Definition:Logical Equivalence/Also denoted as

Logical Equivalence: Also denoted as

Some sources denote $p \dashv \vdash q$ by $p \leftrightarrow q$.

Others use $p \equiv q$.

In modal logic, logical equivalence is expressed as:

$\Box \paren {p \equiv q}$

On $\mathsf{Pr} \infty \mathsf{fWiki}$, during the course of development of general proofs of logical equivalence, the notation $p \leadstoandfrom q$ is used as a matter of course.

The $\LaTeX$ code for \(p \leadstoandfrom q\) is p \leadstoandfrom q .

The $\LaTeX$ code for \(\Box \paren {p \equiv q}\) is \Box \paren {p \equiv q} .

Note that in Distinction between Logical Implication and Conditional, the distinction between $\implies$ and $\leadsto$ is explained.

In the same way, $\leadstoandfrom$ and $\iff$ are not the same -- it makes no sense to write:

$A \iff B \iff C$

when what should be written is:

$A \leadstoandfrom B \leadstoandfrom C$

Sources

• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equivalence: 2. (logical or strict)
• 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equivalence: 2. (logical or strict)