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In hmakholm's answer to this question, the following is written:

If it has no axioms then there is no way to begin a proof in the empty theory, and without rules of inference all that could be proven would be the axioms themselves.

I don't see how either of these assertions can be true. Tackling the first; let's say I have a deductive system which has the following rule, among other:

Γ⊢θ and Γ⊢ϕ, then Γ⊢θ∧ϕ

Well, whenever I wind up with the two premises, I am able to prove θ∧ϕ, right?

Now, to tackle the second; let's say my logic has the following axiom:

P∨¬P

Let's say I find out, or assume, that n is even. Well, now I have to premises; an axiom and a fact/assumption.

P∨¬P2∣n∴¬(2∤n)

Surely, ¬(2∤n) is not an axiom? It follows from premises, unlike axioms, so it cannot be an axiom, right?

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    In your first example, you have a rule and in your second example you have an axiom. How do these contradict the statement that you need rules and axioms?
    – John Douma
    Sep 24, 2022 at 6:03
  • @JohnDouma I think the OP is suggesting that in the first example there is no axiom and in the second, no rule of inference.
    – David
    Sep 24, 2022 at 6:10

2 Answers 2

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(1) Where did Γ come from? Ultimately it must have come from an axiom (if it is not an axiom itself).

(2) You are using a rule of inference: from Φ deduce ¬(¬Φ).

And BTW, "n is even" is expressed by 2∣n, not n∣2.

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  • This is what I suspected. I am just having trouble with this since they are fundamentally the same thing; assumptions. But I guess, if I am to deduce something from an axiom, I must have a rule for how to convert that axiom into an equivalent proposition.
    – user110391
    Sep 24, 2022 at 6:14
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Natural Deduction systems like Fitch-Style systems have no axioms, but plenty of inference rules. However, some proofs in such a system might not be regarded as such to some critics. Take the two following proofs:

  1. p (Hypothesis)
  2. p→p (1-1 → Intro)

and

  1. (p∧¬p) (Hypothesis)
  2. ¬(p∧¬p) (1-1 ¬ intro)

These are valid proofs in such a calculus, and require no axioms. I do think they are a bit convenient, are not very informative, and are still presuppositions that go into ND.

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