Material Implication (other)
Material implication may refer to: * Material conditional, a logical connective * Material implication (rule of inference), a rule of replacement for some propositional logic See also * Implication (other) * Conditional statement (other) A conditional statement may refer to: * A conditional formula in logic and mathematics, which can be interpreted as: ** Material conditional ** Strict conditional ** Variably strict conditional ** Relevance logic, Relevance conditional * A conditio ...

Material Conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. Notation In l ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Material Implication (rule Of Inference)
In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that ''P implies Q'' is logically equivalent to ''not-P or Q'' and that either form can replace the other in logical proofs. In other words, if P is true, then Q must also be true, while if Q is true, then P cannot be true either; additionally, when P is not true, Q may be either true or false. P \to Q \Leftrightarrow \neg P \lor Q Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given logical statements. To illustrate this, consider the following statements: * P: Sam ate an orange for lunch * Q: Sam ate a fruit for lunch Then, to say, "Sam ate an orange for lunch" "Sam ate a fruit for lunch" (P \to Q). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (b ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Implication (other)
Implication may refer to: Logic * Logical consequence (also entailment or logical implication), the relationship between statements that holds true when one logically "follows from" one or more others * Material conditional (also material consequence, or implication), a logical connective and binary truth function typically interpreted as "If ''p'', then ''q''" ** material implication (rule of inference), a logical rule of replacement ** Implicational propositional calculus, a version of classical propositional calculus which uses only the material conditional connective * Strict conditional or strict implication, a connective of modal logic that expresses necessity * ''modus ponens'', or Implication elimination, a simple argument form and rule of inference summarized as "''p'' implies ''q''; ''p'' is asserted to be true, so therefore ''q'' must be true" Linguistics * Implicature, what is suggested in an utterance, even though neither expressed nor strictly implied * Implicati ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]