2.4 Conditional Logic & Evidence-Based Conclusions

Conditional logic and evidence-based conclusions

Translate before evaluating

Logical language becomes easier when you replace details with symbols or set diagrams. For a conditional statement, write:

If P, then Q — P → Q

P is sufficient for Q in the rule. Q is necessary whenever P occurs. Two deductions are valid:

1. P is true, therefore Q is true (modus ponens).
2. Q is false, therefore P is false (modus tollens, or the contrapositive).

Two tempting forms are invalid:

• Q is true, therefore P is true (affirming the consequent).
• P is false, therefore Q is false (denying the antecedent).

If the rule says, “If a booking is approved, a confirmation email is sent,” receiving an email does not prove approval unless the passage says confirmation emails have no other cause. A system notice, reminder, or error could also produce an email. By contrast, if no confirmation was sent and the rule has no exception, the booking was not approved.

If, only if, unless and either

“A only if B” means A → B. B is required for A. It does not say B guarantees A.

“A if B” means B → A. B is sufficient for A.

“A if and only if B” gives both directions: A → B and B → A.

“A unless B” usually means if B does not occur, A occurs: not-B → A. It can also be restated as A or B, subject to ordinary inclusive-or wording. Read any explicit definition in the passage.

“Either A or B” is often inclusive in formal reasoning—at least one may occur—unless the text says “but not both,” “exactly one,” or otherwise establishes exclusivity.

Categorical statements

Represent groups as circles:

• All A are B: the A circle sits inside B.
• No A are B: the circles do not overlap.
• Some A are B: at least one point lies in the overlap.
• Some A are not B: at least one A point lies outside B.

Do not reverse “All A are B” into “All B are A.” All station maps may be documents, but not all documents are station maps. “No A are B” can be converted to “No B are A” because non-overlap is symmetric. “Some A are B” also supports “Some B are A,” because the same object occupies both groups.

The existential caution

In many aptitude conventions, “all A are B” does not prove that any A exists. If the premises say all emergency keys are tagged but never establish an emergency key, you cannot conclude that some tagged item is an emergency key. A “some” premise supplies existence.

Chain rules carefully

Suppose:

1. All approved forms are dated.
2. No undated document is archived.
3. Some archived documents are forms.

Premise 2 is equivalent to all archived documents being dated, but it does not say all dated documents are archived. Premise 3 establishes at least one thing that is both archived and a form. It does not establish that the form is approved. The word “approved” appears only in a one-way rule.

Write each relationship separately. Do not merge categories merely because they share a term.

Ordering and constraint problems

Some verbal logic items provide rules such as “L is before M,” “N cannot be adjacent to P,” and “Q is scheduled immediately after L.” Make a compact notation:

• L < M
• N not next to P
• LQ is a fixed block

Place fixed blocks and endpoints first. Test options against every rule rather than trying to build all possible schedules. One violated constraint eliminates an option.

For “could be true,” find one complete arrangement that satisfies every rule. For “must be true,” try to create a counterexample; if every attempt violates a rule, the statement may be necessary. For “cannot be true,” locate the direct contradiction.

Keep real-world belief separate

Logical questions stipulate a miniature world. If a premise says every blue card is stored upstairs, accept it for the problem even if real procedures differ. Conversely, do not add a sensible workplace rule that is absent. The correct answer follows from the premises, not from what a well-run organisation probably does.

Use a final proof sentence: name the rule and the observed fact. If you cannot state the path without “probably,” the conclusion may not be deductive.