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:

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 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 using the $\implies$ construct, which are still in the process of being amended to use the $\leadsto$ construct as they should.