8.2 Drawing Logical Conclusions from Given Statements

8.2 Drawing Logical Conclusions from Given Statements

Quick Summary: A conclusion is valid when it must be true given the statements, not when it is merely likely or realistic. This is the strictest of the three reasoning item types, and it is also the most learnable, because the valid and invalid moves form a short, fixed list.

Conditional statements

A conditional has the form if A, then B. Administrative prose expresses it in many ways, and recognising them is half the skill:

Notice that only if reverses the direction relative to if. "A only if B" means A → B in the sense that A cannot happen without B; written as an arrow it is A → B, so "requests are queued only if complete" gives queued → complete. Reading only if as though it were if is a frequent and costly slip. Write the arrow down rather than holding the direction in your head.

Chaining

Conditionals link through shared terms.

All certified records are indexed. All indexed records are searchable.

certified → indexed → searchable

Valid: All certified records are searchable.

Chains can run several links. Notate them as arrows and read straight through.

Every approved request is logged. Every logged request receives a tracking number. Any request with a tracking number can be audited.

approved → logged → tracking number → auditable

Valid: every approved request can be audited.

The contrapositive: the most useful valid move

From A → B, it validly follows that not-B → not-A.

If a request is complete, it is queued. Therefore: if a request is not queued, it is not complete.

This is genuinely valid and frequently the key to an item. It also chains backwards through a whole sequence.

certified → indexed → searchable

Valid: not searchable → not indexed → not certified.

So a record that is not searchable cannot be certified.

The two invalid moves

Both are offered as options on almost every conditional item.

Affirming the consequent (invalid)

From A → B and B, you cannot conclude A.

If a request is complete, it is queued. Request 88 is queued. Therefore request 88 is complete. ✗

The statement says completeness is sufficient for queueing. It does not say completeness is the only route to being queued. Perhaps queueing also happens by supervisory override.

Denying the antecedent (invalid)

From A → B and not-A, you cannot conclude not-B.

If a request is complete, it is queued. Request 91 is not complete. Therefore request 91 is not queued. ✗

Same reasoning: some other route to queueing may exist.

The full picture

Memorise this table. Two of the four moves are valid, and the two invalid ones are the standard distractors.

Quantifiers

The some row is the one that costs marks. In ordinary speech, "some records are certified" hints that others are not. In logic, some means at least one, and is fully compatible with all. If all 40 records are certified, it remains true that some are.

Combining quantifiers

All indexed records are searchable. Some archived records are indexed.

Valid: Some archived records are searchable. (Take an archived record that is indexed; being indexed, it is searchable.)

Invalid: All archived records are searchable. (Nothing says every archived record is indexed.) Invalid: Some searchable records are not archived. (Possible, but not established.)

All draft forms are unsigned. No unsigned form is filed.

draft → unsigned, and unsigned → not filed. Valid: No draft form is filed.

A worked item

Statements. Every request routed to the review team is logged. No logged request is discarded. Some requests received on Friday were routed to the review team.

Which must be true?

Notate:

routed -> logged
logged -> NOT discarded
SOME Friday requests -> routed

Chain: routed → logged → not discarded.

Some Friday requests were routed, so some Friday requests are logged, so some Friday requests were not discarded. That must be true.

Check the tempting invalid options:

• No Friday request was discarded. ✗ Only some Friday requests were routed; others may have been handled differently.
• Every logged request was received on Friday. ✗ Affirming the consequent.
• Requests not routed to review were discarded. ✗ Denying the antecedent.

Procedure

1. Write each statement as an arrow or a quantifier line.
2. Circle repeated terms and chain where they link.
3. Add contrapositives if the question involves a negative.
4. Test each option: does it follow necessarily, or merely possibly?
5. Reject anything that affirms the consequent or denies the antecedent.