67 lines
2.4 KiB
Markdown
67 lines
2.4 KiB
Markdown
|
---
|
|||
|
|
tags:
|
|||
|
|
- Logic
|
|||
|
|
- propositional-logic
|
|||
|
|
- 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.
|
|||
|
|
|
|||
|
|
### Demonstration
|
|||
|
|
|
|||
|
|
The following set of sentences 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 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.
|
|||
|
|
|
|||
|
|
### Informal expression
|
|||
|
|
|
|||
|
|
````
|
|||
|
|
The book is blue or the book is brown
|
|||
|
|
The book is brown
|
|||
|
|
````
|
|||
|
|
|
|||
|
|
### Formal expression
|
|||
|
|
|
|||
|
|
````
|
|||
|
|
{P v Q, Q}
|
|||
|
|
````
|
|||
|
|
|
|||
|
|
### Truth-table
|
|||
|
|
|
|||
|
|
````
|
|||
|
|
P Q P ∨ 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 sentence 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.
|
|||
|
|
|
|||
|
|
A [contradiction](Logical%20truth%20and%20falsity.md#logical-falsity) contradiction has very important consequences for reasoning because if a set of propositions is inconsistent, every and all other propositions are derivable from that set.
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
*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%20Elimination.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.
|