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Logic/General_concepts/Consistency.md → Logic/General_concepts/Logical_consistency.md
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---
categories:
- Mathematics
tags: [logic]
- Logic
tags: [propositional-logic]
---
# Logical consistency
## Informal definition
A set of sentences 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 sentences is inconsistent if and only if it is not consistent.
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 sentences form an inconsistent set:
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.
```
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 sentences $\Gamma$ is truth-functionally consistent if and only if there is at least one truth-assignment in which all sentences of $\Gamma$ are true.
> 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
@ -34,12 +34,14 @@ The book is brown
### Formal expression
```
{P v Q, Q}
```
$$
\{P \lor Q, Q\}
$$
### 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 ∨ Q Q
T T T T *
@ -50,7 +52,7 @@ 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 sentence of the $P & \sim P$ is derivable from $\Gamma$. It is **consistent** just if this is not the case.
> 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 $P & \sim P$ 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.