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material conditional (material implication) implies, if P then Q, it is not the case that P and not Q propositional logic, Boolean algebra, Heyting algebra: is false when A is true and B is false but true otherwise.
Then we have by the law of excluded middle [clarification needed] (i.e. either must be true, or must not be true). Subsequently, since P → Q {\displaystyle P\to Q} , P {\displaystyle P} can be replaced by Q {\displaystyle Q} in the statement, and thus it follows that ¬ P ∨ Q {\displaystyle \neg P\lor Q} (i.e. either Q {\displaystyle Q ...
P, as an individual or a class, materially implicates Q, but the relation of Q to P is such that the converse proposition "If Q, then P" does not necessarily have sufficient condition. The rule of inference for sufficient condition is modus ponens , which is an argument for conditional implication:
Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true (or in other words, true by definition). All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence. [1]
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
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, where P and Q are called the statement's disjuncts.
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris ...