Which arguments are valid?
Some scientists are not engineers.
Some astronauts are not engineers.
Hence, some scientists are not astronauts.
My attempt:
$\exists$x, scientist(x) $\wedge$ $\sim$engineer(x)
$\exists$x, astronaut(x) $\wedge$ $\sim$engineer(x)
Hence, $\exists$x, scientist(x) $\wedge$ $\sim$ astronaut(x)
I feel that the argument is invalid, but I couldn't show it.
All astronauts are scientists.
Some astronauts are engineers.
Hence, some engineers are scientists.
My attempt:
$\forall$x, astronaut(x) $\to$ scientist(x)
$\exists$x, astronaut(x) $\wedge$ engineer(x)
$\exists$x, engineer(x) $\wedge$ scientist(x)
I am able to prove this argument is valid if I translate the first sentence to a conjunction, instead of a conditional. But I am unsure if this is allowed.
Some females are not mothers.
Some politicians are not females.
Hence, some politicians are not mothers.
My attempt:
$\exists$x, female(x) $\wedge$ $\sim$mother(x)
$\exists$x, politician(x) $\wedge$ $\sim$female(x)
$\exists$x, politician(x) $\wedge$ $\sim$mother(x)
Performing specialisation (rule of inference) on the first two sentences, $\sim$mother(x) and politician(x). Then using conjunction rule of inference, I get the final sentence, so the argument is valid. But the argument is invalid!!
Much appreciated!