Any inconsistent theory must be complete?

Question

Assume the following definitions:

• $U$ is the set of all sentences in a language

• A theory $T$ is complete if $\forall A \in U$, $A \in T$ or ${\sim} A \in T$ or both.

• A theory is consistent if at most one of $A \in T$ or ${\sim} A \in T$ is true.

My question is:

If a theory is inconsistent, then all of the statements in the language can be derived from that theory, so would this mean that any inconsistent theory is necessarily complete?

Since if $T$ is inconsistent, $T \vdash U\longrightarrow (\forall A \in U$, $A \in T$ and ${\sim} A \in T)$.

Answer 1

To make the question specific, you also need to specify which convention you use for theories:

1. Some people define a theory to be an arbitrary set of sentences.

2. Some people define a theory to be a deductively closed set of sentences. That means: if $\phi$ is provable from a theory $T$ then $\phi$ must already be in $T$, under this definition of a theory.

The choice of definition (1) or (2) then affects many other definitions. Under definition (1), we would say a theory is inconsistent if it proves a contradiction. Under definition (2), we would say that a theory is inconsistent if it contains a contradiction. These are not the same definition.

The definition of "complete" is even more complicated. Many authors define complete theories to be maximal consistent theories. In that case, every complete theory is consistent and deductively closed, and an inconsistent theory is simply excluded from being "complete". For example, when we talk about the "completions" of a theory, we always mean the consistent completions.

On the other hand, if we did allow inconsistent theories to be called "complete" we would have to decide whether this means that for each $\phi$ the theory proves $\phi$ of $\lnot \phi$, or whether it means that for each $\phi$ the theory contains $\phi$ or $\lnot \phi$.

In general, the basic terminology of first-order logic is a real mess in the literature. There are general concepts, which are shared by all presentations. These include the general concepts of theories, completeness, and consistency. But the formal definitions vary wildly from one presentation to another.

One reason for this variation is that first order logic is used in many different fields: math, logic, philosophy, computer science, linguistics, etc. These authors have different particular needs and they adjust their definitions to match. Even inside mathematical logic there is a variation between model theory and proof theory about the definition of a first-order theory. So you have to look at each text individually to answer questions like the one asked above; there is no general answer.

Answer 2

Yes, inconsistent theories are complete. And your argument is correct, since every sentence is provable from an inconsistent theory, then every sentence is in the theory.

I should also point out that your second point is somewhat excessive. In mathematics "or" is always inclusive or, unless stated otherwise explicitly. This means that by saying "$A\in T$ or $\lnot A\in T$" we already include the case "or both".