# Inconsistent systems

• [Hackstaff, 1966] (Doesn't actually cover the systems, but covers (p198-199) the notions of consistency necessary to discuss the systems.

## Background

In studying logic systems we commonly see something like "If you add axiom [whatever] to the system you get an inconsistent system". However, if you are being VERY carefull there are some things that should be said about the inconsistent systems.

An Inconsistent system is, of course, one that is not consistent. But what does this mean? There is a traditional view (Aristole consistenty) that consistent means that it does not have any contradictions (it does not have p&~p as a theorem). But this breaks down when dealing with logics that don't have negation, since some are clearly problematic in a manner similar to an inconsistency and some are not. And what about systems with more than one "negation"? How do you chose which one (or ones) count as a contradiction?

So, in the early part of the 20th Century a closer look was taken at the concept of consistency and attempts were made to generalize the concept.

Note: This page assumes you are at least familiar with the concept of a Well Formed Formula (a wff) of a system.

This page doesn't really make any effort whatsoever to distinguish various possible inconsistent systems, and different inconsistent systems can, for example, have different operators.

## Consistency

There are a number of formal definitions of "consistent", and a number of ways to categorize them. I'll follow the terminology of L.H. Hackstaff's book "Systems of Formal Logic", and his explaination of the viewpoints.

The four definitions he covers are:

### Aristotle Consistency

One of the first writers to formally make a distinction between consistent and inconsistent systems was Αριστοτἑλης (Aristotole). He noticed that direct contradictions lead to problems and (effectively) took the concepts of contradiction and inconsistency as equivalent. Many people to this day take contradictions as being equivalent to inconsistency, since a classical system with one has the other.

However, there is now research into systems (So called "paraconsistent" logics) that admit to some contradictions without everything breaking down.

### Post Consistency

Emil Post was one that took the consistency problem as being that you could prove false statements. And for his work he proposed the definition of consistency that a logical system was consistent if it could not prove a single propositional letter as a theorem ("p", for example). [Since this would allow ANYTHING to be derived by one substitution step]

### Hilbert Consistency

Hilbert, in a manner similar to Post, said that a system was consistent if there is a selected wff that is not a theorem of the system. (If that wff is p&~p this can be the same as Aristotle consistency)

### Absolute Consistency

Absolute consistency says that not all wffs of the system are theorems.

## Inconsistency

A system that is not consistent by any of these definitions is called inconsistent by the definitions. So something that is, for example, not Post consistent is called Post inconsistent.

Classical systems that are Aristotle inconsistent are also Post, Hilbert, and Absoulutely inconsistent.

Any system with negation and conjunction that admits ~p&p as a wff and is Post inconsistent is also Aristotle inconsistent since ~p&p can be obtained by substitution.

A system that doesn't admit operatorless wiff's can be Hilbert and Absolutely inconsistent without being Post inconsistent.

Any system without negation can be Hilbert, Post, and Absolutely inconsistent without being Aristotle inconsistent.

## Basis for:

An Inconsistent systems are the strongest logic system there are... if you add anything to them you still get an inconsistent system.

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