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Paradoxes Of Material Implication
The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formula P \rightarrow Q is true unless P is true and Q is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called ''paradoxes''. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning. math>\neg p \lor (\neg q \lor p) if p is true then it is implied by every q. In other words, Begging the question is based on a paradox. By analogy with \neg p \to (p \to q) being referred to as 'explosion', this could be referred to as 'imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Tautology (logic)
In mathematical logic, a tautology (from ) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Logical Argument
An argument is a series of Sentence (linguistics), sentences, Statement (logic), statements, or propositions some of which are called premises and one is the Logical consequence, conclusion. The purpose of an argument is to give Reason (argument), reasons for one's conclusion via justification, explanation, and/or persuasion. Arguments are intended to determine or show the degree of truth or acceptability of another statement called a conclusion. The process of crafting or delivering arguments, argumentation, can be studied from three main perspectives: the logical, the dialectical and the rhetorical perspective. In logic, an argument is usually expressed not in natural language but in a symbolic formal language, and it can be defined as any group of propositions of which one is claimed to follow from the others through Deductive reasoning, deductively valid inferences that preserve truth from the premises to the conclusion. This logical perspective on argument is relevant for sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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List Of Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. These paradoxes may be due to fallacious reasoning (falsidical), or an unintuitive solution (Veridical paradox, veridical). The term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream viewpoint of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called ''antinomy,'' point out genuine problems in our understanding of the ideas of truth and Definite description, description. Logic * : The supposition that, "if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved" leads to paradoxical conseq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Import-Export (logic)
Import and export or import/export may refer to: * Import and export of goods ** International trade ** Import/export regulations, trade regulations of such goods ** Import/export tariffs, taxes on the trade in such goods * Import and export of data in computing, the moving of data between applications ** Import and export of formats, data conversion from one file type to another * ''Import/Export'', a 2007 Austrian film * An ''import statement'' allows a computer programming Modular programming, module to access the exposed (exported) capabilities of another module * Import–export (logic), a form of deductive argument in classical logic {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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False Dilemma
A false dilemma, also referred to as false dichotomy or false binary, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false premise. This premise has the form of a disjunctive claim: it asserts that one among a number of alternatives must be true. This disjunction is problematic because it oversimplifies the choice by excluding viable alternatives, presenting the viewer with only two absolute choices when, in fact, there could be many. False dilemmas often have the form of treating two contraries, which may both be false, as contradictories, of which one is necessarily true. Various inferential schemes are associated with false dilemmas, for example, the constructive dilemma, the destructive dilemma or the disjunctive syllogism. False dilemmas are usually discussed in terms of deductive arguments, but they can also occur as defeasible arguments. The h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Counterfactuals
Counterfactual conditionals (also ''contrafactual'', ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood. Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Correlation Does Not Imply Causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The idea that "correlation implies causation" is an example of a questionable-cause logical fallacy, in which two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known by the Latin phrase ''cum hoc ergo propter hoc'' ('with this, therefore because of this'). This differs from the fallacy known as '' post hoc ergo propter hoc'' ("after this, therefore because of this"), in which an event following another is seen as a necessary consequence of the former event, and from conflation, the errant merging of two events, ideas, databases, etc., into one. As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not necessarily imply that the resulting c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Connexive Logic
Connexive logic is a class of non-classical logics designed to exclude the paradoxes of material implication. The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's thesis, i.e. the formula, \lnot ( \lnot p \rightarrow p ) as a logical truth. Aristotle's thesis asserts that no statement follows from its own denial. Stronger connexive logics also accept Boethius' thesis, (p \rightarrow q) \rightarrow \lnot(p \rightarrow \lnot q) which states that if a statement implies one thing, it does not imply its opposite. Relevance logic is another logical theory that tries to avoid the paradoxes of material implication. History Connexive logic is arguably one of the oldest approaches to logic. Aristotle's thesis is named after Aristotle because he uses this principle in a passage in the '' Prior Analytics''. It is impossible that the same thing should be necessitated by the being and the not-being of the same thing. I mean, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Importation (logic)
Exportation is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It is the rule that: :((P \land Q) \to R) \Leftrightarrow (P \to (Q \to R)) Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ((P \land Q) \to R) \Rightarrow (P \to (Q \to R)) is the law of exportation, for it "exports" a proposition from the antecedent of (P \land Q) \to R to its consequent. Its converse, the law of importation, (P \to (Q \to R))\Rightarrow ((P \land Q) \to R) , "imports" a proposition from the consequent of P \to (Q \to R) to its antecedent. Formal notation The ''exportation'' rule may be written in sequent notation: :((P \land Q) \to R) \dashv\vdash (P \to (Q \to R)) where \dashv\vdash is a metalogical symbol meaning that (P \to (Q \to R)) is a syntactic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Consistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. 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 there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |