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.
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.
Venn diagrams are sometimes helpful in seeing what’s going on. The conclusion Some fish are amphibians would fit this diagram:
However, the hypothesis fit this diagram just as well, and it describes a world in which no fish are amphibians:
Since the second diagram is consistent with the hypotheses and contradicts the conclusion, the argument cannot be valid.
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.