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.

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    You are confusing two different senses of "unprovable". Axioms are unprovable from outside a system, but within it they are (trivially) provable. In this sense they are tautologies even if in some external sense they are false (which is irrelevant within the system). Godel's Incompleteness is about very different kind of "unprovable" (neither provable nor disprovable). False axioms are not enough to prove "anything", the system needs to be inconsistent for that. Finally, to even talk about "true" or "false" axioms one needs to fix an interpretation, and those are usually not unique. – Conifold Apr 19 '17 at 2:41
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    Asserting something true is not saying it's "trivially provable". To say you can prove the axiom within the system by definition means that that system is be flawed. Godel's point was we cannot prove these things true within the system. A therefore A isn't a proof of A, while it is a tautology, the original axiom still seems to be A and could serve a role as a premise but not a conclusion. – Tatarize Apr 19 '17 at 2:55
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    Asserting that something is a tautology is saying that it is true in every interpretation (equivalently, provable in the system by Godel's completeness theorem). Axioms are (obviously) true in every interpretation, hence they are tautologies. Godel's incompleteness theorem states that some statements true in a particular interpretation can not be proved in any formal system (first order, with arithmetic, etc.) extending the original system, unless it is inconsistent. – Conifold Apr 19 '17 at 3:29
  • What about David Schwartz point with regard to trivially proven axioms with the axioms themselves? "Every proof from within a system is true if, and only if, all the system's axioms are true. But since A is one of the system's axioms, that means you've proven that A is true if and only if A is true. That only proves A if that is A, which it can't be because that would be incoherent self-reference." -- I accept the axioms are true, but doesn't that mean I cannot ultimately prove they are true without running afoul of Godel? I can't assume them true them prove them with themselves, can I? – Tatarize Apr 27 '17 at 12:13
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    Your understanding of Gödel is incorrect: that all theorems are provable is vacuously true because "theorems' are defined as well-formed sentences that are provable. The rest confuses object theory and meta-theory. Löb's theorem only makes sense in meta-theory and "proving the axiom of equality with the axiom of equality when the proof itself would be directly invoking the axiom of equality" mixes language with meta-language (these are two different axioms of equality, one operating in the formal system, another in its meta-theory). – Conifold Apr 28 '17 at 1:19
up vote 1 down vote accepted

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.

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    Every axiom has a proof. If A is an axiom, its proof is A. A one-liner. – user4894 Apr 19 '17 at 17:56
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    @user4894 Every proof from within a system is true if, and only if, all the system's axioms are true. But since A is one of the system's axioms, that means you've proven that A is true if and only if A is true. That only proves A if that is A, which it can't be because that would be incoherent self-reference. A system cannot prove its own axioms, you cannot both assume something is true and prove that it is true, unless the "it" is a self-referential tautology. (Edited to clarify.) – David Schwartz Apr 19 '17 at 18:02
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    You're confused about truth, formal systems, and models. "A" is a proof of A. It's a finite sequence of steps, each one an axiom or a statement that follows from axioms. Truth only applies to models, not axiom systems. A proof can't be true, your first sentence is just wrong. – user4894 Apr 19 '17 at 18:14
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    @user4894 A "proof" is a demonstration that some conclusion follows from some premises. What a proof proves is not its conclusion but that its conclusion follows from its premises. If you could prove your premises, you would not need them as premises. (I apologize for the sloppy wording in my response to you, but my point is quite correct. Within a system, you cannot prove any of that system's premises. That's because you've already assumed them to be true. At best, you can show that the premises follow from themselves.) – David Schwartz Apr 19 '17 at 19:01
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    If you don't understand why "A" is a proof of A, you need to go learn the basics. Truth bro'. – user4894 Apr 19 '17 at 19:41

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.

  • Every theory is axiomatizable -- for example, take the set of all theorems of the theory to be the axioms. Presumably you mean "finitely axiomatizable" or "recursively axiomatizable". – Hurkyl Aug 10 at 17:53

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).

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    Were it true that you could prove a tautology with no axioms, we would not need the axiom of equality. The most common first axiom is "all tautologies are true". – David Schwartz Apr 19 '17 at 0:14
  • @DavidSchwartz I included the second paragraph to cover that side of things. – Cort Ammon Apr 19 '17 at 0:16
  • But I think you kind of didn't. Your second paragraph refutes your first and leaves the question unanswered. (That might be the best we can do, but if so, I think it can be done more clearly.) – David Schwartz Apr 19 '17 at 0:26
  • @DavidSchwartz So wikipedia says "A tautology is a formula that is true in every possible interpretation." The question would be what defines "every possible interpretation." If you are only considering a universe with exactly one interpretation (the one true interpretation with the current set of axioms), then I suppose every statement becomes a tautology. On the other hand, if we choose a proof system such as FOL, then the term starts to take on more meaning. – Cort Ammon Apr 19 '17 at 3:40
  • I think a more useful definition of tautology is, "a phrase or expression in which the same thing is said twice in different words." That is, a tautology is the claim that something is equivalent to itself. But I guess part of the problem is that one can interpret the question to be about formal logic systems, about the philosophy of actual knowledge, or several other ways. – David Schwartz Apr 19 '17 at 3:54

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