Are axioms tautologies?

Question

My understanding is that axioms are the unprovable statements upon which systems are built. Tautologies are in essence things that can't be false.

Godel's Incompleteness Theorem, though, shows that they are just unprovable. Saying something can be proved is not insisting that they are true. Just if they are true, we get a coherent system as a result.

You could have false axioms within a logical system. It would be able to produce systems that could prove anything including their own validity. If they could be false, would they not be tautologies, but rather unprovable statements that we assume to be true?

It would not only seem odd to say, "I assume X is true, therefore X is true," but that could be shown to violate Lob's Theorem.

So are axioms tautologies? I see this asserted a lot.

Answer 1

It is worth separating the logic from the epistemology. Let's start with the logic.

A (first order) theory is a set of sentences. Usually we are interested in deductive systems, so we require a theory to be closed under the relation of provability. A theory T is axiomatizable if there exists a subset of T, the axiom set, such that all of the sentences in T are provable from that axiom set. Not all theories are axiomatizable, and in general there may be many different ways to axiomatize a single theory. As far as the logic goes, there need be nothing special about the members of the axiom set. They just denote one way of representing the content of the theory. Usually we eliminate redundancy from an axiom set by not including sentences that can be proved from the other sentences within it.

As to the epistemology: a theory often has an intended interpretation. We usually want the sentences in T to be statements that are true about some domain. It is not essential that the sentences are true: they could be just uninterpreted formulas with no truth value, or they could even be false under some interpretation. The theory won't explode unless it is inconsistent. But if we do want the sentences to be true relative to some interpretation, we then have reason to choose the members of the axiom set in such a way that they represent the sentences in T of which we are most certain. The epistemological motivation is to represent the theory as flowing from the axioms; the axioms provide a warrant or a justification for accepting the theory as a whole.

In general then, axioms of a theory do not need to be tautologies, and indeed do not even need to be true. But there is an important rider here. Because logicians have tidy minds, we often wish not only to axiomatize our theories but to axiomatize our logic itself. This means expressing everything that can be proved within some logic L as the consequence of some finite set of axioms and rules. When we do this, the sentences that can be proved within our logic are "logical truths". Depending on which authors you read, "logical truth" might be a synonym for "tautology", though I prefer the usage that "tautology" is reserved for logical truths of the propositional calculus. Either way, we can say of a logic L that all of its theorems, including its axioms, are logical truths, or if you wish to be more cautious, that they are true-in-L.

So, if you are talking about the axioms of a logic, as opposed to a theory, then the axioms are logical truths.

Answer 2

An axiom is something you assume to be true without proof. A tautology is a statement which can be proven to be true without relying on any axioms. An axiom is not a tautology because, to prove that axiom, you must assume at least one axiom: itself.

If you wanted to be more pedantic (which is always fun), the idea that you can prove a tautology without any axioms is a bit fun to tug on. After all, you must use rules of inference to prove a tautology, and you could argue those rules are axioms. In doing so, what you would show is that there's two tiers built into the way we prove things. There are the assumptions built into the proof system (such as those in predicate logic), and then there are the assumptions built into the argument (the axioms).

Answer 3

An axiom cannot be proven within the system in which it's an axiom. If it could be proven, we wouldn't need it as an axiom. You can either assume something is true or prove that something is true, but you cannot do both. If it helps, think of everything proven within a system as having an implicit "if all this system's axioms are true, then" before it.

Tautologies can be proven. But, of course, you can't prove anything without axioms, even a tautology.

So you can collapse this difference by arguing that, directly or indirectly, somehow at least one tautology must be accepted as an axiom or the system will not work. Once a tautology is accepted as an axiom, all tautologies are equivalent to that axiom.

One could, I suppose, imagine a system where no axiom is a tautology but all tautologies expressible in that system can be proven from its axioms. In that case, no tautology is an axiom in that system.

One major difference in the way they behave is that you can introduce a tautology, at any time in the proof, without changing assumptions. Introducing a new axiom does change the assumptions made by the argument.