Logical Deductions: MDCAT Logical Reasoning notes

Deduction means "must be true"

A conclusion follows only if it is necessarily true whenever the statements are true. "Possibly true" is not enough. Treat the statements as true even if they sound odd (e.g. "All omnivores are herbivores") and ignore real-world knowledge.

Syllogism rules

Premises	Valid conclusion
All A are B; All B are C	All A are C
All A are B; No B are C	No A are C
Some A are B; All B are C	Some A are C
All A are B; Some B are C	No definite conclusion about A and C
Some A are B; Some B are C	No definite conclusion

Conversions that are always valid:

• "Some A are B" gives "Some B are A".
• "No A are B" gives "No B are A".
• "All A are B" gives "Some B are A" (but not "All B are A").

Use Venn diagrams

Draw circles for each group. "All A are B" puts circle A inside B. "No A are B" keeps them apart. "Some A are B" means they overlap. If you can draw any diagram in which the statements hold but the conclusion fails, the conclusion does not follow.

Worked example 1

Statements: All roses are flowers. All flowers are plants. Some plants are trees.

• I. All roses are plants. Follows (All-All chain).
• II. Some plants are roses. Follows (converting "all roses are plants").
• III. Some roses are trees. Does not follow (the trees may lie outside the flower circle).

Worked example 2

Statements: All pens are tools. No tools are toys. Some toys are cars.

• I. No pens are toys. Follows (All-No rule).
• II. Some cars are toys. Follows (conversion).
• III. Some pens are cars. Does not follow.

Ordering and ranking

Turn every clue into symbols and build a single line.

Example: P is taller than Q but shorter than R. S is shorter than Q but taller than T. Clues: $R>P>Q$ and $Q>S>T$. Combined: $R>P>Q>S>T$. Tallest: R; shortest: T; middle: Q.

Inequality chains

Only combine inequalities that point the same way through a shared term.

• $x>y$ and $y>z$ give $x>z$.
• $p>q$ and $r>q$ tell you nothing about $p$ versus $r$.
• If $m>n$, $n>k$ and $j>m$, then $j>m>n>k$, so $j$ is the greatest and $j>n$ must be true.

Common MDCAT traps

• "All A are B" does not mean "All B are A".
• Two "some" statements never give a definite conclusion.
• A conclusion that is merely possible does not "follow".
• Do not bring in real facts (daffodils are flowers) unless stated.

Quick revision

• All + All = All.
• All + No = No.
• Some + All = Some.
• Draw Venn diagrams to test conclusions.
• Write ranking clues as one inequality chain.