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Lògica de primer ordre Rachunek predykatów pierwszego rzędu First-order logic Första ordningens logik Lehen mailako logika 一階述語論理 一阶逻辑 Lógica de primer orden Логика первого порядка منطق الرتبة الأولى Логіка першого порядку 1차 논리 Prädikatenlogik erster Stufe Teoria del primo ordine Logika predikat tingkat pertama Λογική πρώτου βαθμού Calcul des prédicats Lógica de primeira ordem Predikatkalkulo Predikátová logika prvního řádu
rdfs:comment
Unuaranga logiko, nomita ankaŭ predikatlogiko, aŭ predikatkalkulo, estas formala sistemo desegnita por studi la inferencon en la unuarangaj lingvaĵoj.​ La unuarangaj lingvaĵoj estas siavice formalaj lingvaĵoj kun kvantigiloj kiuj atingas nur unuopajn variablojn, kaj kun predikatoj kaj funkcioj kies argumentoj estas nur konstantoj aŭ unuopaj variabloj. La logiko unuaranga havas espriman povon superan al tiu de la propozicia logiko. Logiko de supera ordo estas formo de predikatkalkulo kiu estas distingata el la unuaranga logiko pere de aldonaj kvantigiloj kaj, foje, per pli forta semantiko. Логика первого порядка — формальное исчисление, допускающее высказывания относительно переменных, фиксированных функций и предикатов. Расширяет логику высказываний. Помимо логики первого порядка существуют также логики высших порядков, в которых кванторы могут применяться не только к переменным, но и к предикатам. Термины логика предикатов и исчисление предикатов могут означать как логику первого порядка, так и логики первого и высшего порядка вместе; в первом случае иногда говорится о чистой логике предикатов или чистом исчислении предикатов. A lógica de primeira ordem (LPO), conhecida também como cálculo de predicados de primeira ordem (CPPO), é um sistema lógico que estende a lógica proposicional (lógica sentencial) e que é estendida pela lógica de segunda ordem. As sentenças atômicas da lógica de primeira ordem têm o formato P (t1,…, tn) (um predicado com um ou mais "argumentos") ao invés de serem símbolos sentenciais sem estruturas. Una lógica de primer orden, también llamada lógica predicativa, lógica de predicados o cálculo de predicados, es un sistema formal diseñado para estudiar la inferencia en los lenguajes de primer orden.​ Los lenguajes de primer orden son, a su vez, lenguajes formales con cuantificadores que alcanzan solo a variables de individuo, y con predicados y funciones cuyos argumentos son solo constantes o variables de individuo.​ La lógica de primer orden tiene un poder expresivo superior al de la lógica proposicional. منطق الرتبة الأولى (First-order logic FOL) أو المنطق الإسنادي عبارة عن نظام للمنطق الرياضي يستخدم في الرياضيات والفلسفة والذكاء الصناعي وعلوم الحاسب. وهو يستخدم في التعبير عن الجمل المنطقية بشكل غير مبهم وبهذا يخالف عن اللغات الطبيعية والتي قد تحتوي على جمل مبهمة. ذلك يسهل الاستنتاج واجراء العمليات المنطقية على الجمل أو المعادلات التي تنشاء باستخدامه.و يعتبر منطق الرتبة الأولى هو تمديد منطق القضايا (منطق العبارات) propositional logic وذلك بإضافة القياس سواء كان عالمي أو وجودي. يعتبر بمنطق الرتبة الثانية تمديد لمنطق الرتبة الأولى وذلك بإضافة القياس على المجموعات. يدعى منطق الرتبة الأولى أحيانا : بمنطق الرتبة الأولى الإسنادي أو first-order predicate calculus (FOPC. First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. Lehen mailako logika, predikatuen logika, logika kuantifikatzailea edo predikatuen kalkulua ere deitzen dena, lehen ordenako hizkuntzen inferentzia aztertzeko diseinatutako sistema formala da. Predikatuak, haien propietateak eta eragiketak aztertzen dituen logika. Aldagai eta kuantifikatzaileen bidez lan egiten du. Predikatuen logikak proposizioen barne-egitura hartzen du kontuan. Lehen ordenako lengoaiak, era berean, banakako aldagaiei bakarrik eragiten dien zenbatzaileak dituzten dira, eta argumentuak, konstanteak edo aldagai indibidualak dituzten predikatuak eta funtzioak baino ez dira. Logika predikat tingkat pertama adalah sistem deduksi formal yang digunakan dalam matematika, filosofi, linguistika, dan ilmu komputer. Jika kalkulus proposisional membahas proposisi sederhana, LTP menambahkan predikat dan kuantor. Misalnya: * Sokrates adalah seorang manusia * Plato adalah seorang manusia Contoh berikut menjabarkan perbedaan kalkulus proposisional dan LTP: * Semua manusia perlu makan * Sokrates adalah manusia * Sokrates perlu makan Dalam kalkulus proposisional, ketiga kalimat di atas diterjemahkan sebagai: * A * B * C ( artinya "maka") * * * Första ordningens logik (FOL) är ett formellt deduktivt system som används i matematik, filosofi, lingvistik och datavetenskap. Det har ett flertal olika namn på engelska: first-order predicate calculus (FOPC), the lower predicate calculus, the language of first-order logic och predicate logic. Till skillnad från naturliga språk, som svenska, använder sig FOL av ett helt otvetydigt formellt språk som tolkas av matematiska strukturer. FOL är ett deduktivt system som går bortom satslogiken genom att tillåta kvantifiering av objekt inom en given domän. Man kan exempelvis med FOL uttrycka satsen "Varje individ har egenskapen P". 一階述語論理(英: first-order predicate logic)とは、個体の量化のみを許す述語論理 (predicate logic) である。述語論理とは、数理論理学における論理の数学的モデルの一つであり、命題論理を拡張したものである。個体の量化に加えて述語や関数の量化を許す述語論理を二階述語論理(英: second-order predicate logic)と呼び、さらなる一般化を加えた述語論理を高階述語論理(英: higher-order predicate logic)という。本項では主に一階述語論理について解説する。二階述語論理や高階述語論理についての詳細はそれぞれの記事を参照。 Η λογική πρώτου βαθμού είναι μια τυπική λογική που χρησιμοποιείται στα μαθηματικά, τη φιλοσοφία, τη γλωσσολογία και την επιστήμη υπολογιστών. Συναντάται με διάφορα ονόματα, όπως κατηγορηματικός λογισμός πρώτου βαθμού ή κατηγορηματική λογική. Η πρωτοβάθμια λογική διαφέρει από την προτασιακή λογική στη χρήση ποσοτικών τελεστών: κάθε ερμηνεία της λογικής πρώτου βαθμού περιλαμβάνει ένα πεδίο τιμών όπου κυμαίνονται οι ποσοτικοί τελεστές. 1차 논리(一次論理, 영어: first-order logic)는 원소에만 한정 기호를 가할 수 있고, 술어에는 한정 기호를 가할 수 없는 술어 논리이다. 명제 논리와 달리 변수에 대하여 한정 기호를 사용할 수 있으나, 2차 논리와 달리 변수들의 집합에 대하여 한정 기호를 사용할 수 없다. 1차 논리의 경우, (2차 논리와 달리) 괴델의 완전성 정리 · 콤팩트성 정리 · 뢰벤하임-스콜렘 정리와 같은 중요한 성질들이 성립한다. 이외에 1차 술어 논리, 1계 논리 등으로도 불린다. 간단히 술어 논리(predicate logic)라 하면 1차 논리를 가리키는 경우가 많다. Логіка першого порядку (числення предикатів) — це формальна система в математичній логіці, в якій допускаються висловлення відносно змінних, фіксованих функцій, і предикатів. Є розширенням логіки висловлювань. В свою чергу є частковим випадком . Le calcul des prédicats du premier ordre, ou calcul des relations, logique du premier ordre, logique quantificationnelle, ou tout simplement calcul des prédicats, est une formalisation du langage des mathématiques, proposée par Gottlob Frege, entre la fin du XIXe siècle et le début du XXe siècle. La logique du premier ordre comporte deux parties : Sur le plan syntaxique, les langages du premier ordre opposent deux grandes classes linguistiques : et . Les traits caractéristiques de la logique du premier ordre sont : Die Prädikatenlogik erster Stufe ist ein Teilgebiet der mathematischen Logik. Sie befasst sich mit der Struktur gewisser mathematischer Ausdrücke und dem logischen Schließen, mit dem man von derartigen Ausdrücken zu anderen gelangt. Dabei gelingt es, sowohl die Sprache als auch das Schließen rein syntaktisch, das heißt ohne Bezug zu mathematischen Bedeutungen, zu definieren.Das dadurch ermöglichte Zusammenspiel von rein syntaktischen Überlegungen einerseits und semantischen Betrachtungen andererseits führt zu wichtigen Erkenntnissen, die Bedeutung für die gesamte Mathematik haben, denn diese lässt sich mittels der Zermelo-Fraenkel-Mengenlehre in der Prädikatenlogik erster Stufe formulieren. Im Unterschied zur Aussagenlogik macht die Prädikatenlogik von Quantoren Gebrauch. Nella logica matematica, una teoria del primo ordine (o calcolo dei predicati) è un particolare sistema formale, cioè una teoria formale, in cui è possibile esprimere enunciati e dedurre le loro conseguenze logiche in modo del tutto formale e meccanico. La teoria del prim'ordine estende di fatto la logica proposizionale con l'introduzione di quantificatori esistenziali e universali, predicati, funzioni, variabili e costanti, che apportano maggiore potenza espressiva al calcolo dei predicati. Come per la logica proposizionale, la teoria del primo ordine può essere scissa in due parti separate: Predikátová logika prvního řádu je používaný v matematice, filozofii, lingvistice a informatice. Často se pro její označení používá kratší a méně přesný termín predikátová logika. Predikátová logika prvního řádu se odlišuje od výrokové logiky zavedením kvantifikovaných proměnných. 一阶逻辑是使用於数学、哲学、语言学及電腦科學中的一种形式系统,也可以稱為:一阶斷言演算、低階斷言演算、量化理論或谓词逻辑。一階邏輯和命題邏輯的不同之處在於,一階邏輯包含量詞。 高階邏輯和一階邏輯不同之處在於,高階邏輯的斷言符號可以有斷言符號或函數符號當做引數,且容許斷言量詞或函數量詞。在一階邏輯的語義中,斷言被解釋為關係。而高階邏輯的語義裡,斷言則會被解釋為集合的集合。 在通常的語義下,一階邏輯是可靠(所有可證的敘述皆為真)且完備(所有為真的敘述皆可證)。雖然一階邏輯的只是半可判定性的,但還是有許多用於一階邏輯上的自動定理證明。一階邏輯也符合一些使其能通過證明論分析的定理,如勒文海姆–斯科倫定理及緊緻性定理。 一階邏輯是數學基礎中很重要的一部份。許多常見的公理系統,如一階皮亞諾公理、冯诺伊曼-博内斯-哥德尔集合论和策梅洛-弗蘭克爾集合論都是一階理論。然而一階邏輯不能控制其無窮模型的基數大小,因根據勒文海姆–斯科倫定理和康托爾定理,可以構造出一種"病態"集合論模型,使整個模型可數,但模型內卻會覺得自己有「不可數集」。類似地,可以證明實數系的普通一階理論既有可數模型又有不可數模型。這類的悖論被稱為斯科倫悖論。但一階的直覺主義邏輯裡,勒文海姆–斯科倫定理不可證明,故不會有以上之現象。 La lògica de primer ordre, també anomenada lògica de predicats o càlcul de predicats, és un sistema formal dissenyat per estudiar la inferència en els llenguatges de primer ordre. Els llenguatges de primer ordre són, al seu torn, llenguatges amb quantificador que arriben només a variables d'individu, i amb funcions els arguments de les quals són només constants o variables d'individu. La lògica de primer ordre té el poder expressiu suficient per definir a pràcticament totes les matemàtiques. Rachunek predykatów pierwszego rzędu (ang. first order predicate calculus) – system logiczny, w którym zmienna, na której oparty jest kwantyfikator, może być elementem pewnej wybranej dziedziny (zbioru), nie może natomiast być zbiorem takich elementów. Tak więc nie mogą występować kwantyfikatory typu „dla każdej funkcji z X na Y...” (gdyż funkcja jest podzbiorem X × Y), „istnieje własność p, taka że...” czy „dla każdego podzbioru X zbioru Z...”. Rachunek ten nazywa się też krótko rachunkiem kwantyfikatorów, ale często używa się też nazwy logika pierwszego rzędu (szczególnie wśród matematyków zajmujących się logiką matematyczną).
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Predicate calculus
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Predikátová logika prvního řádu je používaný v matematice, filozofii, lingvistice a informatice. Často se pro její označení používá kratší a méně přesný termín predikátová logika. Predikátová logika prvního řádu se odlišuje od výrokové logiky zavedením kvantifikovaných proměnných. Teorie o určitém tématu bývá obvykle právě predikátová logika prvního řádu společně se: specifickou univerzální množinou (též. univerzem), ze které jsou brány proměnné, dále pak konečně mnoha funkcemi a predikáty nad touto množinou, a konečně množinou rekurzivních axiomů, jež jsou v rámci teorie pokládány za platné. Někdy pojmem teorie formálně rozumíme množinu vět (sentencí) zapsaných v predikátové logice. Kromě predikátové logiky prvního řádu existují logiky vyšších řádů. Tyto logiky se odlišují tím, že povolují predikáty uvnitř predikátů, kvantifikování predikátu i funkcí (případně predikátů a funkcí zároveň). U teorií predikátové logiky prvního řádu jsou predikáty svázány s teorií množin, kdežto v případě logik vyšších řádů bývají predikáty interpretovány jako množiny množin. Existuje velké množství deduktivních systémů pro predikátovou logiku prvního řádu, které jsou korektní (všechna dokazatelná tvrzení jsou pravdivá) a úplné (všechna pravdivá tvrzení jsou dokazatelná). Velký pokrok byl zaznamenán na poli automatických dokazovačů postavených právě na této logice, a to i přes její semi-rozhodnutelnost v oblasti dokazovačů. A v neposlední řadě splňuje několik vět, např. Löwenheim-Skolemovu větu nebo větu o kompaktnosti. Predikátová logika prvního řádu je nesmírně důležitá již pro samotné základy matematiky, protože je standardní logikou pro . Mnoho běžných axiomatických systémů, jako Peanova aritmetika a axiomatická teorie množin (včetně Zermel-Fraenkelovy teorie množin), lze formalizovat pomocí predikátové logiky. Zato žádná teorie prvního řádu nemá sílu plně a kategoricky popsat struktury s nekonečnou doménou, např. celá čísla nebo reálná čísla. K tomu jsou zapotřebí logiky vyšších řádů. Логіка першого порядку (числення предикатів) — це формальна система в математичній логіці, в якій допускаються висловлення відносно змінних, фіксованих функцій, і предикатів. Є розширенням логіки висловлювань. В свою чергу є частковим випадком . Le calcul des prédicats du premier ordre, ou calcul des relations, logique du premier ordre, logique quantificationnelle, ou tout simplement calcul des prédicats, est une formalisation du langage des mathématiques, proposée par Gottlob Frege, entre la fin du XIXe siècle et le début du XXe siècle. La logique du premier ordre comporte deux parties : * la syntaxe définit le vocabulaire symbolique de base ainsi que les règles permettant de construire des énoncés complexes, * la sémantique interprète ces énoncés comme exprimant des relations entre les éléments d'un domaine, également appelé modèle. Sur le plan syntaxique, les langages du premier ordre opposent deux grandes classes linguistiques : * les constituants servant à identifier ou nommer des éléments du domaine : variables, symboles de constantes, termes ; * les constituants servant à exprimer des propriétés ou des relations entre ces éléments : prédicats et formules. Un prédicat est une expression linguistique qui peut être reliée à un ou plusieurs éléments du domaine pour former une phrase. Par exemple, dans la phrase « Mars est une planète », l'expression « est une planète » est un prédicat qui est relié au nom (symbole de constante) « Mars » pour former une phrase. Et dans la phrase « Jupiter est plus grand que Mars », l'expression « est plus grand que » est un prédicat qui se relie aux deux noms, « Jupiter » et « Mars », pour former une phrase. En logique mathématique, lorsqu'un prédicat est lié à une expression, on dit qu'il exprime une propriété (telle que la propriété d'être une planète), et lorsqu'il est lié à deux ou plusieurs expressions, on dit qu'il exprime une relation (telle que la relation d'être plus grand). Ainsi on peut raisonner sur des énoncés comme « Tout est gentil » et « Il existe un tel que pour tout , est ami avec », ce qui exprimé symboliquement se traduit par la formule : et . Il convient de noter cependant que la logique du premier ordre ne contient aucune relation spécifique (comme telle relation d'ordre, d'inclusion ou d'égalité) ; en fait, il ne s'agit que d'étudier la façon dont on doit parler et raisonner avec les expressions du langage mathématique. Les traits caractéristiques de la logique du premier ordre sont : * l'utilisation de variables comme , etc. pour dénoter des éléments du domaine d'interprétation ; * l'utilisation de prédicats (ou relations) sur les éléments ; * l'utilisation de connecteurs logiques (et, ou, implique etc.) ; * l'utilisation de deux quantificateurs, l'un universel (« Quel que soit », « pour tout » noté ∀) et l'autre existentiel (« il existe au moins un … tel que », noté ∃), appliqués aux variables uniquement. Le calcul des prédicats du premier ordre égalitaire adjoint au calcul des prédicats un symbole de relation, l'égalité, dont l'interprétation est l'affirmation que deux éléments sont les mêmes, et qui est axiomatisée en conséquence. Suivant le contexte, on peut parler simplement de calcul des prédicats pour le calcul des prédicats égalitaire. On parle de logique du premier ordre par opposition aux logiques d'ordre supérieur, où l'on peut aussi appliquer les quantificateurs et les prédicats aux prédicats ou aux fonctions, en plus des variables. En outre, cet article ne traite que de la logique du premier ordre classique, mais on notera qu'il existe aussi une logique du premier ordre intuitionniste. First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. Sometimes, "theory" is understood in a more formal sense as just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic, in which there are predicates having predicates or functions as arguments, or in which quantification over predicates or functions, or both, are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic.No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic. The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce. For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001). Una lógica de primer orden, también llamada lógica predicativa, lógica de predicados o cálculo de predicados, es un sistema formal diseñado para estudiar la inferencia en los lenguajes de primer orden.​ Los lenguajes de primer orden son, a su vez, lenguajes formales con cuantificadores que alcanzan solo a variables de individuo, y con predicados y funciones cuyos argumentos son solo constantes o variables de individuo.​ La lógica de primer orden tiene un poder expresivo superior al de la lógica proposicional. Nella logica matematica, una teoria del primo ordine (o calcolo dei predicati) è un particolare sistema formale, cioè una teoria formale, in cui è possibile esprimere enunciati e dedurre le loro conseguenze logiche in modo del tutto formale e meccanico. La teoria del prim'ordine estende di fatto la logica proposizionale con l'introduzione di quantificatori esistenziali e universali, predicati, funzioni, variabili e costanti, che apportano maggiore potenza espressiva al calcolo dei predicati. Come per la logica proposizionale, la teoria del primo ordine può essere scissa in due parti separate: * la sintassi, che definisce il vocabolario simbolico di base e le regole per la costruzione di enunciati complessi, * la semantica, che interpreta questi enunciati come espressione delle relazioni tra gli elementi di un dominio, aggregati mediante un assegnamento. Un predicato è un'espressione linguistica che può essere collegata a uno o più elementi del dominio per formare una frase. Ad esempio, nella frase "Marte è un pianeta", l'espressione "è un pianeta" è un predicato che è legato al nome (un simbolo costante) "Marte" per formare una frase. Nella frase "Giove è più grande di Marte", l'espressione "è più grande di" è un predicato che collega i due nomi, "Giove" e "Marte", per formare una frase. In logica matematica, quando un predicato è legato a un'espressione, si dice che esprime una proprietà (come la proprietà di essere un pianeta nell'esempio precedente), e quando è legato a due o più espressioni, si dice che esprime una relazione (come la relazione per un pianeta di essere più grande di un altro). Così è ragionare su affermazioni come "Ogni x è bello" e "Esiste un x tale che per ogni y, x è amico di y", che simbolicamente è espresso dalla formula: . Va notato che la teoria del primo ordine non contiene in sé nessuna relazione specifica (come una relazione d'ordine, inclusione o uguaglianza). Logika predikat tingkat pertama adalah sistem deduksi formal yang digunakan dalam matematika, filosofi, linguistika, dan ilmu komputer. Jika kalkulus proposisional membahas proposisi sederhana, LTP menambahkan predikat dan kuantor. Misalnya: * Sokrates adalah seorang manusia * Plato adalah seorang manusia Kedua kalimat di atas dalam kalkulus proposisional adalah dua proposisi yang tidak berhubungan, misalnya dilambangkan dengan p dan q. Dalam LTP, keduanya dihubungkan dengan satu sifat, yaitu Manusia(x), artinya x adalah seorang manusia. Bila x = Socrates kita mendapatkan proposisi pertama, p; dan jika x = Plato kita mendapatkan proposisi kedua, q. Contoh berikut menjabarkan perbedaan kalkulus proposisional dan LTP: * Semua manusia perlu makan * Sokrates adalah manusia * Sokrates perlu makan Dalam kalkulus proposisional, ketiga kalimat di atas diterjemahkan sebagai: * A * B * C ( artinya "maka") Ketiga kalimat di atas tidak dapat dihubungkan dalam kalkulus proposisional. Dalam LTP, kita dapat menerjemahkan ketiga kalimat itu sebagai: * * * Die Prädikatenlogik erster Stufe ist ein Teilgebiet der mathematischen Logik. Sie befasst sich mit der Struktur gewisser mathematischer Ausdrücke und dem logischen Schließen, mit dem man von derartigen Ausdrücken zu anderen gelangt. Dabei gelingt es, sowohl die Sprache als auch das Schließen rein syntaktisch, das heißt ohne Bezug zu mathematischen Bedeutungen, zu definieren.Das dadurch ermöglichte Zusammenspiel von rein syntaktischen Überlegungen einerseits und semantischen Betrachtungen andererseits führt zu wichtigen Erkenntnissen, die Bedeutung für die gesamte Mathematik haben, denn diese lässt sich mittels der Zermelo-Fraenkel-Mengenlehre in der Prädikatenlogik erster Stufe formulieren. Im Unterschied zur Aussagenlogik macht die Prädikatenlogik von Quantoren Gebrauch. Логика первого порядка — формальное исчисление, допускающее высказывания относительно переменных, фиксированных функций и предикатов. Расширяет логику высказываний. Помимо логики первого порядка существуют также логики высших порядков, в которых кванторы могут применяться не только к переменным, но и к предикатам. Термины логика предикатов и исчисление предикатов могут означать как логику первого порядка, так и логики первого и высшего порядка вместе; в первом случае иногда говорится о чистой логике предикатов или чистом исчислении предикатов. A lógica de primeira ordem (LPO), conhecida também como cálculo de predicados de primeira ordem (CPPO), é um sistema lógico que estende a lógica proposicional (lógica sentencial) e que é estendida pela lógica de segunda ordem. As sentenças atômicas da lógica de primeira ordem têm o formato P (t1,…, tn) (um predicado com um ou mais "argumentos") ao invés de serem símbolos sentenciais sem estruturas. O ingrediente novo da lógica de primeira ordem não encontrado na lógica proposicional é a quantificação: dada uma sentença φ qualquer, as novas construções e -- leia "para todo x, φ" e "para algum x, φ", respectivamente—são introduzidas. significa que φ é verdadeiro para todo valor de x e significa que há pelo menos um x tal que φ é verdadeiro. Os valores das variáveis são tirados de um universo de discurso pré-determinado. Um refinamento da lógica de primeira ordem permite variáveis de diferentes tipos, para tratar de diferentes classes de objetos. A lógica de primeira ordem tem poder expressivo suficiente para formalizar praticamente toda a matemática. Uma teoria de primeira ordem consiste em um conjunto de axiomas (geralmente finito ou recursivamente enumerável) e de sentenças dedutíveis a partir deles. A teoria dos conjuntos de Zermelo-Fraenkel é um exemplo de uma teoria de primeira ordem, e aceita-se geralmente que toda a matemática clássica possa ser formalizada nela. Há outras teorias que são normalmente formalizadas na lógica de primeira ordem de maneira independente(embora elas admitam a implementação na teoria dos conjuntos) tais como a aritmética de Peano. Unuaranga logiko, nomita ankaŭ predikatlogiko, aŭ predikatkalkulo, estas formala sistemo desegnita por studi la inferencon en la unuarangaj lingvaĵoj.​ La unuarangaj lingvaĵoj estas siavice formalaj lingvaĵoj kun kvantigiloj kiuj atingas nur unuopajn variablojn, kaj kun predikatoj kaj funkcioj kies argumentoj estas nur konstantoj aŭ unuopaj variabloj. La logiko unuaranga havas espriman povon superan al tiu de la propozicia logiko. Logiko de supera ordo estas formo de predikatkalkulo kiu estas distingata el la unuaranga logiko pere de aldonaj kvantigiloj kaj, foje, per pli forta semantiko. 一阶逻辑是使用於数学、哲学、语言学及電腦科學中的一种形式系统,也可以稱為:一阶斷言演算、低階斷言演算、量化理論或谓词逻辑。一階邏輯和命題邏輯的不同之處在於,一階邏輯包含量詞。 高階邏輯和一階邏輯不同之處在於,高階邏輯的斷言符號可以有斷言符號或函數符號當做引數,且容許斷言量詞或函數量詞。在一階邏輯的語義中,斷言被解釋為關係。而高階邏輯的語義裡,斷言則會被解釋為集合的集合。 在通常的語義下,一階邏輯是可靠(所有可證的敘述皆為真)且完備(所有為真的敘述皆可證)。雖然一階邏輯的只是半可判定性的,但還是有許多用於一階邏輯上的自動定理證明。一階邏輯也符合一些使其能通過證明論分析的定理,如勒文海姆–斯科倫定理及緊緻性定理。 一階邏輯是數學基礎中很重要的一部份。許多常見的公理系統,如一階皮亞諾公理、冯诺伊曼-博内斯-哥德尔集合论和策梅洛-弗蘭克爾集合論都是一階理論。然而一階邏輯不能控制其無窮模型的基數大小,因根據勒文海姆–斯科倫定理和康托爾定理,可以構造出一種"病態"集合論模型,使整個模型可數,但模型內卻會覺得自己有「不可數集」。類似地,可以證明實數系的普通一階理論既有可數模型又有不可數模型。這類的悖論被稱為斯科倫悖論。但一階的直覺主義邏輯裡,勒文海姆–斯科倫定理不可證明,故不會有以上之現象。 一階述語論理(英: first-order predicate logic)とは、個体の量化のみを許す述語論理 (predicate logic) である。述語論理とは、数理論理学における論理の数学的モデルの一つであり、命題論理を拡張したものである。個体の量化に加えて述語や関数の量化を許す述語論理を二階述語論理(英: second-order predicate logic)と呼び、さらなる一般化を加えた述語論理を高階述語論理(英: higher-order predicate logic)という。本項では主に一階述語論理について解説する。二階述語論理や高階述語論理についての詳細はそれぞれの記事を参照。 Första ordningens logik (FOL) är ett formellt deduktivt system som används i matematik, filosofi, lingvistik och datavetenskap. Det har ett flertal olika namn på engelska: first-order predicate calculus (FOPC), the lower predicate calculus, the language of first-order logic och predicate logic. Till skillnad från naturliga språk, som svenska, använder sig FOL av ett helt otvetydigt formellt språk som tolkas av matematiska strukturer. FOL är ett deduktivt system som går bortom satslogiken genom att tillåta kvantifiering av objekt inom en given domän. Man kan exempelvis med FOL uttrycka satsen "Varje individ har egenskapen P". Medan satslogik endast behandlar enkla propositioner så inkluderar första ordningens logik även predikat och kvantifikatorer. Inom satslogiken är de två satserna "Sokrates är en man" och "Platon är en man" helt orelaterade och uttrycks till exempel med p och q. Med FOL uttrycks dock båda dessa satser med samma predikat: Man(x) där Man(x) betyder att x är en man. När x=Sokrates får vi den första satsen, p, och när x=Platon får vi den andra satsen, q. Detta språk blir mycket kraftfullt då man introducerar kvantifikatorer, då man kan uttrycka satser som "för varje x...", som i "för varje x gäller det att, om Man(x), så...". Utan kvantifikatorer är varje giltigt argument i FOL även giltigt i satslogik och vice versa. En första ordningens teori består av en uppsättning axiom (vanligtvis ändlig eller rekursivt räknebar) och de uttryck som går att deducera från dem givet ett antal regler för giltig deduktion inom systemet. Ett första ordningens språk har tillräcklig uttryckskraft för att formalisera två viktiga matematiska teorier: Zermelo-Fraenkels mängdteori och Peanos axiom (första ordningens). Ett första ordningens språk kan emellertid inte kategoriskt uttrycka uppräknelighet. Det kan uttryckas kategoriskt med andra ordningens logik. منطق الرتبة الأولى (First-order logic FOL) أو المنطق الإسنادي عبارة عن نظام للمنطق الرياضي يستخدم في الرياضيات والفلسفة والذكاء الصناعي وعلوم الحاسب. وهو يستخدم في التعبير عن الجمل المنطقية بشكل غير مبهم وبهذا يخالف عن اللغات الطبيعية والتي قد تحتوي على جمل مبهمة. ذلك يسهل الاستنتاج واجراء العمليات المنطقية على الجمل أو المعادلات التي تنشاء باستخدامه.و يعتبر منطق الرتبة الأولى هو تمديد منطق القضايا (منطق العبارات) propositional logic وذلك بإضافة القياس سواء كان عالمي أو وجودي. يعتبر بمنطق الرتبة الثانية تمديد لمنطق الرتبة الأولى وذلك بإضافة القياس على المجموعات. يدعى منطق الرتبة الأولى أحيانا : بمنطق الرتبة الأولى الإسنادي أو first-order predicate calculus (FOPC. Rachunek predykatów pierwszego rzędu (ang. first order predicate calculus) – system logiczny, w którym zmienna, na której oparty jest kwantyfikator, może być elementem pewnej wybranej dziedziny (zbioru), nie może natomiast być zbiorem takich elementów. Tak więc nie mogą występować kwantyfikatory typu „dla każdej funkcji z X na Y...” (gdyż funkcja jest podzbiorem X × Y), „istnieje własność p, taka że...” czy „dla każdego podzbioru X zbioru Z...”. Rachunek ten nazywa się też krótko rachunkiem kwantyfikatorów, ale często używa się też nazwy logika pierwszego rzędu (szczególnie wśród matematyków zajmujących się logiką matematyczną). Na przykład w rachunku predykatów pierwszego rzędu można zapisać zdanie „dla dowolnej liczby rzeczywistej istnieje liczba większa”, jednak nie można zapisać „każdy zbiór liczb rzeczywistych ma kres górny”, gdyż wówczas kwantyfikator ogólny musiałby przebiegać wszystkie możliwe podzbiory zbioru liczb rzeczywistych i potrzebny byłby rachunek predykatów co najmniej drugiego rzędu. Rachunek predykatów pierwszego rzędu w ogólnym przypadku nie jest rozstrzygalny (w przeciwieństwie do rachunku zdań), lecz półrozstrzygalny (czyli rekurencyjnie przeliczalny), ale jeszcze nadaje się do komputerowej analizy (co już niekoniecznie można powiedzieć o rachunku predykatów wyższych rzędów, które dopuszczają kwantyfikatory dla zbiorów). Znaczna część rozważań matematycznych może być sformalizowana na gruncie logiki pierwszego rzędu. Ponadto logika ta ma wiele własności czyniących ją bardziej użyteczną od innych logik, co ma wpływ na pewne preferowanie teorii formalizowalnych na jej gruncie. W literaturze istnieje szereg równoważnych rozwinięć tego tematu. Prezentacja przedstawiona poniżej jest do pewnego stopnia oparta na książce Martina Goldsterna i Haima Judaha. Wśród innych źródeł omawiających te zagadnienia należy wymienić podręcznik Witolda Pogorzelskiego, czy też książkę Zofii Adamowicz i Pawła Zbierskiego. Bardzo popularnym jest też opracowanie Josepha Shoenfielda. La lògica de primer ordre, també anomenada lògica de predicats o càlcul de predicats, és un sistema formal dissenyat per estudiar la inferència en els llenguatges de primer ordre. Els llenguatges de primer ordre són, al seu torn, llenguatges amb quantificador que arriben només a variables d'individu, i amb funcions els arguments de les quals són només constants o variables d'individu. La lògica de primer ordre té el poder expressiu suficient per definir a pràcticament totes les matemàtiques. Com el desenvolupament històric i les aplicacions de la lògica de primer ordre estan molt lligats a la matemàtica, en el que segueix es farà una introducció que contempli i il·lustre aquesta relació, prenent exemples tant de la matemàtica com del llenguatge natural. Primer s'introdueixen cada un dels conceptes bàsics del sistema, i després es mostra com utilitzar-los per analitzar arguments. Lehen mailako logika, predikatuen logika, logika kuantifikatzailea edo predikatuen kalkulua ere deitzen dena, lehen ordenako hizkuntzen inferentzia aztertzeko diseinatutako sistema formala da. Predikatuak, haien propietateak eta eragiketak aztertzen dituen logika. Aldagai eta kuantifikatzaileen bidez lan egiten du. Predikatuen logikak proposizioen barne-egitura hartzen du kontuan. Lehen ordenako lengoaiak, era berean, banakako aldagaiei bakarrik eragiten dien zenbatzaileak dituzten dira, eta argumentuak, konstanteak edo aldagai indibidualak dituzten predikatuak eta funtzioak baino ez dira. Lehen ordenako logikak, logika proposizionala baino adierazkortasun-maila altuagoa du. Η λογική πρώτου βαθμού είναι μια τυπική λογική που χρησιμοποιείται στα μαθηματικά, τη φιλοσοφία, τη γλωσσολογία και την επιστήμη υπολογιστών. Συναντάται με διάφορα ονόματα, όπως κατηγορηματικός λογισμός πρώτου βαθμού ή κατηγορηματική λογική. Η πρωτοβάθμια λογική διαφέρει από την προτασιακή λογική στη χρήση ποσοτικών τελεστών: κάθε ερμηνεία της λογικής πρώτου βαθμού περιλαμβάνει ένα πεδίο τιμών όπου κυμαίνονται οι ποσοτικοί τελεστές. Υπάρχουν πολλά συμπερασματικά συστήματα για την πρωτοβάθμια λογική που είναι συνεπή (παράγουν μόνο σωστά αποτελέσματα) και πλήρη (ικανά να παράγουν οποιαδήποτε ορθή πρόταση). Αν και η σχέση λογικής συνέπειας είναι μόνο ημι-αποφασίσιμη, έχει επιτευχθεί μεγάλη πρόοδος στην αυτόματη απόδειξη θεωρημάτων για τη λογική πρώτου βαθμού. Η λογική πρώτου βαθμού ικανοποιεί επίσης αρκετά θεωρήματα μετα-λογικής που την κάνουν επιδεκτική ανάλυσης στην , όπως το και το . Η λογική πρώτου βαθμού έχει μεγάλη σημασία για τα θεμέλια των μαθηματικών, όπου αποτελεί την πρότυπη τυπική λογική για . Έχει ικανή εκφραστική ισχύ ώστε να μπορεί να διατυπώσει δύο σημαντικές μαθηματικές θεωρίες: τη και την (πρωτοβάθμια) αριθμητική Πεάνο. Εντούτοις, κανένα αξιωματικό σύστημα στην πρωτοβάθμια λογική δεν είναι αρκετά ισχυρό ώστε να περιγράψει κατηγορικά άπειρες δομές όπως οι φυσικοί αριθμοί ή η . Κατηγορικά αξιωματικά συστήματα για τέτοιες δομές μπορούν να παραχθούν σε ισχυρότερες λογικές όπως η λογική δευτέρου βαθμού. 1차 논리(一次論理, 영어: first-order logic)는 원소에만 한정 기호를 가할 수 있고, 술어에는 한정 기호를 가할 수 없는 술어 논리이다. 명제 논리와 달리 변수에 대하여 한정 기호를 사용할 수 있으나, 2차 논리와 달리 변수들의 집합에 대하여 한정 기호를 사용할 수 없다. 1차 논리의 경우, (2차 논리와 달리) 괴델의 완전성 정리 · 콤팩트성 정리 · 뢰벤하임-스콜렘 정리와 같은 중요한 성질들이 성립한다. 이외에 1차 술어 논리, 1계 논리 등으로도 불린다. 간단히 술어 논리(predicate logic)라 하면 1차 논리를 가리키는 경우가 많다.
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dbr:First_Order_Logic
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wikipedia-en:First-order_logic
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dbr:First-order_logic