IMAT Logical Reasoning · Deductive Logic

If, Only If and Unless: Conditionals and Valid Inference

An if-then statement links a condition to a result; only some ways of reasoning from it are valid, and the contrapositive is the one rewording that keeps its meaning.

All Deductive Logic notes

Why this matters

Two ministry questions (2024 and 2025) gave an if-then statement and asked which conclusion can be deduced; both answers were modus tollens. Two Cambridge questions (2021 and 2022) asked for the argument with the same logical structure.

Concept 1 of 4: If P then Q: the converse, the inverse and the contrapositive

"If it is a whale, it is a mammal" is true, but "if it is a mammal, it is a whale" is false: a dog is a mammal. Turning a conditional round changes its meaning. Turning it round AND negating both parts keeps the meaning: if it is not a mammal, it cannot be a whale.

Definition

In if P then Q (written $P \Rightarrow Q$ ), P is the antecedent (the condition) and Q is the consequent (the result).

The statement is false in only one case: P true and Q false. When P is false, it makes no claim.
The contrapositive (if not Q then not P) always has the same meaning as the original.
The converse (if Q then P) and the inverse (if not P then not Q) do not. They are equivalent to each other.
"Whenever P, Q", "every time P, Q" and "all P are Q" are all forms of if P then Q.

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.

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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".

Concept 2 of 4: Necessary and sufficient conditions: only if, unless, if and only if

In "if P then Q", P is enough to guarantee Q, so P is sufficient. Q cannot be missing whenever P happens, so Q is necessary for P. "Only if" points at the necessary condition. Rewrite every wording as an arrow before reasoning with it.

Definition

For $P \Rightarrow Q$ :

P is a sufficient condition for Q: P alone guarantees Q.
Q is a necessary condition for P: without Q, P cannot happen.
"P only if Q" means $P \Rightarrow Q$ . The word after "only if" is the necessary condition.
"P unless Q" means "if not Q, then P".
"P if and only if Q" means both $P \Rightarrow Q$ and $Q \Rightarrow P$ : each is necessary and sufficient for the other.

Wording	As an arrow	What it tells you
If P, then Q	$P \Rightarrow Q$	P is sufficient for Q; Q is necessary for P
Q if P	$P \Rightarrow Q$	Same as above: the word after "if" is the condition
P only if Q	$P \Rightarrow Q$	Q is necessary for P
Only Q can be P (only Qs are Ps)	$P \Rightarrow Q$	Every P is a Q
P unless Q	$\text{not } Q \Rightarrow P$	Without Q, P happens
P if and only if Q	$P \Leftrightarrow Q$	Each is necessary and sufficient for the other

"Only if" and "if" point in opposite directions.

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"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.

Concept 3 of 4: Valid inference: modus ponens and modus tollens, and the two fallacies

Given "if P then Q", only two one-step conclusions are safe. If P happens, Q happens. If Q did not happen, P cannot have happened. The other two moves feel natural but fail, because Q can have causes other than P.

Definition

An argument is valid when the conclusion must be true whenever the premises are true. From $P \Rightarrow Q$ :

Modus ponens: P, so Q. Valid.
Modus tollens: not Q, so not P. Valid (it uses the contrapositive).
Affirming the consequent: Q, so P. Invalid.
Denying the antecedent: not P, so not Q. Invalid.
Chaining: $P \Rightarrow Q$ and $Q \Rightarrow R$ give $P \Rightarrow R$ . Valid.

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.

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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.

Concept 4 of 4: Finding the argument with the same logical structure

Two arguments about different topics can have the same skeleton. Replace each idea with a letter and write the argument as arrows. Then compare skeletons, not topics: the matching option has the same arrows, the same "not"s and the same kind of conclusion, valid or flawed.

Definition

To match structure:

Write the original as a skeleton, for example $A \Rightarrow B$ ; not B; so not A.
Translate "requires", "needs", "only if" into arrows first: "A requires B" is $A \Rightarrow B$ .
Do the same for each option and pick the identical skeleton.
A flawed argument must be matched by an option with the same flaw. A valid option does not match a flawed original.
Ignore the topic and whether the conclusion is true in real life.

Worked example

Original: "Only members can borrow books. Lena is not a member, so Lena cannot borrow books." Which has the same structure? (1) Only adults can vote. Tom is an adult, so Tom can vote. (2) Only licensed drivers may rent a car. Ivo has no licence, so Ivo may not rent a car. (3) Only athletes use this gym. Eva uses this gym, so Eva is an athlete.

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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.

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.

Reference tables (3)

If P then Q: the converse, the inverse and the contrapositive4 rows

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 if6 rows

Wording	As an arrow	What it tells you
If P, then Q	$P \Rightarrow Q$	P is sufficient for Q; Q is necessary for P
Q if P	$P \Rightarrow Q$	Same as above: the word after "if" is the condition
P only if Q	$P \Rightarrow Q$	Q is necessary for P
Only Q can be P (only Qs are Ps)	$P \Rightarrow Q$	Every P is a Q
P unless Q	$\text{not } Q \Rightarrow P$	Without Q, P happens
P if and only if Q	$P \Leftrightarrow Q$	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 fallacies5 rows

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.

Watch out for (4)

The converse is not equivalent
→ If P then Q: the converse, the inverse and the contrapositive
"Only if" introduces the necessary condition
→ Necessary and sufficient conditions: only if, unless, if and only if
Affirming the consequent looks like modus ponens
→ Valid inference: modus ponens and modus tollens, and the two fallacies
Match the skeleton, not the topic
→ Finding the argument with the same logical structure

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