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Consequentia Mirabilis
''Consequentia mirabilis'' (Latin for "admirable consequence"), also known as Clavius's Law, is used in traditional and classical logic to establish the truth of a proposition from the inconsistency of its negation. It is thus related to ''reductio ad absurdum'', but it can prove a proposition using just its own negation and the concept of consistency. For a more concrete formulation, it states that if a proposition is a consequence of its negation, then it is true, for consistency. In formal notation: : (\neg A \rightarrow A) \rightarrow A . Equivalent forms Given P\to Q being equivalent to \neg P\lor Q, the principle is equivalent to :(\neg \neg A \lor A) \rightarrow A . History ''Consequentia mirabilis'' was a pattern of argument popular in 17th-century Europe that first appeared in a fragment of Aristotle's '' Protrepticus:'' "If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the Roman Republic it became the dominant language in the Italian region and subsequently throughout the Roman Empire. Even after the fall of Western Rome, Latin remained the common language of international communication, science, scholarship and academia in Europe until well into the 18th century, when other regional vernaculars (including its own descendants, the Romance languages) supplanted it in common academic and political usage, and it eventually became a dead language in the modern linguistic definition. Latin is a highly inflected language, with three distinct genders (masculine, feminine, and neuter), six or seven noun cases (nominative, accusative, genitive, dative, ablative, and vocative), five declensions, four verb conjuga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Clavius
Christopher Clavius, SJ (25 March 1538 – 6 February 1612) was a Jesuit German mathematician, head of mathematicians at the Collegio Romano, and astronomer who was a member of the Vatican commission that accepted the proposed calendar invented by Aloysius Lilius, that is known as the Gregorian calendar. Clavius would later write defences and an explanation of the reformed calendar, including an emphatic acknowledgement of Lilius' work. In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in and even out of Europe. Early life Little is known about Clavius' early life other than the fact that he was born in Bamberg in either 1538 or 1537. His given name is not known to any great degree of certainty—it is thought by scholars to have perhaps been ''Christoph Clau'' or ''Klau''. There are also some who think that his taken name, ''Clavius'', may be a Latinization of his original Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traditional Logic
In philosophy, 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, so to understand the newer systems, one must be acquainted with the earlier one. Aristotle's system Aristotle's logical work is collected in the six texts that are collectively known as the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency Proof
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductio Ad Absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. Examples The "absurd" conclusion of a ''reductio ad absurdum'' argument can take a range of forms, as these examples show: * The Earth cannot be flat; otherwise, since Earth assumed to be finite in extent, we would find people falling off the edge. * There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one. The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Protrepticus (Aristotle)
''Protrepticus'' ( el, Προτρεπτικός) is a philosophical work by Aristotle that encouraged the young to study philosophy. It survives only in fragments and ancient reports and is considered a lost work. This is likely the origin of the English word Protreptics, which means, “turning or converting someone to a specific end” used in a philosophical sense, a word hardly ever used except in specialized philosophical treatises. Fragments and ancient reports Fragments are preserved in several works by Iamblichus of Calchis. Reconstructions Since the 19th century, when inquiry was initiated by Jakob Bernays (1863), several scholars have attempted to reconstruct the work. Attempted reconstructions include: *A 1961 book by Ingemar Düring *A 1964 book by Anton-Hermann Chroust *2015 ''Protrepticus or Exhortation to Philosophy'' by Hutchinson and Johnson Commentary The book ''The works of Aristotle'' (1908, p. viii) mentioned :The Historia Augusta says that Cicero's Hor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tertium Non Datur
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce's Law
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication. In propositional calculus, Peirce's law says that ((''P''→''Q'')→''P'')→''P''. Written out, this means that ''P'' must be true if there is a proposition ''Q'' such that the truth of ''P'' follows from the truth of "if ''P'' then ''Q''". In particular, when ''Q'' is taken to be a false formula, the law says that if ''P'' must be true whenever it implies falsity, then ''P'' is true. In this way Peirce's law implies the law of excluded middle. Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Propositional Logic
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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