Understanding the Logical Implication ~p → ~q in Propositional Logic

~p → ~q

The expression ~p → ~q is a logical statement in mathematics, specifically in propositional logic

The expression ~p → ~q is a logical statement in mathematics, specifically in propositional logic. It represents the implication or conditional statement that if ~p (not p) is true, then ~q (not q) must also be true.

The symbol “→” represents the logical implication, which means that the statement on the left side of the arrow (~p) is the antecedent or premise, and the statement on the right side (~q) is the consequent or conclusion.

To understand the truth value of the expression, we should consider the truth values of the statements ~p and ~q. If ~p is true (meaning p is false), then the implication statement holds, regardless of whether ~q is true or false. However, if ~p is false (meaning p is true), then the implication is automatically considered true, since any consequence can follow from a false premise.

In simpler terms, the expression ~p → ~q can be summarized as follows:
– If p is false, then the value of q is not relevant in determining the truth of the statement.
– If p is true, then the statement is considered true, regardless of the value of q.

It’s important to note that this explanation assumes the context of propositional logic, where the symbols ~, →, and p, q are representing logical propositions that are either true or false.

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