# Definition:Logical Equivalence

## Definition

If two statements $p$ and $q$ are such that:

then $p$ and $q$ are said to be **(logically) equivalent**.

That is:

- $p \dashv \vdash q$

means:

- $p \vdash q$ and $q \vdash p$

Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign.

In symbolic logic, the notion of **logical equivalence** occurs in the form of provable equivalence and semantic equivalence.

### Provable Equivalence

Let $\mathscr P$ be a proof system for a formal language $\LL$.

Let $\phi, \psi$ be $\LL$-WFFs.

Then $\phi$ and $\psi$ are **$\mathscr P$-provably equivalent** if and only if:

- $\phi \vdash_{\mathscr P} \psi$ and $\psi \vdash_{\mathscr P} \phi$

that is, if and only if they are $\mathscr P$-provable consequences of one another.

The **provable equivalence** of $\phi$ and $\psi$ can be denoted by:

- $\phi \dashv \vdash_{\mathscr P} \psi$

### Semantic Equivalence

Let $\mathscr M$ be a formal semantics for a formal language $\LL$.

Let $\phi, \psi$ be $\LL$-WFFs.

Then $\phi$ and $\psi$ are $\mathscr M$-**semantically equivalent** if and only if:

- $\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$

that is, if and only if they are $\mathscr M$-semantic consequences of one another.

## Also known as

Two **logically equivalent** statements are also referred to as:

## 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:

- $A \leadstoandfrom B \leadstoandfrom C$

## Also see

- Results about
**logical equivalence**can be found**here**.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Theorem $29$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $3$ Truth-Tables: $\text {(v)}$ - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms - 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 - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**equivalence**:**2.**(**logical**or**strict**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**logically equivalent**