74 lines
2.4 KiB
Markdown
74 lines
2.4 KiB
Markdown
---
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categories:
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- Logic
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tags: [propositional-logic]
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---
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# Soundness
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Recall that in the definition of
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[deductive validity](/Logic/General_concepts/Validity_and_entailment.md#validity)
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we do not say: an argument is valid iff if the premises _are true_ and the
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conclusion _is true_. We say _if it is possible for the premises to be true_.
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This is important: we are not interested in the actual truth of the premises or
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the conclusion.
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#### Demonstration
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Therefore this argument is valid:
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```
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(P1) Oranges are the same colour as bananas.
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(P2) Bananas are yellow.
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____________________________________________
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(C) Oranges are yellow.
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```
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Of course oranges are not yellow but _were_ (P1) true, then given (P2), the
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conclusion must also be true.
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This argument is also valid:
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```
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(P1) Oranges are the same colour as carrots.
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(P2) Carrots are orange.
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____________________________________________
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(C) Oranges are orange in colour.
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```
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The difference here is that the premises happen to be true and, given that the
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argument is valid, the conclusion must also be true. What we have defined here
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is **soundness**: the argument is said to be sound as well as valid. This is an
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additional and stronger criterion of evaluation.
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> An argument is sound if and only if it is deductively valid and all its
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> premises are true.
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We must not forget that truth alone is not the sole condition for soundness. We
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can have arguments whose conclusion and premises are all true without the
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argument being sound:
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```
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(P1) London is the capital of the United Kingdom
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(P2) The capital of the United Kingdom is in the southern part of the United Kingdom.
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(P3) Cambridge is not the capital of the United Kingdom
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____________________________________________
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(C) London is south of Cambridge
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```
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All propositions are true here but the argument is not deductively valid: the
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premises are all true but the conclusion is false.
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We can also have arguments which are valid but which are not sound:
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```
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(P1) Vitamin C prevents colds.
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(P2) Vitamin C does not prevent colds.
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____________________________________________
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(C) Vitamin C is harmless
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```
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This argument is valid because we cannot consistently assert the premises and
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deny the conclusion. In either case, the conclusion can be said to follow from
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the premises. The problem is that we cannot consistently assert both premises:
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it is not possible for both propositions to be true at the same time.
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