# What does it mean for an axiom to be logical?

I have recently been hearing the phrase logical axiom being thrown around in reference to the philosophy of mathematics and I'm having a hard time understanding what one might mean when they are using it.

By the use of the word logical I am tempted to believe that whatever word is to follow has been somehow obtained through a logical sequence of steps.

If an axiom is something that is assumed true in absence of a logical sequence of steps, it is a bit difficult for me to understand what one means by qualifying it as logical.

• A logical axiom can be obtained through a logical sequence of steps. The number of steps just zero. Aug 28, 2013 at 5:05
• @Joseph, thanks for the title. That's exactly what I wanted to change it to, but didn't dare. Aug 29, 2013 at 1:15
• @HunanRostomyan no worries! In general, please feel free to dive in! :) I actually wrote about this a little while ago Aug 29, 2013 at 1:17

An axiom is simply a primitive sentence of a language system. It's often used in two different contexts:

Context 1. Sentence s is an axiom if and only if ∅ ⊢ s.

Context 2. Sentence s is an axiom if and only if ∅ |= s.

The first context is that of syntactical language systems (i.e. proof systems). There, the main logical relation is that of syntactic consequence ( ⊢ ). A sentence in such systems is said to be an axiom just in case it's a syntactic consequence of the null set of premises.

The second context is that of semantical language systems. There, the main logical relation is that of semantic consequence ( |= ). A sentence in such systems is said to be an axiom just in case it is a semantic consequence of (or is logically entailed by) the null set of premises.

As David H already mentioned: in both of these contexts, an axiom is a sentence that does follow from a set of other sentences (namely: ∅) by a number of logical steps (namely: 0). This, of course, trivially.

Lastly, in the context of philosophy of mathematics, a certain set of axioms might be called "logical" simply to distinguish them from axioms governing the use of what we ordinarily take to be 'extra-logical' notions. For example, a set of axioms for propositional logic may be designated as "logical axioms" to distinguish them from a set of axioms for arithmetic or set theory; these other sets of axioms in this context may then be designated respectively as "arithmetical axioms" and "set theoretic axioms."

Don't they just mean an axiom of logic as opposed to an axiom of something else? The law of the excluded middle is a logical axiom. The Peano axioms are axioms of the natural numbers. For all x, y, xy = yx is an axiom of Abelian groups.

I would say that what makes an axiom -logical- is that it has no particular subject matter. So 'things equal to the same thing are equal to each other' is a logical axiom; 'a circle can be drawn with any point as center and with any radius' is NOT a logical axiom, as it has geometrical terms as subject matter.

In symbolic logic, I believe this translates into the following: a logical axiom is universally valid (i.e true in all models), a non-logical axiom is only true in some models (esp. in the intended model).