[Math] Can we deduce anything from the empty set of axioms

logicpredicate-logic

If the set of logical axioms is empty and so is the set of non-logical axioms, then it seems we can't make any deduction.As the first formula in the deduction must belong to either sets or it it's deduced from previous formulas by means of a rule of inference (no such previous formulas even exists!)

In Friendly introduction to logic, it's stated that,

Actually, after we set up our rules of inference, there will be some deductioms from the empty set of axioms, but that comes later.

Is that even possible? my answer is yes since I can think of the possibility when we have a rule of the inference of the form: $(\emptyset,\phi)$ for some tautlology $\phi$. So we can start the deduction using this particular $\phi$, Is that right?

But, If we don't have even any rule of inference at hand, Can we still have some deductions? It seems to be impossible for this to happen, right?

Best Answer

See Definition 2.4.5. [page 61] :

If $\Gamma$ is a finite set of $\mathcal L$-formulas, $\phi$ is an $\mathcal L$-formula, and $\phi$ is a propositional consequence of $\Gamma$, then $\langle \Gamma, \phi \rangle$ is a rule of inference of type (PC).

[...] notice that if $\phi$ is a formula such that $\phi_p$ is a tautology, rule (PC) allows us to assert $\phi$ in any deduction, using $\Gamma = \emptyset$.

Thus, we can start a deduction simply "writing down" a tautology.

See Ch.2.3 The Logical Axioms : the logical axioms are :

  • the three equality axioms [see page 56]

  • two quantifiers axioms [page 57].

You can compare with the very similar deductive calculus of Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), page 112 : the logical axioms are the axioms for equality, three axioms for the (universal) quantifier plus all the tautologies.

Enderton's choice and Leary's one are quite similar; in both cases, at each step in a derivation we may "introduce" a tautology.

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