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Single Idea 13628

[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness ]

Full Idea

We can live without completeness in logic, and live well.

Gist of Idea

We can live well without completeness in logic

Source

Stewart Shapiro (Foundations without Foundationalism [1991], Pref)

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.-14

A Reaction

This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps.

The 14 ideas with the same theme [all the truths of a system are formally deducible]:

9544 A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised [Hughes/Cresswell]
19065 Soundness and completeness proofs test the theory of meaning, rather than the logic theory [Dummett]
9720 A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton]
10763 Completeness and compactness together give axiomatizability [Tharp]
10834 Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos]
10069 A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
10598 A theory is 'negation complete' if it proves all sentences or their negation [Smith,P]
10597 'Complete' applies both to whole logics, and to theories within them [Smith,P]
13628 We can live well without completeness in logic [Shapiro]
13698 In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider]
10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
10161 If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman]
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
10761 Completeness can always be achieved by cunning model-design [Rossberg]