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Paradoxes Of Material Implication
The paradoxes of material implication are a group of tautology (logic), 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 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 (philosophy), meaning and reasoning. Paradox of entailment As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction. In natural language, an instance of the paradox of entailment ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. 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. It cannot be untrue. 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 Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments 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 deductively valid inferences that preserve truth from the premises to the conclusion. This logical perspective on argument is relevant for scientific fields such as mathematics and computer science. Logic is the study of the forms of reasoning in arguments and the development of standards and criteria to evaluate arguments. Deductive arguments can be valid, and the valid ones can be sound: in a valid argument, premisses necessitate the conclusion, even if one or more of the premises is false ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or 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 closely related to another valid form of argument, ''modus tollens''. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of ''modus ponens''. Hypothetical syllogism is closely related to ''modus ponens'' and sometimes thought of as "double ''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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Although considered paradoxes, some of these are simply based on fallacious reasoning ( falsidical), or an unintuitive solution (veridical). Informally, the term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream perception 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 description. Logic * : The supposition that, 'if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved' leads to paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' used as a form or analog of an include directive Many programming languages and other computer files have a directive, often called include (sometimes copy or import), that causes the contents of the specified file to be inserted into the original file. These included files are called copybooks ... in some programming languages * Import–export (logic), a form of deductive argument in classical logic {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. For example, a false dilemma is committed when it is claimed that "Stacey spoke out against capitalism; therefore, she must be a communist". One of the options excluded is that Stacey may be neither communist nor capitalist. 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterfactuals
Counterfactual conditionals (also ''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 possible worlds where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connexive Logic
Connexive logic names one class of alternative, or 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, * ~(~p → 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 → q) → ~(p → ~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, for example, that it is imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 \land ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistent
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] |
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