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Most logical systems will have two distinct forms of entailment, one is system-based entailment (logical consequence), and the other is proof-based entailment (derivability). In the former, an entailment follows from showing how, in all cases, a designated value is preserved. In the latter, a set of rules is used to prove that an entailment holds.

Completeness is the step that allows us to claim that system-based entailment implies proof-based entailment; 𝛤⊨P ⇒ 𝛤⊢P.

Are systems that lack this rule considered as having failed? Are there any Philosophers who consider completeness, or a lack of completeness as a virtue of a system?

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Are systems that lack this rule considered as having failed?

Not necessarily. It depends on what you take to be the most important aspects of a system.

Are there any Philosophers who consider completeness, or a lack of completeness as a virtue of a system?

Yes. Some philosophers give pride of place to deductive power. Here I've got someone maybe from the Dummettian inferentialist tradition in mind. By their lights, what logic is for is formalizing arguments (at first-order). Given this goal, semantic incompleteness is undesirable, because it means that some entailments cannot be captured by your proof system. These philosophers will regard a failure of semantic completeness as problematic (as in the case of second-order logic, as Mauro points out), because their project requires a tight connection between logical entailment and deductive validity.

Other philosophers, however, give pride of place to expressive power. Mathematical structuralists are an example here. By these philosopher's lights, what matters is that we pin down structures - of the natural numbers, say. Since you can only do that at second-order, you have to give up semantic completeness to do it. These philosophers won't regard second-order logic as having 'failed', even though it's semantically incomplete, because it does precisely what they want it to: precisely characterize structures.

  • I appreciate your answer. With regards to your first paragraph, could you recommend any papers or chapters expressing such a notion? – BeingOfNothingness Apr 18 '18 at 13:10
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    It's been a while since I've looked at this literature, but Crispin Wright is one figure that might be useful here. Unfortunately I don't have any particular pieces in mind. Of course you can always start with Dummett, but that can be heavy going. The relevant book would be "The Logical Basis of Metaphysics". – possibleWorld Apr 18 '18 at 13:27
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    @BeingOfNothingness Oh, also, you might look at Quine's 'Philosophy of Logic' for some philosophical reasons to prefer first-order logic (along with its semantic completeness). I don't know how persuasive Quine is on this point, but it's an influential position. – possibleWorld Apr 18 '18 at 14:51
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Techinal terms: derivability () versus logical consequence ().

The completeness of the proof system is a very very nice property: it ensures that the proof system fully capture our "natural" understanding of the basic relation "to follow from".

For sure, the lack of completeness is not a "virtue", but there are interesting logics that are not complete.

See Second-Order Logic: deductive systems for second-order logic cannot be complete for the standard semantics.

  • Is it categorically so that completeness is not a "virtue" of a system? That is, is this something that people have argued for/against or is it generally assumed by all that completeness is favourable in any logical system? – BeingOfNothingness Apr 18 '18 at 12:47

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