Which of the following statements are logically related? In each option, read…

Concept:

A set of three categorical statements forms a valid syllogism only when the first two act as premises and the third, stated in that same order, is a conclusion that follows necessarily from them — most commonly via transitivity of the universal affirmative (“All X are Y” + “All Y are Z” implies “All X are Z”). Swapping which statement plays premise or conclusion, or including a statement that contradicts an earlier one, breaks the syllogism even when the same statements are involved.

Application:

1. Label the five given statements A–E in the order they appear.

2. Test A + B → E: premises “All cows are goats” and “All goats are dogs” combine by transitivity to give exactly “All cows are dogs” — the stated conclusion. This ordering is valid.

3. Test A + E → B (the same three statements, different order): both premises only place “cows” inside goats and inside dogs, making cows a common subset of both — but that does not force every goat to be a dog, since goats outside the cows subset are unconstrained. The conclusion does not follow. Invalid.

4. Test D + A + B: D (“No goats are dogs”) directly contradicts premise B (“All goats are dogs”). Invalid.

5. Test A + B + C: C (“No goats are cows”) directly contradicts premise A (“All cows are goats”). Invalid.

Cross-check:

Picture the premises as nested sets: cows ⊆ goats ⊆ dogs. The only new fact this nesting proves is cows ⊆ dogs, i.e. “All cows are dogs” — confirming that the combination pairing “All cows are goats” and “All goats are dogs” as premises with “All cows are dogs” as the conclusion, in that order, is the only non-contradictory, logically valid set.