MATERIALIMPLICATION

Material implication

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction if and only if the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and can replace each other in logical proofs.

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material implication

Noun

  1. An implication as defined in classical propositional logic, leading to the truth of such as <math>Q \vdash P \to Q</math>, to be read as "any proposition whatsoever is a sufficient condition for a true proposition".
    In the truth table in Figure 1, the first row corresponds to modus ponens, the last row corresponds to modus tollens, the second row could be taken to represent an invalid argument (where P→Q is the argument and P is a premise or conjunction of premises), and the third row helps ensure that an argument of the form <math> P \rightarrow Q, \neg P \vdash \neg Q </math> is invalid.
    The following paradox (and also axiom) of material implication: <math> P \rightarrow (Q \rightarrow P) </math> could be taken to mean the monotonicity of entailment, that is, if 'P' is true then no other or new fact 'Q' should be able to arise which would imply the nullification of P's truth, i.e., it could not be the case, for any 'Q', that Q → ¬P.


The above text is a snippet from Wiktionary: material implication
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