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Full Idea
A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics.
Gist of Idea
'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.8
A Reaction
Similarly, 'weakly complete' is when every logical truth is a theorem.
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| 10765 | Soundness would seem to be an essential requirement of a proof procedure [Tharp] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 18757 | Soundness theorems are uninformative, because they rely on soundness in their proofs [McGee] |
| 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
| 16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
| 16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |