Conditional Logic

Conditional statements — "if...then" reasoning — are fundamental to the UCAT Decision Making subtest. They underpin syllogisms, logical puzzles, and assumption questions. This lesson teaches you the four related forms of any conditional statement, which transformations are logically valid, and how to apply this knowledge to UCAT questions rapidly and accurately.

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The Basic Conditional

A conditional statement has the form:

If P, then Q.

Where:

• P is the antecedent (the "if" part)
• Q is the consequent (the "then" part)

Examples

Conditional	P (antecedent)	Q (consequent)
If it rains, the ground gets wet.	It rains	The ground gets wet
If a patient has Type 1 diabetes, they require insulin.	Patient has Type 1 diabetes	They require insulin
If you score above 700, you will be invited to interview.	Score above 700	Invited to interview

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The Four Related Forms

Every conditional "If P, then Q" has three related statements. Understanding which are valid and which are invalid is essential.

Form	Statement	Valid?
Original	If P, then Q	— (given)
Contrapositive	If not Q, then not P	✓ Always valid
Converse	If Q, then P	✗ Not necessarily valid
Inverse	If not P, then not Q	✗ Not necessarily valid

The Contrapositive (VALID)

The contrapositive reverses and negates both parts. It is logically equivalent to the original — always true whenever the original is true.

Original: If a student passes all modules, they graduate. Contrapositive: If a student does not graduate, they did not pass all modules.

These two statements say exactly the same thing. If one is true, the other must be true.

The Converse (INVALID)

The converse simply swaps P and Q without negating. It is not logically equivalent.

Original: If a student passes all modules, they graduate. Converse: If a student graduates, they passed all modules.

This might seem correct, but consider: perhaps students can also graduate through a compensatory route or with mitigating circumstances. The original statement does not rule this out.

UCAT Trap: The converse is the most commonly tested invalid form. Many answer options present the converse of a given conditional and ask whether it follows logically. It does not.

The Inverse (INVALID)

The inverse negates both parts without swapping. It is not logically equivalent.

Original: If a student passes all modules, they graduate. Inverse: If a student does not pass all modules, they do not graduate.

Again, this might not be true — there could be other routes to graduation.

Key Relationship

Notice that:

• The converse is the contrapositive of the inverse (and vice versa).
• So the converse and inverse are logically equivalent to each other — but neither is equivalent to the original.

Equivalent pairs
Original ↔ Contrapositive
Converse ↔ Inverse

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Applying Conditional Logic to UCAT Questions

Scenario 1: Direct Application

Given: If a hospital has an Accident & Emergency department, it operates 24 hours a day.

Which of the following must be true?

A. If a hospital operates 24 hours a day, it has an A&E department. B. If a hospital does not have an A&E department, it does not operate 24 hours a day. C. If a hospital does not operate 24 hours a day, it does not have an A&E department. D. All hospitals that operate 24 hours a day have A&E departments.

Analysis:

• A = Converse → Invalid ✗
• B = Inverse → Invalid ✗
• C = Contrapositive → Valid ✓
• D = Converse (rephrased) → Invalid ✗

Answer: C

Scenario 2: Chaining Conditionals

Sometimes the UCAT presents a chain of conditionals:

If P, then Q. If Q, then R.

From this, you can validly conclude:

If P, then R.

And by contrapositive:

If not R, then not P.