Definition:Logical Equivalence
Contents
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 $\mathcal P$ be a proof system for a formal language $\mathcal L$.
Let $\phi, \psi$ be $\mathcal L$-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 $\mathcal L$.
Let $\phi, \psi$ be $\mathcal L$-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, iff they are $\mathscr M$-semantic consequences of one another.
Also known as
Two logically equivalent statements are also referred to as:
- interderivable
- equivalent
- coimplicant
Also denoted as
Some sources denote $p \dashv \vdash q$ by $p \leftrightarrow q$.
Others use $p \equiv q$.
Also see
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 1.5$: Further Proofs: Résumé of Rules: Theorem $29$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2.4$: 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
- 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
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 2.3$: Truth-Tables: $\text {(v)}$