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Single Idea 9719
[filed under theme 5. Theory of Logic / K. Features of Logics / 3. Soundness
]
Full Idea
If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'.
Gist of Idea
A proof theory is 'sound' if its valid inferences entail semantic validity
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.2
The
12 ideas
with the same theme
[whether all formal deductions always lead to truth]:
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19123
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If soundness can't be proved internally, 'reflection principles' can be added to assert soundness
[Gödel, by Halbach/Leigh]
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9719
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A proof theory is 'sound' if its valid inferences entail semantic validity
[Enderton]
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10765
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Soundness would seem to be an essential requirement of a proof procedure
[Tharp]
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10070
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If everything that a theory proves is true, then it is 'sound'
[Smith,P]
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10086
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Soundness is true axioms and a truth-preserving proof system
[Smith,P]
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10596
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A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation)
[Smith,P]
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13635
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'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
[Shapiro]
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10120
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Soundness is a semantic property, unlike the purely syntactic property of consistency
[George/Velleman]
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18757
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Soundness theorems are uninformative, because they rely on soundness in their proofs
[McGee]
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16342
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You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
[Halbach]
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16341
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Normally we only endorse a theory if we believe it to be sound
[Halbach]
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16344
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Soundness must involve truth; the soundness of PA certainly needs it
[Halbach]
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