Why is this inference incorrect?

Question

I wasn't sure if this was the correct place to post my question, but here goes:

Formally, why are these inferences incorrect?

All men are people;

Half of all people are women;

Therefore, half of all men are women.

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Penguins are black and white;

Some old TV shows are black and white;

Therefore, some penguins are old TV shows.

Answer 1

See Aristotle’s logic for the notion of deduction (sullogismos).

Aristotle says:

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics I.2, 24b18–20)

Each of the “things supposed” is a premise of the argument, and what “results of necessity” is the conclusion.

This is the source of the notion of valid argument :

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion to be false.

The above examples, being both examples of true premises and false conclusion shows that the corresponding arguments are not valid.

For the second one, see Fallacy of the undistributed middle.