Syllogism with truth-tellers and sometimes-liars Statements: (a) Mary: “Ann and I both have cats.” (b) Ann: “I do not have a cat.” Facts: Mary always tells the truth; Ann sometimes lies. Conclusions to test: I) Ann has a cat. II) Mary has a cat. III) Ann is lying.

Introduction / Context:
This problem mixes categorical reasoning with truthfulness constraints. You must reconcile two statements where one speaker always tells the truth and the other may sometimes lie. Consistency with the 'always truthful' speaker determines the status of both claims.

Given Data / Assumptions:

• Mary invariably tells the truth.
• Ann sometimes lies (so her statement could be true or false on any given occasion).
• Mary asserts: “Ann and I both have cats.”
• Ann asserts: “I do not have a cat.”

Concept / Approach:
If Mary always tells the truth, then her whole conjunctive statement must be true: both Mary and Ann own cats. Any statement contradicting this truth must be false on this occasion. Ann’s statement contradicts Mary’s claim about Ann having a cat, so Ann must be lying here.

Step-by-Step Solution:

Verification / Alternative check:
Try to assume Ann tells the truth here (that she has no cat). That would immediately contradict Mary’s guaranteed truthfulness. Since Mary’s reliability is absolute, Ann’s contrary statement must be the false one in this episode.

Why Other Options Are Wrong:

• II only or I and II only fail to recognize the necessary falsity of Ann’s claim in this instance.
• I only drops Mary’s ownership, which is part of Mary’s true statement.

Common Pitfalls:
Mistaking “sometimes lies” for “always lies,” or overlooking that an “always truthful” conjunctive statement commits you to both parts.

Final Answer:
I, II and III.