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Single Idea 19065

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

Full Idea

A proof of soundess or completeness is a test, not so much of the logical theory to which it applies, but of the theory of meaning which underlies the semantics.

Gist of Idea

Soundness and completeness proofs test the theory of meaning, rather than the logic theory

Source

Michael Dummett (The Justification of Deduction [1973], p.310)

Book Ref

Dummett,Michael: 'Truth and Other Enigmas' [Duckworth 1978], p.310

A Reaction

These two types of proof concern how the syntax and the semantics match up, so this claim sounds plausible, though I tend to think of them as more like roadworthiness tests for logic, checking how well they function.

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]