Statements: All ways are waters. Some waters are boats. Conclusions: I. Some…

Concept: In a categorical syllogism, a conclusion linking the two end terms is valid only if it holds true in EVERY possible diagram consistent with the given statements, and only if the term common to both statements (the middle term) is distributed — i.e., it refers to the whole of that category — in at least one of the two statements. A ‘some’ (particular) statement never distributes its subject or its predicate, and an ‘all’ (universal affirmative) statement distributes only its subject, never its predicate.

Application: The first statement places the entire ‘ways’ category inside ‘waters’. The second statement says only a portion of ‘waters’ overlaps with ‘boats’, without specifying which portion.

• One valid diagram draws that boats-overlap entirely outside the ways-region (waters drawn larger than ways, with the boats-overlap sitting in the leftover part of waters). In that diagram, no boats are ways at all, so a conclusion claiming an overlap between boats and ways does not hold in every case.

• Because the statement connecting waters and boats is only a ‘some’ statement, it can never support a conclusion claiming the relationship holds for ALL of waters — a particular premise can only ever yield a particular conclusion, never a universal one.

Cross-check: The middle term ‘waters’ is the predicate of the first (universal affirmative) statement and the subject of the second (particular affirmative) statement, so it is undistributed in both. A term undistributed in both statements cannot license any definite conclusion about how the two end terms relate — the classical ‘undistributed middle’ situation.

Since neither proposed conclusion holds across every valid diagram, neither one follows from the given statements.