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I would like to know whether the following argument is valid.

Some amphibians live in the water
All fish live in the water
Therefore, some fish are amphibians.

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it isn't valid. Is any animal in the water a fish? – Vittorio Patriarca Jun 4 '12 at 23:26
up vote 4 down vote accepted

Venn diagrams are sometimes helpful in seeing what’s going on. The conclusion Some fish are amphibians would fit this diagram:

enter image description here

However, the hypothesis fit this diagram just as well, and it describes a world in which no fish are amphibians:

enter image description here

Since the second diagram is consistent with the hypotheses and contradicts the conclusion, the argument cannot be valid.

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This argument is not valid. Notice that some violins make sounds and all pianos make sounds, but this does not mean that some pianos are violins.

The argument is essentially

$\exists x ( \operatorname{Amphibian} (x) \land \operatorname{Water} (x) )$

$\forall x ( \operatorname{Fish} (x) \implies \operatorname{Water} (x) )$

$\therefore \exists x ( \operatorname{Fish} (x) \land \operatorname{Amphibian} (x) )$

The argument would be valid IF the second statement were changed to:

$\forall x ( \operatorname{Water} (x) \implies \operatorname{Fish} (x) )$ (All things that live in water are fish)

In its current form, the argument is an example of affirming the consequent.

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how do i show it. – count Jun 4 '12 at 23:28
Try to think with sets. It helps. – Vittorio Patriarca Jun 4 '12 at 23:32

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