5.2 Deductive Reasoning: Categorical Logic, Syllogisms, and Conditionals

5.2 Deductive Reasoning: Categorical Logic, Syllogisms, and Conditionals

Quick Answer: The Aptitude Test's 8 Deductive Reasoning (演繹推理) questions are set in Chinese. Each gives a short passage that you must assume is true and asks for the inference, or set of inferences, that must follow. Key competencies include: (1) evaluating categorical syllogisms via Euler/Venn diagrams while remembering that some means at least one, possibly all; (2) applying the contrapositive rule ($P \rightarrow Q \iff \neg Q \rightarrow \neg P$) across chained conditionals; (3) rejecting invalid inferences generated by Affirming the Consequent ($Q \rightarrow P$) or Denying the Antecedent ($\neg P \rightarrow \neg Q$); and (4) translating administrative directives where if marks sufficiency, only if marks necessity, and unless establishes a negated antecedent ($\neg B \rightarrow A$).

The Architecture of Deductive Reasoning

Unlike Verbal Reasoning—which evaluates factual adherence to an informational text—Deductive Reasoning (演繹推理) tests formal, structural validity. In deductive logic, if the stated premises are assumed to be true, the valid conclusion must be true by necessity. The truth of the premises guarantees the truth of the conclusion; there is no room for probability, induction, or real-world plausibility.

On the CRE Aptitude Test, the 8 Deductive Reasoning items are set in Chinese (the English Verbal Reasoning items are covered in Section 5.1). The CSB's directions read: 「請根據以下短文的內容，選出一個或一組推論。請假定短文的內容都是正確的。」 That is: based on the passage, choose one inference or a set of inferences, and assume everything in the passage is true. The logic involved includes:

• Categorical Syllogisms: Deductions based on set inclusions and exclusions (all, no, some, some... not).
• Conditional Reasoning: Logical implications involving if... then, only if, unless, and if and only if.
• Chained Deductions: Multi-step relational chains where three or four regulatory conditions must be linked together to evaluate an individual case.

The Official Sample and Chinese Logic Words

The CSB's sample:

龍門鎮中的每一個人都是亞華的親戚，亞華只有一個兒子，銘希在龍門鎮裡生活，志南是亞華的丈夫。由此可推論： A 銘希是亞華的女兒。　B 亞華生活在龍門鎮裡。　C 銘希是亞華的親戚。　D 志南生活在龍門鎮裡。 答案：C

• C follows: everyone in the town is Ah Wa's relative (所有 S 都是 P), and Ming Hei lives in the town (Ming Hei is S), so Ming Hei is Ah Wa's relative.
• A does not follow: being a relative does not make Ming Hei a daughter.
• B and D do not follow: the passage says people in the town are relatives; it does not say relatives live in the town. Inferring that Ah Wa or her husband lives there reverses the rule (所有 S 是 P does not give 所有 P 是 S).

Because the items are in Chinese, translate each logic word into its formal meaning before testing the options:

When an item offers sets of inferences (for example, "(1) and (3) only"), test each numbered inference separately and then pick the option that lists exactly the valid ones.

Categorical Logic: Aristotle's Four Standard Forms

Categorical logic deals with relationships between categories or classes of objects. Every categorical proposition connects a Subject class ($S$) with a Predicate class ($P$) through one of four standard structures established in classical logic:

The Crucial "Some" Distinction in Formal Logic

In everyday English and Cantonese conversation, using the word "some" carries a conversational implicature of limitation: if someone says "Some committee members attended the briefing", listeners naturally infer that not all members attended.

Testing Syllogisms with Euler Diagrams

A categorical syllogism consists of two premises (Major and Minor) and a conclusion connecting three terms: the Major Term, the Minor Term, and the Middle Term (which appears in both premises but never in the conclusion).

To test whether a syllogistic deduction is valid under CRE exam conditions:

1. Draw the spatial relationships of the premises using overlapping circles (Euler diagrams).
2. Attempt actively to construct a counterexample diagram—a valid configuration of the circles that satisfies both premises but falsifies the candidate conclusion.
3. The Golden Rule of Deductive Validity: A conclusion is valid if and only if it holds true across every single possible Euler diagram configuration. If even one valid spatial arrangement violates the conclusion, the deduction is invalid.

The Fallacy of the Undistributed Middle

Consider this classic syllogistic error:

• Premise 1: All Administrative Officers are degree holders ($A \subseteq D$).
• Premise 2: All Town Planners are degree holders ($T \subseteq D$).
• Candidate Conclusion: All Administrative Officers are Town Planners ($A \subseteq T$).

Euler Analysis: Circle $A$ is inside Circle $D$. Circle $T$ is also inside Circle $D$. However, Circle $A$ and Circle $T$ can be completely separate from each other while both remaining inside $D$. Because an arrangement exists where $A \cap T = \emptyset$, the conclusion is invalid. The middle term (degree holders) was never "distributed" across the entire class in a way that forces $A$ and $T$ to touch.

Conditional Logic: The Law of the Contrapositive

Conditional statements govern the vast majority of civil service regulations, legal codes, and administrative criteria. A standard conditional is expressed as: $\text{If } P \text{, then } Q \quad (P \rightarrow Q)$ Where:

• $P$ is the Antecedent (the condition or trigger).
• $Q$ is the Consequent (the guaranteed result or required property).

The Four Forms of Conditional Inference

Given the rule $P \rightarrow Q$:

1. Modus Ponens (Affirming the Antecedent - VALID):
   • Given: $P$ is True.
   • Deduction: Therefore, $Q$ must be True.
   • Example: "If an application is submitted late, it is rejected. Application X was submitted late. Therefore, Application X is rejected."
2. Modus Tollens (Denying the Consequent - VALID):
   • Given: $Q$ is False ($\neg Q$).
   • Deduction: Therefore, $P$ must be False ($\neg P$).
   • Example: "Application Y was not rejected. Therefore, Application Y was not submitted late."
3. Fallacy of Affirming the Consequent (INVALID):
   • Given: $Q$ is True.
   • Faulty Deduction: Therefore, $P$ must be True. (ERROR)
   • Why it fails: An application could be rejected for another reason, such as missing documents. Knowing the result $Q$ occurred does not prove it was triggered by $P$.
4. Fallacy of Denying the Antecedent (INVALID):
   • Given: $P$ is False ($\neg P$).
   • Faulty Deduction: Therefore, $Q$ must be False ($\neg Q$). (ERROR)
   • Why it fails: Denying the specific condition $P$ does not prevent $Q$ from occurring via an alternative path.

The Contrapositive Equivalence

The fundamental law of deductive logic is: $P \rightarrow Q \iff \neg Q \rightarrow \neg P$ A conditional statement and its contrapositive are logically identical in truth value. If a conditional rule is true, its contrapositive is unconditionally true. However, its converse ($Q \rightarrow P$) and its inverse ($\neg P \rightarrow \neg Q$) are formal fallacies.

Translating Bureaucratic and Statutory Language

On the CRE Aptitude Test, premises rarely arrive in tidy mathematical symbols like $P \rightarrow Q$. They are embedded in dense bureaucratic prose. You must instantly translate these linguistic formulations into proper directional conditionals.

1. Sufficient vs. Necessary Conditions

• Sufficient Condition Indicator Words: if, when, whenever, every time, all who, any person who, provided that.
  • These words introduce the Antecedent (the left side: $P$).
  • Sentence: "Any applicant who submits the form before the deadline is entered into the ballot."
  • Symbolic Form: $\text{On time} \rightarrow \text{Entered}$.
• Necessary Condition Indicator Words: only if, only when, requires, must, is a prerequisite for, is a mandatory condition.
  • These words introduce the Consequent (the right side: $Q$).
  • Sentence: "A candidate is eligible for the Administrative Officer interview only if they attain Level 2 in Use of English."
  • Symbolic Form: $\text{Eligible for AO Interview} \rightarrow \text{Level 2 UE}$.

2. The Logic of "Unless"

In administrative regulations, the word "unless" appears constantly:

• "An application will be rejected unless it is countersigned by a directorate supervisor."

The Universal Translation Formula for "Unless": $\text{A unless B} \iff \neg B \rightarrow A \iff \neg A \rightarrow B$ Replace "unless" with "if not":

• "If an application is not countersigned by a directorate supervisor, it will be rejected" ($\neg \text{Countersigned} \rightarrow \text{Rejected}$).
• Contrapositive: "If an application is not rejected, it must have been countersigned" ($\neg \text{Rejected} \rightarrow \text{Countersigned}$).

3. Biconditionals: "If and Only If"

A biconditional ($P \iff Q$) signifies that $P$ and $Q$ are both sufficient and necessary for each other. It means $P \rightarrow Q$ and $Q \rightarrow P$. If one is true, the other must be true; if one is false, the other must be false.

• Indicator phrases: if and only if, exactly when, precisely in cases where.

Chained Conditionals and Syllogistic Deduction Walkthrough

To observe how multi-step deductive logic operates under exam conditions, examine the following worked case study.

Fictional Regulatory Stimulus: Staff Benefit Rules

1. All Grade A officers ($S$) are Band 1 officers ($D$).
2. No Band 1 officer ($D$) is eligible for standard overtime cash compensation ($O$).
3. An officer is granted the Housing Benefit ($H$) only if they are a Band 1 officer ($D$).
4. An officer is entitled to the Travel Concession ($T$) if and only if they receive the Housing Benefit ($H$) and have completed at least 8 years of continuous civil service ($Y$).
5. Officer Chan is entitled to the Travel Concession ($T$ is True).

Step-by-Step Formalization

1. $S \rightarrow D$
2. $D \rightarrow \neg O$
3. $H \rightarrow D$ (from "only if": receiving allowance requires Band 1 status)
4. $T \iff (H \wedge Y)$ (biconditional: entitlement implies both $H$ and $Y$)
5. $T = \text{True}$

Deriving Inescapable Deductions

• Deduction A (from Premise 4 and 5): Because $T$ is True and $T \iff (H \wedge Y)$, both $H$ and $Y$ must be strictly True. Therefore, Officer Chan receives the Housing Benefit ($H$), and Officer Chan has completed at least 8 years of continuous service ($Y$).
• Deduction B (from Premise 3 and Deduction A): We know $H \rightarrow D$. Since $H$ is True, by Modus Ponens, Officer Chan is a Band 1 officer ($D$).
• Deduction C (from Premise 2 and Deduction B): We know $D \rightarrow \neg O$. Since Chan is a Band 1 officer ($D$ is True), by Modus Ponens, Officer Chan is NOT eligible for standard overtime cash compensation ($\neg O$).

Testing Distractor Traps on Officer Chan

• Trap Statement 1: "Officer Chan is a Grade A officer ($S$)."
  • Analysis: We know $S \rightarrow D$, and we know Chan is $D$. Inferring $S$ from $D$ commits the Fallacy of Affirming the Consequent. Chan could be a an officer in any other Band 1 grade. This statement is not logically guaranteed.
• Trap Statement 2: "If Officer Chan had completed only 7 years of service, she could still receive the Travel Concession."
  • Analysis: The rule specifies $T \iff (H \wedge Y)$. If $Y$ is false (7 years $< 8$ years), the conjunction $(H \wedge Y)$ is false, which necessitates that $T$ is false. This statement directly contradicts the biconditional rule.

+--------------------------------------------------------------------------+
|                     DEDUCTIVE DERIVATION AUDIT TRAIL                     |
+--------------------------------------------------------------------------+
| Premise / Step       | Symbolic Translation        | Derived Truth State |
|----------------------+-----------------------------+---------------------|
| Given Fact           | T is True                   | T = 1               |
| Premise 4 (Bicond.)  | T <-> (H AND Y)             | H = 1, Y = 1        |
| Premise 3 (Cond.)    | H -> D                      | D = 1               |
| Premise 2 (Negative) | D -> NOT O                  | O = 0 (Not eligible)|
| Premise 1 (Trap)     | S -> D (Affirming Conseq.)  | S = UNKNOWN         |
+--------------------------------------------------------------------------+

Independent Preparation Advisory

This guide is published independently by OpenExamPrep for self-study and examination preparation. It is not affiliated with, authorized by, sponsored by, or endorsed by the Civil Service Bureau of the HKSAR Government. Candidates should confirm examination formats and regulatory standards against official CSB documentation.