What does it mean for an axiom to be logical?

Question

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.

Answer 1

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."

Hope this helps. For questions/corrections, please leave a comment or simply edit this post.

Answer 2

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.

Answer 3

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).