IMAT Logical Reasoning · Deductive Logic

Statements, Quantifiers and Negation

A statement is either true or false; words like all, some, none and at least one decide exactly what it claims, and what its opposite claims.

All Deductive Logic notes

Why this matters

The ministry papers asked which conclusion follows from a single fact (2025) and which of five statements is true (2026). The older papers asked which two comparison statements mean the same thing (2011) and what must follow from a chain of comparisons (2012).

Concept 1 of 3: Propositions: statements that are either true or false

Logic works only with sentences that have a truth value. "Close the door" is neither true nor false, but "The door is closed" is one or the other. The size of a claim matters: one example is enough to show that something exists, but no number of examples proves that something holds for every case.

Definition

A proposition is a sentence that is either true or false, never both.

A universal claim speaks about every member of a group: "all", "every", "no", "none".
An existential claim says at least one member exists: "some", "at least one", "there is".
In logic, some means at least one, and possibly all. It does not mean "not all".
One example proves an existential claim. One counterexample disproves a universal claim.
A conclusion is deducible only if the given statements force it. Being true in the real world is not enough.

Worked example

You are told only this: "Vesuvius is an active volcano in Italy." Which of these can be deduced? (i) At least one active volcano is in Italy. (ii) Every volcano in Italy is active. (iii) Some active volcanoes are outside Italy.

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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".

Concept 2 of 3: Negating statements with all, some, none and at least one

The negation of a statement is the statement that is true exactly when the original is false. To break "all A are B", you need only one A that is not B; you do not need every A to fail. So the negation of a strong claim is usually a weak one.

Definition

The negation of a statement S is true in every case where S is false, and false in every case where S is true.

"All" and "some ... not" are negations of each other.
"No" and "some" are negations of each other.
Contraries such as "all A are B" and "no A are B" cannot both be true, but they can both be false. So a contrary is not a negation.
The negation of "P and Q" is "not P, or not Q (or both)". The negation of "P or Q" is "not P and not Q".

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.

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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.

Concept 3 of 3: Comparison statements: equivalent wordings and ordering chains

Comparison words hide a symbol. "Not more than" is the same as "at most", which allows equality; "fewer than" does not. Once every statement is written as a symbol, equivalent statements look identical. For an ordering, link the comparisons into a chain: only items joined by a chain can be compared for certain.

Definition

Translate each comparison into a symbol before comparing:

"more than" is $>$ ; "fewer than" or "less than" is $<$ .
"at least", "not fewer than", "no less than" are $\ge$ .
"at most", "not more than", "no more than" are $\le$ .
"not (x > y)" is $x \le y$ : negating a strict inequality brings in equality.
Ordering is transitive: if $a > b$ and $b > c$ , then $a > c$ . Two items each greater than a third cannot be ordered against each other.

Negating a comparison

\text{not}(x > y) \;\Leftrightarrow\; x \le y \qquad \text{not}(x < y) \;\Leftrightarrow\; x \ge y

$\le$ at most, not more than
$\ge$ at least, not fewer than

Worked example

A crate holds apples (A) and pears (P). Which two of these statements say the same thing? (1) There are not fewer apples than pears. (2) There are fewer pears than apples. (3) The pears do not outnumber the apples.

Practice this conceptself-check · 4 quick reps

"Not more than" allows equality

"There are not more cats than dogs" means cats

\le

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".

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Comparison statements: equivalent wordings and ordering chains
Negating a comparison
$\text{not}(x > y) \;\Leftrightarrow\; x \le y \qquad \text{not}(x < y) \;\Leftrightarrow\; x \ge y$

Reference tables (1)

Negating statements with all, some, none and at least one7 rows

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.

Watch out for (4)

True is not the same as deducible
→ Propositions: statements that are either true or false
The negation of all is not none
→ Negating statements with all, some, none and at least one
"Not more than" allows equality
→ Comparison statements: equivalent wordings and ordering chains
Items not joined by a chain cannot be compared
→ Comparison statements: equivalent wordings and ordering chains

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