eolas/Logic/Logical_equivalence.md

---
categories:
  - Mathematics
tags: [logic]
---

> 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

$$
P \supset Q \equiv \sim P \lor Q
$$

### Truth-tables

```
P	Q				P	⊃	Q
T	T					T
T	F					F
F	T					T
F	F					T
```

```
P	Q				~	P	∨	Q
T	T						T
T	F						F
F	T						T
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%20Introduction.md):

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

//TODO: Add demonstration of this by deriving two equivalents from one of DeMorgan's Laws