Statements: All roads are waters. Some waters are boats. Conclusions: Some…

Concept

A syllogism conclusion linking two premises can be validly drawn only if the middle term — the term common to both premises — is distributed (taken in its full extent) in at least one of them. A term is distributed only as the subject of a universal statement (A-type: 'All X are Y') or the predicate of a negative statement (E/O-type); it is never distributed in a particular affirmative (I-type: 'Some X are Y'). If the middle term stays undistributed in both premises, no valid conclusion follows — this is the fallacy of the undistributed middle.

Applying it here

1. Premise 1, 'All roads are waters,' is an A-type statement. It distributes only its subject, 'roads'; the middle term 'waters' sits in the predicate position, so it remains undistributed.

2. Premise 2, 'Some waters are boats,' is an I-type statement, and I-type statements distribute neither term. Here the middle term 'waters' is the subject, so it stays undistributed a second time.

3. Because 'waters' is undistributed in both premises, no definite relationship between 'roads' and 'boats' can be fixed — the fallacy of the undistributed middle applies.

Checking the conclusions

• Conclusion I, 'Some boats are roads,' would need a valid chain linking boats to roads through the middle term; since that link was never established, this conclusion cannot be drawn.

• Conclusion II, 'All waters are boats,' cannot be derived either — the only premise connecting 'waters' and 'boats' is the particular statement 'Some waters are boats', and a particular (I-type) premise can never be strengthened into a universal (A-type) conclusion.

• The 'either…or' pattern applies only when two conclusions are complementary — drawn from the very same subject-predicate pair so that if one fails the other must hold. I and II here involve different term pairings (boats-roads vs. waters-boats), so that pattern does not apply, independent of each conclusion's own validity.

Result

Neither conclusion follows, because the middle term 'waters' is undistributed in both given premises.