Axioms and rules of inference are fundamentally different:

Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences.

As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.

[1] A mathematical introduction to logic, by H. Enderton

[2] Untersuchungen über das logische Schliessen I, by G. Gentzen

> Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences.

Yes, but the derivations themselves will change.

> As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.

Yep. I'm aware of the phenomenon that a single mathematical object of type T (say, infinity-categories) may admit multiple presentations by objects of type T' (say, model categories). But, just because two objects of type T' present the same object of type T (e.g., two Quillen-equivalent categories), it doesn't mean that they are equal in all respects (e.g., the category of simplicial sets is much nicer than the category of topological spaces).

Exactly. What are axioms and what are rules of inference ends up being a distinction without a difference.

I'm not sure I would go that far.

There is a difference, but it's not clear quite what it is.

For example, it becomes progressively harder to get nice (with cut elimination and finite rule schemata) sequent calculi for richer axiomatic systems. For example I have not come across a nice (in the above sense) sequent style formalisation of ZFC set theory. Have you?

No, but why is that a problem?

It's an indication, a hint, that rules and axioms are not always completely exchangable.

If you have different rules, you have a different object and the meaning of the axiom is different. Even if it is written using the same symbols.

To continue the compiler analogy..

Just because the ASCII sequence "int c=0;" means different things in C and Java, doesn't imply "int c=0;" is meaningless when specifically talking about only C.

The problem with your analogy is that C and Java have completely different abstract syntaxes. A better analogy would be taking the syntax of an existing programming language, and completely changing its meaning. For instance, consider the effect of making Racket use lazy evaluation (Lazy Racket), or making Haskell use strict evaluation (the upcoming -XStrict pragma). Strict and lazy languages validate different sets of equational laws (and hence compiler optimizations!), neither of which is a subset of the other, so this is perhaps a more interesting example than switching between intuitionistic and classical logic.

But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.

What you're saying is more or less equivalent to “two C implementations targeting different [architectures / operating systems / whatever] are actually implementations of two different programming languages”.

No, I'm saying the meaning behind statements are different. On some architectures, an int is 16-bit, on others 32-bit.

Anyway, this analogy was pushed too far a long time ago.

Ah, lovely! You just arrived exactly where I was trying to get.

> I'm saying the meaning behind statements are different.

Yes, exactly! And, just like a single C program can have two different meanings under two different implementations, the same axiomatic system can have two different meanings when deducing its consequences (proving theorems) using different rules of inference.

Not necessarily. It's very possible for one system to embed inside the other. In this case you can often talk about exactly the same axiom/theorem/whatever within different inferential systems.

An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic. That said, we can use flattening to embed classical logic inside of intuitionistic logic and then recover a whole family of propositions of which AC is one and also see why it has a difficult time holding in IPL.

> An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic.

Naïvely encoded in Martin-Löf type theory, that is.

Still, "axiom of choice is a theorem in intuitionistic logic" is absolutely true whether or not you accept intuitionistic logic.

But thas is not Mathematics, they call it "metamathematics" for something.

This seems to be a technical objection that dodges the meat of the parent's comment. In what sense is it true that a given theorem follows from a given set of axioms and a given choice of inference rules?

A “theorem” is by definition a statement in a mathematical theory that has a proof. In what sense is this true? Well, that depends on what “true” means in your model... :-p

I agree with parent comment. At the heart of your disagreement is that you seem to use definition from logic of what axioms are, while your opponents use "definitions" from metamathematics.

Umm. I've been of the understanding that we never change the inference rules, Can you point to some branch of mathematics that involves a change of inference rules?

At my level of understanding, it seems a reasonable characterization of intuitionistic logic to say that it operates with a more restricted set of inference rules. It was the parent who made the stronger claim about differences here, and they speak a bit more to the topic in nearby comments.