"De Morgan's law" meaning in All languages combined

{
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "propositional logic"
      },
      "expansion": "propositional logic",
      "name": "l"
    }
  ],
  "etymology_text": "Named after the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal propositional logic.",
  "forms": [
    {
      "form": "De Morgan's laws",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "head": "De Morgan's law"
      },
      "expansion": "De Morgan's law (plural De Morgan's laws)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Logic",
          "orig": "en:Logic",
          "parents": [
            "Formal sciences",
            "Philosophy",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "27 27 9 9 29",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "ref": "2004 August, J. L. Schellenberg, “The Atheist’s Free Will Offence”, in International Journal for Philosophy of Religion, volume 56, № 1, pages 11-12",
          "text": "Let ‘F’ stand for the state of affairs that consists in finite persons possessing and exercising free will. Let ‘p’ stand for ‘God exists’; ‘q’ for ‘F obtains’; ‘r’ for ‘F poses a serious risk of evil’; and ‘s’ for ‘There is no option available to God that counters F.’ With this in place, the argument may be formalized as follows:\n(1) [(p & q) & r] → s Premiss\n(2) ~s Premiss\n(3) ~[(p & q) & r] 1, 2 MT\n(4) ~(p & q) v ~r 3 DM\n(5) r Premiss\n(6) ~(p & q) 4, 5 DS\n(7) ~p v ~q 6 DM\n(3) follows from the conjunction of (1) and (2) by modus tollens; De Morgan’s law applied to (3) yields (4); (4) and (5) together lead to (6) by disjunctive syllogism; and another application of De Morgan’s law takes us from (6) to the final conclusion, according to which either God exists or there is free will (but not both).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Either of two laws in formal logic which state that:",
        "The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)"
      ],
      "id": "en-De_Morgan's_law-en-noun-CnPE8vXX",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "logic",
          "logic"
        ],
        [
          "logic",
          "logic#English"
        ],
        [
          "negation",
          "negation#English"
        ],
        [
          "conjunction",
          "conjunction#English"
        ],
        [
          "disjunction",
          "disjunction#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics, logic) Either of two laws in formal logic which state that:",
        "The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)"
      ],
      "topics": [
        "human-sciences",
        "logic",
        "mathematics",
        "philosophy",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Logic",
          "orig": "en:Logic",
          "parents": [
            "Formal sciences",
            "Philosophy",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "27 27 9 9 29",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "Either of two laws in formal logic which state that:",
        "The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)"
      ],
      "id": "en-De_Morgan's_law-en-noun-vo80-DKR",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "logic",
          "logic"
        ],
        [
          "logic",
          "logic#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics, logic) Either of two laws in formal logic which state that:",
        "The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)"
      ],
      "topics": [
        "human-sciences",
        "logic",
        "mathematics",
        "philosophy",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "glosses": [
        "Either of two laws in set theory which state that:",
        "The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′"
      ],
      "id": "en-De_Morgan's_law-en-noun-3g-ivh2v",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "set theory",
          "set theory#English"
        ],
        [
          "complement",
          "complement#English"
        ],
        [
          "union",
          "union#English"
        ],
        [
          "intersection",
          "intersection#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Either of two laws in set theory which state that:",
        "The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "glosses": [
        "Either of two laws in set theory which state that:",
        "The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′"
      ],
      "id": "en-De_Morgan's_law-en-noun-7UZ1ijK3",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "set theory",
          "set theory#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Either of two laws in set theory which state that:",
        "The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "27 27 9 9 29",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        },
        {
          "_dis": "21 21 9 9 40",
          "kind": "other",
          "name": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        },
        {
          "_dis": "24 24 11 11 31",
          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)"
      ],
      "id": "en-De_Morgan's_law-en-noun-v57V0I7R",
      "links": [
        [
          "mathematics",
          "mathematics"
        ]
      ],
      "raw_glosses": [
        "(mathematics, loosely) Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)"
      ],
      "synonyms": [
        {
          "_dis1": "6 6 7 7 74",
          "tags": [
            "initialism"
          ],
          "word": "DM"
        }
      ],
      "tags": [
        "broadly"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "sounds": [
    {
      "ipa": "/dɨˈmɔɹɡɪnzˌlɔ/",
      "tags": [
        "US"
      ]
    },
    {
      "enpr": "dēmôrʹgĭnz.lô'",
      "tags": [
        "US"
      ]
    }
  ],
  "translations": [
    {
      "_dis1": "42 42 5 5 6",
      "code": "cs",
      "lang": "Czech",
      "sense": "law of formal logic",
      "tags": [
        "masculine"
      ],
      "word": "De Morganův zákon"
    },
    {
      "_dis1": "42 42 5 5 6",
      "code": "nl",
      "lang": "Dutch",
      "sense": "law of formal logic",
      "tags": [
        "feminine"
      ],
      "word": "wet van De Morgan"
    },
    {
      "_dis1": "42 42 5 5 6",
      "code": "fi",
      "lang": "Finnish",
      "sense": "law of formal logic",
      "word": "De Morganin laki"
    },
    {
      "_dis1": "42 42 5 5 6",
      "code": "de",
      "lang": "German",
      "sense": "law of formal logic",
      "tags": [
        "neuter"
      ],
      "word": "De Morgen'sches Gesetz"
    },
    {
      "_dis1": "42 42 5 5 6",
      "code": "pl",
      "lang": "Polish",
      "sense": "law of formal logic",
      "tags": [
        "neuter"
      ],
      "word": "prawo De Morgana"
    },
    {
      "_dis1": "4 4 35 35 22",
      "code": "cs",
      "lang": "Czech",
      "sense": "law of set theory",
      "tags": [
        "masculine"
      ],
      "word": "De Morganův zákon"
    },
    {
      "_dis1": "4 4 35 35 22",
      "code": "nl",
      "lang": "Dutch",
      "sense": "law of set theory",
      "tags": [
        "feminine"
      ],
      "word": "wet van De Morgan"
    },
    {
      "_dis1": "4 4 35 35 22",
      "code": "fi",
      "lang": "Finnish",
      "sense": "law of set theory",
      "word": "De Morganin laki"
    },
    {
      "_dis1": "4 4 35 35 22",
      "code": "de",
      "lang": "German",
      "sense": "law of set theory",
      "tags": [
        "neuter"
      ],
      "word": "De Morgen'sches Gesetz"
    },
    {
      "_dis1": "4 4 35 35 22",
      "code": "pl",
      "lang": "Polish",
      "sense": "law of set theory",
      "tags": [
        "neuter"
      ],
      "word": "prawo De Morgana"
    }
  ],
  "word": "De Morgan's law"
}

{
  "categories": [
    "English countable nouns",
    "English entries with incorrect language header",
    "English entries with language name categories using raw markup",
    "English eponyms",
    "English lemmas",
    "English multiword terms",
    "English nouns",
    "English terms with IPA pronunciation",
    "English terms with non-redundant non-automated sortkeys"
  ],
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "propositional logic"
      },
      "expansion": "propositional logic",
      "name": "l"
    }
  ],
  "etymology_text": "Named after the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal propositional logic.",
  "forms": [
    {
      "form": "De Morgan's laws",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "head": "De Morgan's law"
      },
      "expansion": "De Morgan's law (plural De Morgan's laws)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English terms with quotations",
        "Quotation templates to be cleaned",
        "en:Logic",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "2004 August, J. L. Schellenberg, “The Atheist’s Free Will Offence”, in International Journal for Philosophy of Religion, volume 56, № 1, pages 11-12",
          "text": "Let ‘F’ stand for the state of affairs that consists in finite persons possessing and exercising free will. Let ‘p’ stand for ‘God exists’; ‘q’ for ‘F obtains’; ‘r’ for ‘F poses a serious risk of evil’; and ‘s’ for ‘There is no option available to God that counters F.’ With this in place, the argument may be formalized as follows:\n(1) [(p & q) & r] → s Premiss\n(2) ~s Premiss\n(3) ~[(p & q) & r] 1, 2 MT\n(4) ~(p & q) v ~r 3 DM\n(5) r Premiss\n(6) ~(p & q) 4, 5 DS\n(7) ~p v ~q 6 DM\n(3) follows from the conjunction of (1) and (2) by modus tollens; De Morgan’s law applied to (3) yields (4); (4) and (5) together lead to (6) by disjunctive syllogism; and another application of De Morgan’s law takes us from (6) to the final conclusion, according to which either God exists or there is free will (but not both).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Either of two laws in formal logic which state that:",
        "The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "logic",
          "logic"
        ],
        [
          "logic",
          "logic#English"
        ],
        [
          "negation",
          "negation#English"
        ],
        [
          "conjunction",
          "conjunction#English"
        ],
        [
          "disjunction",
          "disjunction#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics, logic) Either of two laws in formal logic which state that:",
        "The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)"
      ],
      "topics": [
        "human-sciences",
        "logic",
        "mathematics",
        "philosophy",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Logic",
        "en:Mathematics"
      ],
      "glosses": [
        "Either of two laws in formal logic which state that:",
        "The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "logic",
          "logic"
        ],
        [
          "logic",
          "logic#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics, logic) Either of two laws in formal logic which state that:",
        "The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)"
      ],
      "topics": [
        "human-sciences",
        "logic",
        "mathematics",
        "philosophy",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Mathematics"
      ],
      "glosses": [
        "Either of two laws in set theory which state that:",
        "The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "set theory",
          "set theory#English"
        ],
        [
          "complement",
          "complement#English"
        ],
        [
          "union",
          "union#English"
        ],
        [
          "intersection",
          "intersection#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Either of two laws in set theory which state that:",
        "The complement of a union is the intersection of the complements; as expressed by: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Mathematics"
      ],
      "glosses": [
        "Either of two laws in set theory which state that:",
        "The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "set theory",
          "set theory#English"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Either of two laws in set theory which state that:",
        "The complement of an intersection is the union of the complements; as expressed by: (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Mathematics"
      ],
      "glosses": [
        "Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ]
      ],
      "raw_glosses": [
        "(mathematics, loosely) Any of various laws similar to De Morgan’s laws for set theory and logic; for example: ¬∀𝑥 𝑃(𝑥) ⇔ ∃𝑥 ¬𝑃(𝑥)"
      ],
      "tags": [
        "broadly"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "sounds": [
    {
      "ipa": "/dɨˈmɔɹɡɪnzˌlɔ/",
      "tags": [
        "US"
      ]
    },
    {
      "enpr": "dēmôrʹgĭnz.lô'",
      "tags": [
        "US"
      ]
    }
  ],
  "synonyms": [
    {
      "tags": [
        "initialism"
      ],
      "word": "DM"
    }
  ],
  "translations": [
    {
      "code": "cs",
      "lang": "Czech",
      "sense": "law of formal logic",
      "tags": [
        "masculine"
      ],
      "word": "De Morganův zákon"
    },
    {
      "code": "nl",
      "lang": "Dutch",
      "sense": "law of formal logic",
      "tags": [
        "feminine"
      ],
      "word": "wet van De Morgan"
    },
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "law of formal logic",
      "word": "De Morganin laki"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "law of formal logic",
      "tags": [
        "neuter"
      ],
      "word": "De Morgen'sches Gesetz"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "law of formal logic",
      "tags": [
        "neuter"
      ],
      "word": "prawo De Morgana"
    },
    {
      "code": "cs",
      "lang": "Czech",
      "sense": "law of set theory",
      "tags": [
        "masculine"
      ],
      "word": "De Morganův zákon"
    },
    {
      "code": "nl",
      "lang": "Dutch",
      "sense": "law of set theory",
      "tags": [
        "feminine"
      ],
      "word": "wet van De Morgan"
    },
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "law of set theory",
      "word": "De Morganin laki"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "law of set theory",
      "tags": [
        "neuter"
      ],
      "word": "De Morgen'sches Gesetz"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "law of set theory",
      "tags": [
        "neuter"
      ],
      "word": "prawo De Morgana"
    }
  ],
  "word": "De Morgan's law"
}