Statements: All cats are dogs some pigs are cats. All dogs are tigers…

Concept: In syllogism, two universal affirmative (All A are B) statements sharing a common middle term chain by transitivity into a further universal statement (All A are C); a universal affirmative statement converts validly to its particular form (Some B are A) whenever the subject class A is known to be non-empty; and a particular statement (Some A are B) combined with a universal statement covering B (All B are C) yields a further particular statement (Some A are C). A particular-negative conclusion (Some A are not B) can never be valid alongside a universal-affirmative conclusion already established between the same two terms, since the two contradict each other.

Application:

1. From “All cats are dogs” and “All dogs are tigers,” the middle term “dogs” links the two, giving the chained universal: all cats are tigers.

2. Since “Some pigs are cats” confirms the class of cats is non-empty, the universal “All cats are tigers” converts validly to the particular “Some tigers are cats.”

3. Combining “Some pigs are cats” with “All cats are tigers” (particular + universal on the shared term “cats”) gives “Some pigs are tigers.”

4. The listed conclusion “Some cats are not tigers” is a particular-negative statement between exactly the same two terms (cats, tigers) for which the chain already establishes a universal-affirmative relationship, so it directly contradicts step 1 and cannot also be valid.

Cross-check: Picture nested circles — tigers as the outer circle, dogs entirely inside it, and cats entirely inside dogs (from the two universal premises); pigs overlaps cats. This single diagram forces every cat to sit inside the tigers circle, so a claim treating cats and tigers as anything other than fully contained is automatically excluded, while the “some” claims about tigers-cats and pigs-tigers must hold since the required overlaps are guaranteed by the nesting.

So the conclusions that validly follow are: some tigers are cats, some pigs are tigers, and all cats are tigers — matching conclusions 1, 2, and 3 from the list above.