4.3 Formal Syllogisms, Categorical Logic & Deductive Proofs

Ateneo College Entrance Test ACET

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1 Chapter 1: Mathematics

1.1 Real Numbers, Ratios, Rates & Percentages 1.2 Algebraic Expressions, Polynomials & Linear Equations 1.3 Functions, Domain/Range, Graphs & Transformations 1.4 Quadratic, Polynomial, Exponential & Logarithmic Equations 1.5 Plane Geometry: Triangles, Circles, Polygons & Perimeter/Area 1.6 Coordinate Geometry: Lines, Distance, Midpoints & Conic Sections 1.7 Trigonometry: Right Triangle Ratios, Unit Circle & Basic Identities 1.8 Arithmetic & Geometric Sequences, Series & Word Patterns 1.9 Permutations, Combinations, Probability & Statistics

2 Chapter 2: English Proficiency

2.1 Subject-Verb Agreement, Verb Tenses & Aspect 2.2 Pronoun-Antecedent Agreement, Case, Misplaced & Dangling Modifiers 2.3 Sentence Structure, Run-Ons, Fragments & Parallelism 2.4 Contextual Vocabulary, Word Roots, Prefixes & Suffixes 2.5 Advanced Synonyms, Antonyms & Nuanced Vocabulary 2.6 Verbal Analogies: Relationship Types & Solving Strategies 2.7 Punctuation, Capitalization, Mechanics & Common Errors 2.8 Diction, Idiomatic Expressions, Prepositions & Frequently Confused Words

3 Chapter 3: Reading Comprehension

3.1 Identifying Main Ideas, Central Themes & Paragraph Summaries 3.2 Locating Explicit Details, Supporting Evidence & Data 3.3 Drawing Logical Inferences & Unstated Conclusions 3.4 Analyzing Author's Purpose, Point of View, Tone & Attitude 3.5 Interpreting Figurative Language, Tone Words & Context Clues 3.6 Text Structure, Rhetorical Devices & Evaluating Arguments

4 Chapter 4: Logical and Abstract Reasoning

4.1 Number & Letter Series Patterns & Formulas 4.2 Abstract Reasoning: Matrix Patterns & Figure Rotations 4.3 Formal Syllogisms, Categorical Logic & Deductive Proofs 4.4 Coding-Decoding, Cipher Patterns & Symbol Operations 4.5 Spatial Visualization, Paper Folding & Venn Diagram Logic

5 Chapter 5: General Knowledge

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6 Chapter 6: ACET Essay Writing & Exam Strategies

6.1 ACET Essay: Thesis Development, Structuring & Argumentation 6.2 Time Allocation, Pacing Strategies & Subtest Tactics 6.3 Guessing Strategies (No Penalty), Test Anxiety & Exam Day Readiness

Free exam prep ACET (Ateneo College Entrance Test) Practice Questions 108 questions Study Guide 35 topics Cheat Sheet 56 facts Flashcards 50 cards

4.3 Formal Syllogisms, Categorical Logic & Deductive Proofs

Key Takeaways Explain This Topic AI

Distinguish between valid and invalid categorical syllogisms by converting statements into standard categorical propositions (A, E, I, O).

Master logical equivalence operations: Contrapositive (if not Q then not P is valid), Converse (if Q then P is invalid), and Inverse (if not P then not Q is invalid).

Recognize classical formal fallacies such as Undistributed Middle, Affirming the Consequent, Denying the Antecedent, and Four Terms Fallacy.

Translate natural language conditional claims ('If', 'Only if', 'Unless', 'Necessary and Sufficient') into formal propositional notation for rigorous deduction. Last updated: July 2026

Formal Syllogisms, Categorical Logic & Deductive Proofs Deductive logic on the ACET tests your ability to evaluate arguments based strictly on their structural validity, independent of real-world truth or empirical plausibility. Questions present premises and require you to identify necessary conclusions, detect logical fallacies, or establish statement equivalences. Mastering categorical syllogisms and conditional proposition calculus is essential for securing a high score in this section. 1. Truth vs. Validity: The Deductive Baseline In formal logic, a clear distinction exists between Truth and Validity : Validity is a property of argument structure. An argument is valid if and only if it is logically impossible for its premises to be true while its conclusion is false. Truth is a property of individual statements regarding factual reality. Soundness: An argument is sound if and only if it is valid AND all of its premises are factually true. ACET Rule: Never use external domain knowledge or real-world facts to evaluate a syllogism. If a premise states "All cats are reptiles," you must accept that statement as absolute truth within the scope of the problem. 2. Standard Categorical Propositions (The AEIO Framework) Categorical logic analyzes relationships between classes of objects. Every categorical statement maps to one of four standard form propositions: UNIVERSAL PARTICULAR +----------------------------+----------------------------+ | Universal Affirmative (A) | Particular Affirmative (I) | AFFIRMATIVE | "All S are P" | "Some S are P" | | Formula: S ⊆ P | Formula: S ∩ P ≠ ∅ | +----------------------------+----------------------------+ | Universal Negative (E) | Particular Negative (O) | NEGATIVE | "No S are P" | "Some S are not P" | | Formula: S ∩ P = ∅ | Formula: S \ P ≠ ∅ | +----------------------------+----------------------------+ Distribution of Terms A term is distributed if the proposition makes a claim about every member of the class represented by that term.

Proposition Type Symbol Quantity / Quality Subject Term ( S S S ) Predicate Term ( P P P ) All S S S are P P P A Universal Affirmative Distributed Undistributed No S S S are P P P E Universal Negative Distributed Distributed Some S S S are P P P I Particular Affirmative Undistributed Undistributed Some S S S are not P P P O Particular Negative Undistributed Distributed 3. Rules of Syllogistic Validity A categorical syllogism consists of three propositions containing exactly three terms: the Major Term (predicate of conclusion), the Minor Term (subject of conclusion), and the Middle Term (appears in both premises, but NOT in conclusion). To determine if a syllogism is valid, verify these five fundamental rules: Rule of the Distributed Middle: The middle term must be distributed in at least one premise. Violation ⟹ \implies ⟹ Fallacy of Undistributed Middle . Rule of Retained Distribution: Any term distributed in the conclusion MUST be distributed in the premise where it occurs. Violation ⟹ \implies ⟹ Illicit Major or Illicit Minor fallacy. Rule of Negative Premises: Two negative premises ( E E E or O O O ) yield NO valid conclusion. Violation ⟹ \implies ⟹ Exclusive Premises Fallacy . Rule of Negative Quality: If either premise is negative, the conclusion MUST be negative; if the conclusion is negative, one premise MUST be negative. Existential Rule: Two universal premises ( A A A or E E E ) cannot produce a particular conclusion ( I I I or O O O ) in modern Boolean logic. 4. Conditional Propositional Calculus ( p → q p \rightarrow q p → q ) A conditional statement asserts that if an antecedent ( p p p ) is true, then a consequent ( q q q ) must also be true: p → q p \rightarrow q p → q . Truth Table for Implication

p p p q q q p → q p \rightarrow q p → q Logical Interpretation T T T Antecedent occurs, consequent holds (Valid) T F F Antecedent occurs, consequent fails (Broken Promise) F T T Vacuously True (Consequent can occur independently) F F T Vacuously True Statement Transpositions & Equivalences For a given conditional proposition p → q p \rightarrow q p → q : Original Implication: p → q p \rightarrow q p → q Converse: q → p q \rightarrow p q → p ( INVALID equivalence — Fallacy of Affirming the Consequent) Inverse: ¬ p → ¬ q \neg p \rightarrow \neg q ¬ p → ¬ q ( INVALID equivalence — Fallacy of Denying the Antecedent) Contrapositive: ¬ q → ¬ p \neg q \rightarrow \neg p ¬ q → ¬ p ( VALID EQUIVALENCE! Always logically identical to p → q p \rightarrow q p → q ) EQUIVALENT Original: p --------> q <===============> Contrapositive: ~q --------> ~p | | | | INVALID | | INVALID INVALID | | INVALID v v v v Converse: q --------> p <===============> Inverse: ~p --------> ~q EQUIVALENT Valid Argument Forms vs. Fallacies Modus Ponens (Affirming Antecedent - VALID): p → q , p ⊢ q p \rightarrow q, \quad p \quad \vdash \quad q p → q , p ⊢ q Modus Tollens (Denying Consequent - VALID): p → q , ¬ q ⊢ ¬ p p \rightarrow q, \quad \neg q \quad \vdash \quad \neg p p → q , ¬ q ⊢ ¬ p Hypothetical Syllogism (Chain Rule - VALID): p → q , q → r ⊢ p → r p \rightarrow q, \quad q \rightarrow r \quad \vdash \quad p \rightarrow r p → q , q → r ⊢ p → r Affirming the Consequent (FALLACY - INVALID): p → q , q ⊢ p (ERROR!) p \rightarrow q, \quad q \quad \vdash \quad p \quad \text{(ERROR!)} p → q , q ⊢ p (ERROR!) Denying the Antecedent (FALLACY - INVALID): p → q , ¬ p ⊢ ¬ q (ERROR!) p \rightarrow q, \quad \neg p \quad \vdash \quad \neg q \quad \text{(ERROR!)} p → q , ¬ p ⊢ ¬ q (ERROR!) 5. Translating Natural Language into Formal Logic ACET word problems disguise conditional operators using subtle English idioms. Use this translation key:

English Expression Formal Logic Translation Equivalence / Contrapositive "If A A A , then B B B " A → B A \rightarrow B A → B ¬ B → ¬ A \neg B \rightarrow \neg A ¬ B → ¬ A " A A A only if B B B " A → B A \rightarrow B A → B ¬ B → ¬ A \neg B \rightarrow \neg A ¬ B → ¬ A " A A A is necessary for B B B " B → A B \rightarrow A B → A ¬ A → ¬ B \neg A \rightarrow \neg B ¬ A → ¬ B " A A A is sufficient for B B B " A → B A \rightarrow B A → B ¬ B → ¬ A \neg B \rightarrow \neg A ¬ B → ¬ A " A A A unless B B B " ¬ B → A \neg B \rightarrow A ¬ B → A (or ¬ A → B \neg A \rightarrow B ¬ A → B ) A ∨ B A \lor B A ∨ B " A A A if and only if B B B " A ↔ B A \leftrightarrow B A ↔ B ( A → B ) ∧ ( B → A ) (A \rightarrow B) \land (B \rightarrow A) ( A → B ) ∧ ( B → A )

Test Your Knowledge

Which of the following statements is logically equivalent to: 'If an applicant passes the ACET, then the applicant receives an admission letter'?

If an applicant receives an admission letter, then the applicant passed the ACET.

If an applicant does not pass the ACET, then the applicant does not receive an admission letter.

If an applicant does not receive an admission letter, then the applicant did not pass the ACET.

An applicant receives an admission letter only if the applicant does not pass the ACET. Check Answer Need a hint? AI

Test Your Knowledge

Consider the following premises: Premise 1: 'All engineers are analytical.' Premise 2: 'Some analytical people are artists.' Which conclusion can be validly deduced?

Some engineers are artists.

All artists are engineers.

Some artists are not analytical.

No valid conclusion can be drawn connecting engineers and artists. Check Answer Need a hint? AI

Test Your Knowledge

Premise 1: 'If the weather is stormy, the flight will be canceled.' Premise 2: 'The flight was not canceled.' What valid conclusion follows?

The weather was not stormy.

The weather was stormy.

The flight was delayed rather than canceled.

The weather will remain stormy tomorrow. Check Answer Need a hint? AI

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