a sentence of the form "''Mary believes that...''" is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply ''operator'' since it is unary) is non-truth-functional.
The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulas may be interpreted as either true or false.
== Table of binary truth functions ==
In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.
抄文引用元・出典: フリー百科事典『 In mathematical logic, a truth function is a function from a set of truth values to truth values. Classically the domain and range of a truth function are , but they may have any number of truth values, including an infinity of these.A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true ''if, and only if'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence:"''Mary believes that Al Gore was President of the USA on April 20, 2000''"is true while:"''Mary believes that the moon is made of green cheese''"is false. In both cases, each component sentence (i.e. "''Al Gore was president of the USA on April 20, 2000''" and "''the moon is made of green cheese''") is false, but each compound sentence formed by prefixing the phrase "''Mary believes that''" differs in truth-value. That is, the truth-value of a sentence of the form "''Mary believes that...''" is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply ''operator'' since it is unary) is non-truth-functional.The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulas may be interpreted as either true or false.== Table of binary truth functions ==In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.">ウィキペディア（Wikipedia）』
■In mathematical logic, a truth function is a function from a set of truth values to truth values. Classically the domain and range of a truth function are , but they may have any number of truth values, including an infinity of these.A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true ''if, and only if'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence:"''Mary believes that Al Gore was President of the USA on April 20, 2000''"is true while:"''Mary believes that the moon is made of green cheese''"is false. In both cases, each component sentence (i.e. "''Al Gore was president of the USA on April 20, 2000''" and "''the moon is made of green cheese''") is false, but each compound sentence formed by prefixing the phrase "''Mary believes that''" differs in truth-value. That is, the truth-value of a sentence of the form "''Mary believes that...''" is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply ''operator'' since it is unary) is non-truth-functional.The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulas may be interpreted as either true or false.== Table of binary truth functions ==In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.">ウィキペディアで「Material conditional.'' IMHO by no means the material conditional may not be referred to or abbreviated as the adjective "truth functional", omitting a noun like "conditional" or "implication". Seems more like an internal link spamming rather than an appropriate dab hatnote. --Incnis Mrsi -->In mathematical logic, a truth function is a function from a set of truth values to truth values. Classically the domain and range of a truth function are , but they may have any number of truth values, including an infinity of these.A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true ''if, and only if'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence:"''Mary believes that Al Gore was President of the USA on April 20, 2000''"is true while:"''Mary believes that the moon is made of green cheese''"is false. In both cases, each component sentence (i.e. "''Al Gore was president of the USA on April 20, 2000''" and "''the moon is made of green cheese''") is false, but each compound sentence formed by prefixing the phrase "''Mary believes that''" differs in truth-value. That is, the truth-value of a sentence of the form "''Mary believes that...''" is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply ''operator'' since it is unary) is non-truth-functional.The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulas may be interpreted as either true or false.== Table of binary truth functions ==In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.」の詳細全文を読む
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