2.2 Some Statements and If-Then Logic

Some means at least one

The official preparation guide treats words such as some, many, most, and a few as statements about a portion of a group. The crucial warning is that a statement about part of a group says nothing definite about the remaining part unless the passage adds another fact.

For exam purposes, "Some A are B" guarantees at least one overlap between A and B. It can be safely restated as "Some B are A." If some morning arrivals are contractors, then at least one contractor is a morning arrival.

Do not add "but not all." In everyday speech, "some gates are open" may hint that others are closed. In strict passage logic, it proves only that at least one gate is open. All the gates might be open. Whether any are closed is Indeterminable.

A some-not statement also proves an existing example: "Some checked bags are not oversized" means at least one object belongs to checked bags and does not belong to oversized items. It does not prove that most checked bags are ordinary-sized or that any oversized item is checked.

Combining some with universal rules

Example facts:

• Some technicians are evening employees.
• All evening employees attend briefing.

A necessary conclusion is that some technicians attend briefing. Follow the known overlapping member through the universal arrow.

Now change the facts:

• All technicians are employees.
• Some employees attend briefing.

You cannot conclude that some technicians attend briefing. The employees who attend may all come from a different subgroup. This is the shared-category trap: two smaller groups connected to the same larger group need not overlap.

If P, then Q

A conditional states that whenever the first event or condition occurs, the second follows. Write it as P → Q:

• If a package is logged, then it receives a tracking number.
• Package K is logged.
• Therefore, Package K receives a tracking number.

This is the direct valid move. The words whenever and every time can express the same structure. "Whenever the east entrance closes, arrivals use the west entrance" means east closes → arrivals use west.

The valid contrapositive

A conditional is also equivalent to reversing and negating both parts: not Q → not P.

• If a package is logged, it receives a tracking number.
• Package K did not receive a tracking number.
• Therefore, Package K was not logged.

The official guide explicitly demonstrates this rule. The contrapositive works because a logged package without a tracking number would violate the original fact.

Two invalid moves

Reversing the statement

From P → Q, the fact Q does not prove P. A tracking number might also be assigned through another process. Concluding "tracking number, therefore logged" reverses the arrow. This is often called affirming the consequent, but the label matters less than seeing the directional error.

Negating in the same direction

The fact not P does not prove not Q. A package that was not logged might still receive a tracking number for another reason. Concluding "not logged, therefore no number" denies the first condition and improperly carries the negation forward.

Multi-step conditional chains

Conditionals can be linked when the endpoint of one is the starting point of another:

• If the briefing begins, the doors close.
• If the doors close, late arrivals wait outside.
• The briefing begins.

Therefore the doors close and late arrivals wait outside. The combined chain is briefing begins → doors close → late arrivals wait. Its contrapositive also works: if late arrivals are not waiting outside, then the doors did not close, so the briefing did not begin.

Do not link statements merely because they share a word. The direction must connect exactly. Also separate an exam fact from a realistic exception. If the passage says an event always triggers an outcome, accept that rule for the item; do not invent equipment failure, discretionary waivers, or ordinary practice.

Quick decision routine

1. Circle some/many/most/a few and mark the guaranteed overlap only.
2. Turn if/whenever/every time into an arrow.
3. Follow arrows forward.
4. If reasoning backward, negate both sides.
5. Reject conclusions that reverse without negating or negate without reversing.
6. If a relationship remains possible but not required, select Indeterminable rather than forcing True or False.