# Is “all swans are white” equivalent to “if it is not white, then it is not a swan”?

More formally, is "All As are Bs" equivalent to "if it is not a B, then it is not an A"?

• You might find it interesting to read about the Raven Paradox – lulu Jun 8 '16 at 20:15
• Don't lose your quantifier: "Everything that is not an B is also not a A". – DanielV Jun 8 '16 at 21:37

Yes. This is called the contrapositive of the statement. $A \Rightarrow B$ is equivalent to the statement $\neg B \Rightarrow $$\neg A See here as well. • I would prefer saying "A\implies B is equivalent to saying \neg B\implies \neg A", which is stronger. – yo' Jun 8 '16 at 19:32 • You're right. I'll edit the post. – M10687 Jun 8 '16 at 19:32 • I see in that link I can say they are "logically equivalent". Can I also say they are "materially equivalent", that is "all swans are white if and only if, if they are not white, they are not swans"? – kaspersky Jun 8 '16 at 19:39 • The statement you have in quotes is logical equivalence. Try writing it out in quantifiers to see this. – M10687 Jun 8 '16 at 19:41 Yes, it's true. Let p_A(x) be "x is A" and p_B(x) be "x is B". Then "All As are Bs" is$$\forall\, x: \; p_A(x) \rightarrow p_B(x).$$Statement p_A(x) \rightarrow p_B(x) is false in only one case: when p_A(x) is true and p_B(x) is false. Similary statement \neg p_B(x) \rightarrow \neg p_A(x) is false only if \neg p_B(x) is true and \neg p_A(x) is false, i.e. when p_A(x) is true and p_B(x) is false i.e. only when firs statement is false. Thus these two statements are equal because they are false and true in the same time:$$\forall\, x: \; p_A(x) \rightarrow p_B(x) \Leftrightarrow \forall\,x : \neg p_B(x) \rightarrow \neg p_A(x).$\$