eolas/Logic/Consistency.md

categories	tags
Logic	propositional-logic

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 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 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.