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Proof System
In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Formal language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order logic. * Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. * Axioms: Formulas in ''L'' assumed to be valid. All theorems are derived from axioms. A formal proof of a well-formed formula in a proof system is a set of axioms and rules of inference of proof system that infers that the well-formed formula is a theorem of proof system. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerhard Gentzen
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died of starvation in a Czech prison camp in Prague in 1945. Life and career Gentzen was a student of Paul Bernays at the University of Göttingen. Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor. Gentzen joined the Sturmabteilung in November 1933, although he was by no means compelled to do so. Nevertheless, he kept in contact with Bernays until the beginning of the Second World War. In 1935, he corresponded with Abraham Fraenkel in Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the Chosen People." In 1935 and 1936, Hermann Weyl, head of the Göttingen mathematics department in 1933 until his resignation under Nazi pressure, made strong efforts t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first-order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, for pure thought." Frege's motivation for developing his formal approach to logic resembled Gottfried Wilhelm Leibniz, Leibniz's motivation for his ''calculus ratiocinator'' (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter-century. This is the first work in Analytical Philosophy, a field that later British and Anglo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, philosophy of logic, logic, and Philosophy of mathematics, mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the History of logic#Rise of modern logic, development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it. Aristotle's system Aristotle's logical work is collected in the six texts that are collectively known as the '' Organon''. Two of these texts in particular, namely th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aegis
The aegis ( ; ''aigís''), as stated in the ''Iliad'', is a device carried by Athena and Zeus, variously interpreted as an animal skin or a shield and sometimes featuring the head of a Gorgon. There may be a connection with a deity named Aex, a daughter of Helios and a nurse of Zeus or alternatively a mistress of Zeus (Gaius Julius Hyginus, Hyginus, ''Astronomica'' 2. 13). The modern concept of doing something "under someone's ''aegis'' means doing something under the protection of a powerful, knowledgeable, or benevolent source. The word ''aegis'' is identified with protection by a strong force with its roots in Greek mythology and adopted by the Roman mythology, Romans; there are Comparative mythology, parallels in Norse mythology and in Egyptian mythology as well, where the Greek word ''aegis'' is applied by extension. Etymology The Greek language, Greek ''aigis'' has many meanings, including: # "violent windstorm", from the verb ''aïssō'' (word stem ''aïg-'') = "I ru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syllogism
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BC book '' Prior Analytics''), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This article is concern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Organon
The ''Organon'' (, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics, who maintained against the Stoics that Logic was "an instrument" of Philosophy. Aristotle never uses the title ''Organon'' to refer to his logical works. The book, according to M. Barthélemy St. Hilaire, was not called "Organon" before the 15th century, and the treatises were collected into one volume, as is supposed, about the time of Andronicus of Rhodes; and it was translated into Latin by Boethius about the 6th century. The six works of Organon are as follows: Constitution of the texts The order of the works is not chronological (which is now hard to determine) but was deliberately chosen by Theophrastus to constitute a well-structured system. Indeed, parts of them seem to be a scheme of a lecture on logic. The arrangement of the works was made by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syllogistic
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BC book '' Prior Analytics''), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This article is concerne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, and the arts. As the founder of the Peripatetic school of philosophy in the Lyceum (classical), Lyceum in Athens, he began the wider Aristotelianism, Aristotelian tradition that followed, which set the groundwork for the development of modern science. Little is known about Aristotle's life. He was born in the city of Stagira (ancient city), Stagira in northern Greece during the Classical Greece, Classical period. His father, Nicomachus (father of Aristotle), Nicomachus, died when Aristotle was a child, and he was brought up by a guardian. At around eighteen years old, he joined Plato's Platonic Academy, Academy in Athens and remained there until the age of thirty seven (). Shortly after Plato died, Aristotle left Athens and, at the request ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
