# Definition:Logical Implication/Distinction with Conditional

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## Distinction between Logical Implication and Conditional

It is important to understand the difference between:

and:

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

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic