More formally, is "All As are Bs" equivalent to "if it is not a B, then it is not an A"?
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2$\begingroup$ You might find it interesting to read about the Raven Paradox $\endgroup$– luluJun 8, 2016 at 20:15
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$\begingroup$ Don't lose your quantifier: "Everything that is not an B is also not a A". $\endgroup$– DanielVJun 8, 2016 at 21:37
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3 Answers
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Yes. This is called the contrapositive of the statement. $A \Rightarrow B$ is equivalent to the statement $\neg B \Rightarrow $$\neg A$
See here as well.
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$\begingroup$ I would prefer saying "$A\implies B$ is equivalent to saying $\neg B\implies \neg A$", which is stronger. $\endgroup$– yo'Jun 8, 2016 at 19:32
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$\begingroup$ I see in that link I can say they are "logically equivalent". Can I also say they are "materially equivalent", that is "all swans are white if and only if, if they are not white, they are not swans"? $\endgroup$ Jun 8, 2016 at 19:39
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$\begingroup$ The statement you have in quotes is logical equivalence. Try writing it out in quantifiers to see this. $\endgroup$– M10687Jun 8, 2016 at 19:41
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Yes, it's true. Let $p_A(x)$ be "$x$ is $A$" and $p_B(x)$ be "$x$ is B". Then "All $A$s are $B$s" is $$\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).$$
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Yes. Or: "If it is not white and it is still a swan, then it is not true that 'All swans are white'."
