Syllogism with particular and universal premises: given 'Some pastries are toffees' and 'All toffees are chocolates', determine which conclusion(s) necessarily follow about chocolates, toffees, and pastries

Given data

• Premise 1: Some pastries are toffees (∃ Pastry ∩ Toffee).
• Premise 2: All toffees are chocolates (Toffee ⊆ Chocolate).
• Conclusions: (I) Some chocolates are toffees. (II) Some toffees are not pastries.

Concept/Approach (why this method)

Push the existential statement through the inclusion: if some A are B and all B are C, then those B are C, yielding an existential on C.

Step-by-Step calculation (logical derivation)1) From Premise 1, pick at least one element x with x ∈ Pastry and x ∈ Toffee.2) From Premise 2, all toffees are chocolates ⇒ x ∈ Chocolate.3) Therefore there exists at least one chocolate that is a toffee ⇒ Conclusion I is true.4) Conclusion II ('Some toffees are not pastries') is not forced; it could be that all toffees are also pastries. Hence II does not follow.

Verification/Alternative

Set diagram: Toffees wholly inside Chocolates; intersection area between Pastries and Toffees non-empty. II requires a non-overlapping part of Toffees outside Pastries, which is not guaranteed.

Common pitfalls

• Assuming existential negatives without evidence.
• Confusing 'some A are B' with 'some A are not B'.

Final AnswerOnly conclusion I follows.