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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?

FNH
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  • In the usual axiom/rule of inference setup, from nothing we can derive nothing. – André Nicolas Sep 30 '14 at 04:17
  • @AndréNicolas, What is the usual axiom/rule of inference setup? – FNH Sep 30 '14 at 04:18
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    The details vary an awful lot. But we have axioms and rules of inference, and a proof is a finite sequence such that every sentence in the sequence is either an axiom or follows from a subcollection earlier in the sequence by a rule of inference. A theorem is the last sentence of a proof. – André Nicolas Sep 30 '14 at 04:22
  • @AndréNicolas, Ok, Logically, Is it ok to consider a rule of inference of the form I mentioned in the questions and hence to use it as a first formula in the deduction? I'm asking about whether is is logically correct. here it's trivailly true that the first formula follows from the previous ones by means of a rule of inference since there is no previous ones and our rule of inference doesn't need previous ones.. – FNH Sep 30 '14 at 04:30
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    Sure, we can have rules of inference only, no axioms, by having fules of inference that allow certain inferences from an empty collection. – André Nicolas Sep 30 '14 at 04:31
  • Notice they said "our rules of inference." As Andre said, the answer to this question highly depends on the choice of logic. – DanielV Sep 30 '14 at 04:39
  • @DanielV, What do you mean by the choice of logic? Do you mean by logic here? the choice of axioms and rules of inferences? or different kind of logics (modals elc ... ) ? – FNH Sep 30 '14 at 05:48

3 Answers3

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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.

Mauro ALLEGRANZA
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There are many, many, many ways of defining the proof system for first-order logic. Some authors may consider the logical axioms to be "axioms"; some may consider them to be "rules of inference". In general, I would read the quote in the question to mean there will be deductions without non-logical axioms, because this is almost certainly what the author intended.

In other deduction systems, such as the tableau method, it is possible to prove all logically valid formulas with no axioms at all - there are no "logical axioms" in these systems. I mention the tableau system in particular because it is so different from the Hilbert-style system that you refer to. A tableau-style deduction is a tree with the negation of the desired formula at its root, and other formulas labeling the other nodes of the tree.

Similarly, the proof systems that are known as "natural deduction" have a very different structure than Hilbert-style systems. A deduction in a natural deduction system is a different sort of tree, not just a list of formulas.

Carl Mummert
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Your reasoning is correct. However, t's difficult to find any technical or even philosophical motivations for the conception of such logic system you are describing. Such system would be as useful as a trivial one, i.e. an inconsistent system. For practical purposes, we are interested in systems we can derive something from something.

Bruno Bentzen
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