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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 \to A) \to A . Weaker variants of the principle are provable in minimal logic, but the full principle itself is not provable even in intuitionistic logic. 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 justify this vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area around Rome, Italy. Through the expansion of the Roman Republic, it became the dominant language in the Italian Peninsula and subsequently throughout the Roman Empire. It has greatly influenced many languages, Latin influence in English, including English, having contributed List of Latin words with English derivatives, many words to the English lexicon, particularly after the Christianity in Anglo-Saxon England, Christianization of the Anglo-Saxons and the Norman Conquest. Latin Root (linguistics), roots appear frequently in the technical vocabulary used by fields such as theology, List of Latin and Greek words commonly used in systematic names, the sciences, List of medical roots, suffixes and prefixes, medicine, and List of Latin legal terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Identity
In logic, the law of identity states that each thing is identical with itself. It is the first of the traditional three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws. History Ancient philosophy The earliest recorded use of the law appears in Plato's dialogue '' Theaetetus'' (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing: It is used explicitly only once in Aristotle, in a proof in the '' Prior Analytics'': Medieval philosophy Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (''Met.'' IV, lect. 6) and Duns Scotus (''Quaest. sup. Met.'' IV, Q. 3) follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (''Omne Ens est Ens'', Qq. in Met ... [...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 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/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ex Falso Quodlibet
In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion is the law according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion. The proof of this principle was first given by 12th-century French philosopher William of Soissons. Due to the principle of explosion, the existence of a contradiction ( inconsistency) in a formal axiomatic system is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity. Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the mode ... [...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#Mathematics, 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'' Logical consequence, follows from the truth of "if ''P'' then ''Q''". 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/cc in Scheme (programming language), Scheme. v \end \end and where Peirce's law as a formula can be simplified to: : \begin ((u \mathrel v) \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Syllogism
In classical logic, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises. An example in English: # I will choose soup or I will choose salad. # I will not choose soup. # Therefore, I will choose salad. Propositional logic In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid rule of inference. If it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. Equivalently, if ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunction Elimination
In propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction ''A and B'' is true, then ''A'' is true, and ''B'' is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself. An example in English: :It's raining and it's pouring. :Therefore it's raining. The rule consists of two separate sub-rules, which can be expressed in formal language as: :\frac and :\frac The two sub-rules together mean that, whenever an instance of "P \land Q" appears on a line of a proof, either "P" or "Q" can be placed on a subsequent line by itself. The above example in English is an application of the first sub-rule. Formal notation The ''conjunction elimination'' sub-rules may be written in sequent notation: : (P \land Q) \vdash P and : (P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Identity
In logic, the law of identity states that each thing is identical with itself. It is the first of the traditional three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws. History Ancient philosophy The earliest recorded use of the law appears in Plato's dialogue '' Theaetetus'' (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing: It is used explicitly only once in Aristotle, in a proof in the '' Prior Analytics'': Medieval philosophy Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (''Met.'' IV, lect. 6) and Duns Scotus (''Quaest. sup. Met.'' IV, Q. 3) follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (''Omne Ens est Ens'', Qq. in Met. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must also be true." ''Modus ponens'' is a mixed hypothetical syllogism and is closely related to another valid form of argument, '' modus tollens''. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of ''modus ponens''. The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with '' modus tollens'', is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal. Explanation The form of a ''modus ponens'' argument is a mixed hypothetical syllogism, with two premises and a con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Non-contradiction
In logic, the law of noncontradiction (LNC; also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that for any given proposition, the proposition and its negation cannot both be simultaneously true, e.g. the proposition "''the house is white''" and its negation "''the house is not white''" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one of two propositions like "the house is white" and "the house is not white" holds. One reason to have this law is the principle of explosion, which states that anything follows from a contradiction. The law is employed in a ''reductio ad absurdum'' proof. To express the fact that the law is tenseless and to avoid equivocation, sometimes the law is amended to say "contradictory propositions cannot both be true 'at the same time and in the same sense'" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implication Introduction
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. Overview The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that ''if it were true'' it would lead to the consequent. Conditional proofs are of great importance in mathematics. Conditional proofs exist linking several otherwise unproven conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently. A famous network of conditional proofs is the NP-complete class of complexity theory. There is a large number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation Introduction
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction. Formal notation This can be written as: :\Big((P \rightarrow Q) \land (P \rightarrow \neg Q)\Big) \rightarrow \neg P An example of its use would be an attempt to prove two contradictory statements from a single fact. For example, if a person were to state "Whenever I hear the phone ringing I am happy" and then state "Whenever I hear the phone ringing I am ''not'' happy", one can infer that the person never hears the phone ringing. Many proofs by contradiction use negation introduction as reasoning scheme: to prove ¬''P'', assume for contradiction ''P'', then derive from it two contradictory inferences ''Q'' and ¬''Q''. Since the latter contradiction renders ''P'' impossible, ¬''P'' must hold. Proof With \neg P identifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
