necessity (logic)

In logic, necessity defines a relationship between two logical statements one of which can only be true if the other is true as well. When we say “p is necessary for q,” we mean that statement q cannot be true unless statement p is as well. If p is false then so is q. We can write this as an implication: ¬p→¬q.

Written as ¬p→¬q captures the dependence (necessity) of q‘s truth value on p, but taking the contrapositve yields a simpler conditional statement: qp, q implies p.

For example, take the statement, “If something is a canoe it needs to be a boat.” This means that if something is not a boat then it is not a canoe, ¬b→¬c. The contrapositive statement is, “If something is a canoe then it is a boat,” cb.

 

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