Overview
A conditional rule says what must happen when its premise holds; it usually does not say that the premise is the only way the consequence can happen. Logic questions exploit that distinction. We will chain implications, inspect small sets of allowed cases, and count interleavings while enforcing a nonadjacency rule. In every setting, a “must” answer must survive every allowed case, not just one example.
Use a contrapositive rather than a converse
If an archive is open, a staff member is present; if a staff member is present, the lights are on. Chaining the rules gives archive open implies lights on. Therefore, if the lights are off, the archive must be closed. This is the contrapositive: denying a necessary consequence rules out its premise.
The converse “lights on implies archive open” does not follow. Staff and lights could be present while the archive is closed, because neither rule forbids that situation. Likewise, closing the archive does not force staff to leave or lights to switch off. A useful way to reject such alternatives is to describe one allowed situation where they fail. For the valid conclusion, however, you need the implication argument showing that no allowed situation can violate it.
List valid selections to identify a necessary disjunction
Exactly three of A, B, C, D, E and F are selected. A requires B, C requires D, E and F cannot both be selected, and F is selected. F excludes E and fills one place, leaving two places among A, B, C and D. The only valid pairs are AB, BD and CD.
Every pair contains B or D, so at least one of B and D must be selected. Neither individually is necessary: AB lacks D and CD lacks B. Pair BD also shows why at least one of A and C is not required. This is a case where a disjunction is certain even though neither component is certain alone. Writing the complete small case list prevents you from choosing a statement that is true in your first valid selection but false in another.
Combine “at least one” with “at most one”
At least one of P and Q occurred. If P occurred then R did, and if Q occurred then S did. At most one of R and S occurred. If both P and Q had occurred, the implications would force both R and S, contradicting the last rule. Thus P and Q cannot both occur.
Because at least one is already given, exactly one must have occurred. The premises do not identify which one. An answer naming P alone or Q alone therefore goes beyond the evidence, even though either case is possible. Treat the quantifiers precisely: “at least one” permits both, “at most one” permits neither, and together for the same pair they give exactly one. Here the implications transfer the at-most-one restriction from the consequences back to the two possible causes.
Count valid orders by removing a forbidden block
Five presentations must satisfy A before E before C and B before D, with B and D not consecutive. Ignoring nonadjacency temporarily, choose the two positions occupied by B and D; their internal order is fixed, and the remaining positions hold A, E and C in order. This gives ten interleavings. Four put B and D adjacent: treating BD as a block, place that block before, between or after the three members of the other chain. Removing those four leaves six.
The answer ten omits the final condition, while four counts the forbidden arrangements. For transfer, distinguish an implication, an ordering and a prohibition before combining them. Use counterexamples to reject “must” statements, complete case lists when the space is small, and a count of violations when a simple base count exists. In practice, explicitly check the condition most likely to be postponed; a final exclusion is still part of the original problem.
Apply the method to a new case
Every grant recipient has submitted a proposal, and every person who submits a proposal has signed it. Lia has not signed a proposal. Can Lia be a grant recipient? The first premise makes being a recipient imply having submitted a proposal; the second makes submitting a proposal imply signing it. Chaining them shows that every recipient has a signature. The contrapositive therefore excludes Lia from the recipient group. Reversing the chain to claim that every signer receives a grant would add a rule absent from the premises. Check with a countermodel: a person may sign and submit a proposal that is rejected; the two premises still hold. The conclusion about Lia is necessary, while a conclusion about a signer receiving a grant would be merely possible. Preserve implication direction and distinguish a denial of the necessary condition from evidence about the sufficient condition.