# 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. – David H Aug 28 '13 at 5:05
• @Joseph, thanks for the title. That's exactly what I wanted to change it to, but didn't dare. – Hunan Rostomyan Aug 29 '13 at 1:15
• @HunanRostomyan no worries! In general, please feel free to dive in! :) I actually wrote about this a little while ago – Joseph Weissman Aug 29 '13 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."