Logical biconditional (original) (raw)
Als Bikonditional, Bisubjunktion oder materiale Äquivalenz, manchmal (aber mehrdeutig) einfach nur Äquivalenz bezeichnet man * eine zusammengesetzte Aussage, die genau dann wahr ist, wenn ihre beiden Teilaussagen denselben Wahrheitswert haben, also entweder beide wahr oder beide falsch sind; * die entsprechend definierte Wahrheitswertfunktion; * das sprachliche Zeichen (den Junktor), mit dem diese beiden Teilaussagen zusammengesetzt werden.
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| dbo:abstract | Als Bikonditional, Bisubjunktion oder materiale Äquivalenz, manchmal (aber mehrdeutig) einfach nur Äquivalenz bezeichnet man * eine zusammengesetzte Aussage, die genau dann wahr ist, wenn ihre beiden Teilaussagen denselben Wahrheitswert haben, also entweder beide wahr oder beide falsch sind; * die entsprechend definierte Wahrheitswertfunktion; * das sprachliche Zeichen (den Junktor), mit dem diese beiden Teilaussagen zusammengesetzt werden. (de) In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective used to conjoin two statements P and Q to form the statement "P if and only if Q", where P is known as the antecedent, and Q the consequent. This is often abbreviated as "P iff Q". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔ or ⇔ may be represented in Unicode in various ways), a prefixed E "Epq" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to both and , and the XNOR (exclusive nor) boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis is false but the conclusion is true. In this case, the result is true for the conditional, but false for the biconditional. In the conceptual interpretation, P = Q means "All P's are Q's and all Q's are P's". In other words, the sets P and Q coincide: they are identical. However, this does not mean that P and Q need to have the same meaning (e.g., P could be "equiangular trilateral" and Q could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate). In the propositional interpretation, means that P implies Q and Q implies P; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as P could be "the triangle ABC has two equal sides" and Q could be "the triangle ABC has two equal angles". In general, the antecedent is the premise, or the cause, and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment, the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis. A common way of demonstrating a biconditional of the form is to demonstrate that and separately (due to its equivalence to the conjunction of the two converse conditionals). Yet another way of demonstrating the same biconditional is by demonstrating that and . When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the hypothesis and whose consequent is the thesis of the theorem. It is often said that the hypothesis is the sufficient condition of the thesis, and that the thesis is the necessary condition of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time. (en) Na Lógica e Matemática, a Lógica bicondicional (também conhecida como bicondicional material) é o Conectivo lógico de duas proposições afirmando "p se e somente se q", onde q é uma Hipótese (ou antecedente) e p é um conclusão (ou consequente). Isso é frequentemente abreviado p sse q. O operador é denotado usando uma seta de dupla implicação (↔), a prefixed E (Epq), um sinal de igualdade (=),um sinal de equivalência (≡), ou EQV. Isso é logicamente equivalente a (p → q) ∧ (q → p), ou o XNOR (nor exclusivo) operador da Álgebra_booleana.Isto é equivalente a "(não p ou q) e (não q ou p)". Também é logicamente equivalente a "(p e q) ou (não p e não q)",significando "os dois ou nenhum".A única diferença paraCondicional_material é o caso no qual a hipótese é falsa mas a conclusão é verdadeira. Neste caso, na condicional, o resultado é verdadeiro, contudo, na bicondicional o resultado é falso.Na interpretação conceitual, a = b significa "Todos os a 's são b 's e todos os b 's são a 's"; Em outras palavras, os conjuntos a e b coincidem: eles são idênticos. Isso não significa que todos os conceitos têm o mesmo significado. Exemplos: "triângulo" e "trilateral", "triângulo equiangular" e "triângulo equilátero". O antecedente é o "sujeito" e o consequente é o e predicado de uma afirmativa/ Proposição universal. Na interpretação proposicional, a ⇔ b significa que a implica b e b implica a; em outras palavras, que as proposições são equivalentes, o que é dizer, ambas são verdadeiras ou falsas ao mesmo tempo. Isso não significa que elas tem o mesmo significado. Exemplo: "O triângulo ABC tem dois lados iguais", e "O triângulo ABC tem 2 ângulos iguais". O antecedente é a premissa ou a causa e o consequente é a consequência. Quando uma implicação é traduzida por um julgamento hipotético (ou condicional) O antecedente é chamado de "hipótese (ou de condição) e o consequente é chamado de tese. Uma forma comum de se demonstrar um bicondicional é usar sua equivalência para a conjunção de duas condicionais ,em que há uma troca entre a hipótese e a conclusão, as demonstrando separadamente. (pt) |
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| rdfs:comment | Als Bikonditional, Bisubjunktion oder materiale Äquivalenz, manchmal (aber mehrdeutig) einfach nur Äquivalenz bezeichnet man * eine zusammengesetzte Aussage, die genau dann wahr ist, wenn ihre beiden Teilaussagen denselben Wahrheitswert haben, also entweder beide wahr oder beide falsch sind; * die entsprechend definierte Wahrheitswertfunktion; * das sprachliche Zeichen (den Junktor), mit dem diese beiden Teilaussagen zusammengesetzt werden. (de) In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective used to conjoin two statements P and Q to form the statement "P if and only if Q", where P is known as the antecedent, and Q the consequent. This is often abbreviated as "P iff Q". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔ or ⇔ may be represented in Unicode in various ways), a prefixed E "Epq" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to both and , and the XNOR (exclusive nor) boolean operator, which means "both or neither". (en) Na Lógica e Matemática, a Lógica bicondicional (também conhecida como bicondicional material) é o Conectivo lógico de duas proposições afirmando "p se e somente se q", onde q é uma Hipótese (ou antecedente) e p é um conclusão (ou consequente). Isso é frequentemente abreviado p sse q. O operador é denotado usando uma seta de dupla implicação (↔), a prefixed E (Epq), um sinal de igualdade (=),um sinal de equivalência (≡), ou EQV. Isso é logicamente equivalente a (p → q) ∧ (q → p), ou o XNOR (nor exclusivo) operador da Álgebra_booleana.Isto é equivalente a "(não p ou q) e (não q ou p)". Também é logicamente equivalente a "(p e q) ou (não p e não q)",significando "os dois ou nenhum".A única diferença paraCondicional_material é o caso no qual a hipótese é falsa mas a conclusão é verdadeira. Nes (pt) |
| rdfs:label | Bikonditional (de) Logical biconditional (en) Conectivo lógico bicondicional (pt) |
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