In logic we are taught the following definition.
$$A\to B$$
that yields false whenever $A$ is true and $B$ is false.
Who defined this and why was it defined that way? What inspired him/her?
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Think about what it would mean for $A \Rightarrow B$ to be false: you want $A$ to be true and $B$ not to be true. That is, $A \Rightarrow B$ being false should be the same as “$A$ and not B” being true. So $A \Rightarrow B$ being true should be the same as “$A$ and not $B$” being false, which is the same as its negation “(not $A$) or $B$“ being true. In other words, $A \Rightarrow B$ should have the same truth value as “(not $A$) or $B$”, which is another way of saying these statements should be equivalent. – KCd Aug 17 '20 at 11:23
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Hi, given both the lack of a worldwide standard for symbology in quantificational logic and the vagaries of HTML representation, please consider either providing a translation or replacing the symbols with their text (words) equivalent. – Carl Witthoft Aug 17 '20 at 11:38
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I agree with @KCd that this is not a "definition" but rather a logical conclusion reached based on definition of set theory. – Carl Witthoft Aug 17 '20 at 11:38
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@CarlWitthoft: I am trying to mitigate the confusion what I want to ask. I have no idea about the symbol. I am using the symbol that I learnt when I was in high school. – Display Name Aug 17 '20 at 11:43
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I think the most "fun" bit in the collection of truth tables is the rule that the statement $A \rightarrow B$ is "truth-functionally true" whenever $A$ is FALSE regardless of the value of $B$ – Carl Witthoft Aug 17 '20 at 11:56
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@conifold: Thank for the link! – Display Name Aug 17 '20 at 12:31