TITLE: Difference(s) between an axiom scheme and an axiom
QUESTION [7 upvotes]: The basic question which motivated me to write this post is the following,

What is(are) the difference(s) between an axiom scheme and an axiom?

In Margaris's book First Order Mathematical Logic we have the following, 
However, the difference between an axiom and an axiom scheme is not clearly stated there in the sense that in the book it is not clearly specified exactly what is(are) the property (or properties)  that help us to distinguish between an axiom and an axiom scheme.
Question

What is an axiom scheme? Is an axiom scheme different from one of its instances? If so, then in what sense exactly? 


Disclaimer

I had some discussion with user21820 regarding this question and one can see our discussion here.
For a different and more philosophical version of the question see here.

REPLY [2 votes]: Sequent logics have axioms and inferences.  These are mechanically different things.  $\dfrac{P \to Q,~P}{Q}$ is an inference, where as $\lnot \exists y ~:~ y + 1 = 0$ is an axiom.  
Even though $P \to Q \to P$ looks like an axiom, it isn't (see caveat below).  It is the inference $\dfrac{P \text{ is Prop},~Q \text{ is Prop}}{P \to Q \to P}$, where $P$ and $Q$ are just placeholders for any arbitrary propositional expression.  $\vdash (A \to B) \to (C \land D) \to (A \to B)$ is a valid instance of the inference A1.  If $P \to Q \to P$ was an axiom, then it wouldn't affect the provability of $(A \to B) \to (C \land D) \to (A \to B)$ for the same reason that the axiom "all squares have 4 sides" can't be instantiated as "all triangles have 4 sides".  Axioms mean exactly what they say and nothing else.
But it looks a lot like an axiom.  Also, every proof only uses a finite number of instances of (for example) A1.  Suppose a proof uses 70 instances of A1.  You could say it is uses 1 inference, or you could say it is uses 70 different axioms and doesn't use the inference.  Both would be correct.  Thus the term "axiom-schema".  As far as why you would ever want to look at things that way:  consider the principle of mathematical induction:
$$\forall P() ~:~ \bigg(P(0) \land (P(k) \to P(k + 1)) \vdash \forall n~:~ P(n)\bigg)$$
This can be treated as a rule of inference, or as 1 second order axiom, or as finite number of first order axioms for any given proof, or as a countably infinite number of axioms in general.  All of those ways of looking at induction are correct.  So it is often called an "axiom-schema" to reference how it spans the ideas of inferences and sets of axioms.  Furthermore, many results of first order logic (without restrictions to finite sets of expressions) will apply to this despite it also being a second order axiom since it could also be described as infinitely many first order axioms.
Caveat:  It is possible to introduce an inference known as something like "universal propositional substitution" (UPS), which basically says "If P is proven, and A is a propositional variable and B is any proposition, then all instances of A can be replaced by B to create a new proven expression P[B/A]".  If you have UPS then A1 etc can be treated as axioms, and some logicians like Jaśkowski did that instead of having axiom-schemas.  The pros and cons of using UPS instead of axiom schemas are probably for another question though.