# Definition:Logical Implication/Distinction with Conditional

## 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:

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

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

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.