eolas/zk/Logical_equivalence.md

---
tags: [propositional-logic]
---

# Logical equivalence

> Two sentences, P and Q, are truth-functionally equivalent if and only if there
> is no truth assignment in which P is true and Q is false

### Informal expression

P: If it is raining then the pavement will be wet.

Q: The pavement is not wet unless it is raining.

### Formal expression

$$
(P \rightarrow Q) \longleftrightarrow (\lnot P \lor Q)
$$

### Truth-tables

| $P$ | $Q$ | $ P \rightarrow Q $ | $ \lnot P \lor Q$ |
| --- | --- | ------------------- | ----------------- |
| T   | T   | T                   | T                 |
| T   | F   | T                   | F                 |
| F   | T   | T                   | T                 |
| F   | F   | F                   | T                 |

### Derivation

> Propositions $P$ and $Q$ are equivalent in a system of
> [derivation](Formal%20proofs%20in%20propositional%20logic.md) for
> propositional logic if $Q$ is derivable from $P$ and $P$ is derivable from
> $Q$.

Note that the property of equivalence stated in terms of derivablity above is
identical to the derivation rule for the
[material biconditional](Biconditional_Introduction.md):

![bi-intro.png](bi-intro.png)