If A is "1+1=2" and B is "apple starts with a", does $A\implies B$?
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2Does this answer your question? Are all true statements equivalent? – José Carlos Santos Apr 12 '21 at 18:19
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In a logic system $A \implies B$ simply means $\lnot A \lor B$. As the statement "Either $1 + 1 \ne 2$ or Apple starts with A" is certainly true (because apple does start with a) then, yes, the statement is true. – fleablood Apr 12 '21 at 19:22
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In a word, yes.
This is a language thing - the word “implies,” in standard usage, usually means that the two are related causally.
But in mathematics “implies” is a funnier thing. $P\implies Q$ is true whenever $Q$ is true - you don’t even need $P$ to be true. It is also true if both $P$ and $Q$ is false.
The only time it is false is if $P$ is true and $Q$ is false.
Thomas Andrews
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In symbolic logic, if B is true then A-> B is true no matter whether A is true or false. So if A and B are both true then A->B and B->A are both true.
Outside of symbolic logic we often take "A implies B" to mean "A causes B". That's whole different matter! A and B can both be true without one "causing" the other.
user247327
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