I see part of the problem. You're calling "if P then Q" a proposition. I would call it the major premise. And it looks to me like you've got it backwards.

The syllogism would go:

Major Premise: If P then Q

Minor Premise: P

Conclusion: Q

From this it is valid to say:

If not Q then not P

but not:

If not P then not Q

Now. Are you talking about the last Table on the page?

a. I would call the normal AND

b. is the standard inclusive OR

c. looks like P AND NOT Q

d. is the standard exclusive OR (XOR)

The standard colloquial defintion of "or" is d. The conflict between b. and d. is the basis for the joke question, "Did you take the bus or bring your lunch?"