IMAT Logical Reasoning and Problem Solving · Formula sheet
Deductive Logic formulas
2 formulas, 5 reference tables and 14 common traps for IMAT Logical Reasoning and Problem Solving Deductive Logic, grouped by subtopic.
Statements, Quantifiers and Negation
Learn this subtopic in the notesComparison statements: equivalent wordings and ordering chains
Negating a comparison
- at most, not more than
- at least, not fewer than
Negating statements with all, some, none and at least one
| Statement | Its negation | Not the negation |
|---|---|---|
| All A are B | Some A are not B (at least one A is not B) | No A are B |
| No A are B | Some A are B | All A are B |
| Some A are B | No A are B | Some A are not B |
| Some A are not B | All A are B | No A are B |
| At least two came | At most one came (one or none) | Nobody came |
| P and Q | Not P, or not Q, or both | Not P and not Q |
| P or Q | Not P and not Q | Not P, or not Q |
Test a candidate negation: in every possible situation, exactly one of the pair must be true.
Common traps
True is not the same as deducible
IMAT often places a statement that is true in the real world among the options. If it does not follow from the given statements alone, it is not the answer to "which can be deduced".
The negation of all is not none
"Not all A are B" means at least one A is not B. It does not mean that no A is B. Choosing the "none" option confuses the negation with the contrary, which is a much stronger claim.
"Not more than" allows equality
"There are not more cats than dogs" means cats dogs, so the numbers may be equal. It is not the same as "there are fewer cats than dogs", which rules equality out. IMAT builds wrong options on exactly this difference.
Items not joined by a chain cannot be compared
If two items are each greater than a third, nothing fixes their order against each other. An option that orders them is "could be true", never "must be true".
If, Only If and Unless: Conditionals and Valid Inference
Learn this subtopic in the notesIf P then Q: the converse, the inverse and the contrapositive
| Form | Pattern | Example | Same meaning as the original? |
|---|---|---|---|
| Original | If P then Q | If it is a whale, it is a mammal. | Yes |
| Converse | If Q then P | If it is a mammal, it is a whale. | No |
| Inverse | If not P then not Q | If it is not a whale, it is not a mammal. | No (it matches the converse) |
| Contrapositive | If not Q then not P | If it is not a mammal, it is not a whale. | Yes |
Only the contrapositive can replace the original statement.
Necessary and sufficient conditions: only if, unless, if and only if
| Wording | As an arrow | What it tells you |
|---|---|---|
| If P, then Q | P is sufficient for Q; Q is necessary for P | |
| Q if P | Same as above: the word after "if" is the condition | |
| P only if Q | Q is necessary for P | |
| Only Q can be P (only Qs are Ps) | Every P is a Q | |
| P unless Q | Without Q, P happens | |
| P if and only if Q | Each is necessary and sufficient for the other |
"Only if" and "if" point in opposite directions.
Valid inference: modus ponens and modus tollens, and the two fallacies
| Name | Premises | Conclusion | Valid? |
|---|---|---|---|
| Modus ponens | If P then Q; P | Q | Yes |
| Modus tollens | If P then Q; not Q | Not P | Yes |
| Affirming the consequent | If P then Q; Q | P | No |
| Denying the antecedent | If P then Q; not P | Not Q | No |
| Chain | If P then Q; if Q then R | If P then R | Yes |
The two invalid forms are the converse and the inverse in disguise.
Common traps
The converse is not equivalent
"If P then Q" does not give "if Q then P". All whales are mammals, but not all mammals are whales. The only rewording with the same meaning is the contrapositive, "if not Q then not P".
"Only if" introduces the necessary condition
"P only if Q" means if P then Q. Reading it as "if Q then P" turns a necessary condition into a sufficient one, which is the most common wrong option on these items.
Affirming the consequent looks like modus ponens
From "if it rains, the street is wet" and "the street is wet", you cannot conclude that it rained: a burst pipe also wets the street. Only "the street is dry, so it did not rain" is valid.
Match the skeleton, not the topic
An option on the same subject as the original, or with a conclusion that sounds sensible, is often the wrong one. Only the pattern of arrows and "not"s decides the match.
Syllogisms: Reasoning with All, Some and No
Learn this subtopic in the notesDrawing all, some and no statements as Venn diagrams
| Statement | How to draw it | Region you know about |
|---|---|---|
| All A are B | A drawn inside B | A outside B is empty |
| No A are B | A and B drawn apart | The overlap of A and B is empty |
| Some A are B | A and B overlapping, cross in the overlap | The overlap has a member |
| Some A are not B | Cross in A, outside B | A outside B has a member |
| Only As are Bs | B drawn inside A | B outside A is empty |
Draw empty regions first, then place the crosses where they are forced to go.
Common traps
"All A are B" does not reverse
All cats are mammals, yet many mammals are not cats. An option that swaps the two groups of an "all" statement is the converse and does not follow.
Two "some" statements prove nothing
"Some A are B" and "some B are C" can be drawn with the A members and the C members in different parts of B. So "some A are C" is only a could-be. Options built from two "some" premises are never "must be true".
Some does not mean not all
"Some students passed" is still true if every student passed. An option that reads "some" as "some but not all" adds a claim that was never made.
Certainty, Truth-tellers and Assignment Puzzles
Learn this subtopic in the notesWhat must happen: worst-case reasoning and the pigeonhole principle
Guarantee by the worst case
- items into boxes force a shared box
Common traps
"Could happen" is not "must happen"
An option that is true in the lucky case ("the next one will be mint") is not certain. For "must" or "definitely" questions, test each option against the worst case that the rules allow.
A liar's "both" statement is false, not reversed word by word
If a liar says "we are both truth-tellers", it does not follow that both are liars. It only follows that at least one is a liar. Negate the whole statement, as for any negation.
Check every clue against your chosen option
An option often satisfies the first two clues and fails a later one. Before choosing, run the option through every clue; a single broken clue rules it out.