Definition:Logical Implication/Distinction with Conditional

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Distinction between Logical Implication and Conditional

It is important to understand the difference between:

$A \implies B$: If we assume the truth of $A$, we can deduce the truth of $B$

and:

$A \leadsto B$: $A$ is asserted to be true, therefore it can be deduced that $B$ is true


When $A$ is indeed true, the distinction is less important than when the truth of $A$ is in question, but it is a bad idea to ignore it.


Compare the following:

\((1):\quad\) \(\displaystyle x > y\) \(\implies\) \(\displaystyle \paren {x^2 > x y \text { and } x y > y ^2}\)
\(\displaystyle \) \(\implies\) \(\displaystyle x^2 > y^2\)


\((2):\quad\) \(\displaystyle x\) \(>\) \(\displaystyle y\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2\) \(>\) \(\displaystyle x y\)
\(\, \displaystyle \text { and } \, \) \(\displaystyle x y\) \(>\) \(\displaystyle y^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2\) \(>\) \(\displaystyle y^2\)


We note that $(1)$ is a conditional statement of the form:

$A \implies B \implies C$

This can mean either:

$\paren {A \implies B} \implies C$

or:

$A \implies \paren {B \implies C}$

instead of what is actually meant:

$\paren {A \implies B} \text { and } \paren {B \implies C}$


Hence on $\mathsf{Pr} \infty \mathsf{fWiki}$ we commit to using the form $A \leadsto B$ rigorously in our proofs.


The same applies to $\iff$ and $\leadstoandfrom$ for the same reasons.


Note that there are many pages on $\mathsf{Pr} \infty \mathsf{fWiki}$ using the $\implies$ construct, which are still in the process of being amended to use the $\leadsto$ construct as they should.


Sources