Deductive Reasoning: Syllogisms, Conditional (If-Then) Logic, and Valid Conclusion Identification

Deductive Reasoning on the CAT

Deductive reasoning tests whether you can determine whether a conclusion must be true given a set of premises. The CAT presents syllogisms and conditional (if-then) statements, then asks which conclusion is valid, which is invalid, or which cannot be determined. The key rule: a valid conclusion is one that must be true whenever all the premises are true — not one that merely could be true or seems plausible.

Syllogism Structure

A categorical syllogism has three parts:

1. Major premise — a general statement (e.g., "All sworn officers must complete firearms training.")
2. Minor premise — a specific statement (e.g., "Officer Diaz is a sworn officer.")
3. Conclusion — what follows necessarily ("Therefore, Officer Diaz must complete firearms training.")

The test for validity: if the premises are true, can the conclusion possibly be false? If yes, the conclusion is invalid.

Worked syllogism:

• Premise 1: All recruits who fail the fitness test must retake it within 30 days.
• Premise 2: Candidate Okafor failed the fitness test.
• Conclusion A: Candidate Okafor must retake the fitness test within 30 days. ✓ Valid
• Conclusion B: Candidate Okafor will pass on the second attempt. ✗ Invalid — premises say nothing about passing.
• Conclusion C: All recruits who retake the test failed it the first time. ✗ Invalid — the rule goes one direction; some retakers could be taking it for other reasons the premises do not mention.

The trap in Conclusion C is converting an "All A are B" statement into "All B are A." "All who fail must retake" does not mean "All who retake failed." Direction matters.

Venn Diagram Check

For syllogisms, sketch two circles:

• Circle 1 = the group named in the major premise.
• Circle 2 = the individual or subgroup in the minor premise.

If the minor premise places the individual inside Circle 1, and the major premise says something about everyone in Circle 1, the conclusion applies. If the individual is outside Circle 1, or the major premise only describes a subset, the conclusion does not follow.

Another Worked Syllogism

• Premise 1: No officer convicted of a disqualifying offense may be appointed.
• Premise 2: Candidate Lee was convicted of a disqualifying offense.
• Conclusion A: Candidate Lee may not be appointed. ✓ Valid
• Conclusion B: Candidate Lee was an officer. ✗ Invalid — the premise restricts appointment, not current status.
• Conclusion C: Everyone who may not be appointed was convicted of a disqualifying offense. ✗ Invalid — there may be other reasons someone may not be appointed; the premise does not say this is the only disqualifier.

Conditional (If-Then) Logic

A conditional statement has the form If P, then Q (P → Q):

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

The CAT asks you to identify which conclusions follow from a given conditional. There are exactly two valid forms and two common fallacies.

Valid Form 1: Modus Ponens (Affirming the Antecedent)

• If P, then Q.
• P is true.
• Therefore, Q is true.

Worked example:

• Rule: If an officer uses deadly force, a Use of Force report must be filed within 24 hours.
• Fact: Officer Chen used deadly force.
• Valid conclusion: A Use of Force report must be filed within 24 hours.

Valid Form 2: Modus Tollens (Denying the Consequent)

• If P, then Q.
• Q is false.
• Therefore, P is false.

Worked example:

• Rule: If a suspect is armed with a firearm, deadly force may be justified.
• Fact: Deadly force was not justified (Q is false).
• Valid conclusion: The suspect was not armed with a firearm (P is false).

Fallacy 1: Affirming the Consequent (INVALID)

• If P, then Q.
• Q is true.
• Therefore, P is true. ✗

Worked example:

• Rule: If an officer uses deadly force, a report is filed within 24 hours.
• Fact: A Use of Force report was filed within 24 hours.
• Invalid conclusion: Therefore, deadly force was used. ✗ — other force levels also require reports; the report alone does not prove deadly force.

Fallacy 2: Denying the Antecedent (INVALID)

• If P, then Q.
• P is false.
• Therefore, Q is false. ✗

Worked example:

• Rule: If a suspect is armed with a firearm, deadly force may be justified.
• Fact: The suspect is not armed with a firearm.
• Invalid conclusion: Therefore, deadly force is not justified. ✗ — deadly force could be justified for other reasons (e.g., edged weapon, vehicle as weapon).

How to Identify the Valid Conclusion on the CAT

1. Translate the rule into "If P, then Q" form.
2. Determine what the fact establishes — P true, P false, Q true, or Q false.
3. Apply the matching form:
   • P true → Q true (modus ponens)
   • Q false → P false (modus tollens)
   • Q true → no valid conclusion about P
   • P false → no valid conclusion about Q
4. Eliminate answer choices that match the two fallacies or add new information not in the premises.

Chain of Conditionals

Some CAT items chain two conditionals: If P, then Q. If Q, then R. Fact: P is true. Conclusion: R is true.

Worked example:

• Rule 1: If an officer is on administrative leave, the officer's firearm is surrendered.
• Rule 2: If the firearm is surrendered, a receipt is issued.
• Fact: Officer Patel is on administrative leave.
• Valid conclusion: A receipt was issued. (P → Q → R, P true, so R true.)

The trap is reversing the chain: "If a receipt was issued, the officer is on administrative leave" is invalid — it affirms the consequent twice over.