eolas/zk/Logical_consistency.md

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

# Logical consistency

## Informal definition

A set of propositions is consistent if and only if **it is possible for all the
members of the set to be true at the same time**. A set of propositions is
inconsistent if and only if it is not consistent.

### Demonstration

The following set of propositions form an inconsistent set:

1. Anyone who takes astrology seriously is a lunatic.
2. Alice is my sister and no sister of mine has a lunatic for a husband.
3. David is Alice's husband and he read's the horoscope column every morning.
4. Anyone who reads the horoscope column every morning takes astrology
   seriously.

The set is inconsistent because not all of them can be true. If (1), (3), (4)
are true, (2) cannot be. If (2), (3),(4) are true, (1) cannot be.

## Formal definition

> A finite set of propositions $\Gamma$ is truth-functionally consistent if and
> only if there is at least one truth-assignment in which all propositions of
> $\Gamma$ are true.

### Informal expression

```
The book is blue or the book is brown
The book is brown
```

### Formal expression

$$
\{P \lor Q, Q\}
$$

### Truth table

$ \{P, Q\} $ form a consistent set because there is at least one assignment when
both propositions are true. In fact there are two (corresponding to each
disjunct) but one is sufficient.

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

## Derivation

> In terms of logical derivation, a finite $\Gamma$ of propositions is
> **inconsistent** in a system of derivation for propositional logic if and only
> if a proposition of the form $P \& \lnot P$ is derivable from $\Gamma$. It is
> **consistent** just if this is not the case.

In other terms, if you can derive a contradiction from the set, the set is
logically inconsistent.

A
[contradiction](Logical_truth_and_falsity.md#logical-falsity)
has very important consequences for reasoning because if a set of propositions
is inconsistent, any other proposition is derivable from it.

![](/img/derivation_from_contradiction.png)

_A demonstration of the the consequences of deriving a contradiction in a
sequence of reasoning._

Here we want to derive some proposition $Q$. If we can derive a contradiction
from its negation as an assumption then, by the
[negation elimination](Negation_Elimination.md)) rule, we can
assert $Q$. This is why contradictions should be avoided in arguments, they
'prove' everything which, by association, undermines any particular premise you
are trying to assert.