Syllogisms (Logical Deduction)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your HAT-UG exam.

What is a Syllogism? A syllogism is a logical argument consisting of two premises and a conclusion. If both premises are true and the argument is valid, the conclusion must be true.

The Four Types of Statements:

Type	Form	Example
A (Universal Affirmative)	All A are B	All cats are mammals
E (Universal Negative)	No A are B	No cats are dogs
I (Particular Affirmative)	Some A are B	Some cats are black
O (Particular Negative)	Some A are not B	Some cats are not domestic

Syllogism Structure:

• Major premise: Contains the predicate of the conclusion (B)
• Minor premise: Contains the subject of the conclusion (A)
• Middle term: Appears in both premises but not the conclusion (M)
• Conclusion: States a relationship between subject (S) and predicate (P)

Mood and Figure: The mood is determined by the types of statements (A, E, I, O). The figure is determined by the position of the middle term in the premises.

⚡ HAT-UG Tip: In any syllogism, check whether the middle term is distributed (mentioned in all of) in at least one premise — if not, the argument is invalid (undistributed middle fallacy).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for HAT-UG Analytical Reasoning students with a few days to months.

The 15 Valid Syllogism Forms:

Figure 1 (M–P, S–M → S–P):

• Barbara: All P are M. All M are S. ∴ All P are S. (AAA)
• Celarent: No P are M. All M are S. ∴ No P are S. (EAE)
• Darii: All P are M. Some M are S. ∴ Some P are S. (AII)
• Ferio: No P are M. Some M are S. ∴ Some P are not S. (EIO)

Figure 2 (P–M, S–M → S–P):

• Cesare: No P are M. All S are M. ∴ No S are P. (EAE)
• Camestres: All P are M. No S are M. ∴ No S are P. (AEE)
• Festino: No P are M. Some S are M. ∴ Some S are not P. (EIO)
• Baroco: All P are M. Some S are not M. ∴ Some S are not P. (AOO)

Figure 3 (M–P, M–S → S–P):

• Darapti: All P are M. All M are S. ∴ Some S are P. (AAI)
• Felapton: No P are M. All M are S. ∴ Some S are not P. (EAO)
• Disamis: Some P are M. All M are S. ∴ Some P are S. (IAI)
• Datisi: All P are M. Some M are S. ∴ Some P are S. (AII)
• Bocardo: Some P are not M. All M are S. ∴ Some P are not S. (OAO)
• Ferison: No P are M. Some M are S. ∴ Some P are not S. (EIO)

Testing Validity with Venn Diagrams:

1. Draw three overlapping circles: S, P, M
2. Represent both premises by shading (universal) or placing X (particular)
3. Read off what can be definitively stated about S and P
4. If the conclusion follows, it is valid

Example: All teachers are professionals. Some professionals are authors. Therefore, Some teachers are authors.

• Shade: Teachers inside Professionals
• X in: overlap of Professionals and Authors
• Can we definitively place X in overlap of Teachers and Authors? No — the X could be in the part of Professionals that doesn’t overlap with Teachers. NOT VALID.

The Square of Opposition:

• A and O are contradictories (one true, one false)
• E and I are contradictories
• A and E are contraries (cannot both be true, can both be false)
• I and O are subcontraries (cannot both be false, can both be true)
• A → I and E → O (subalternation: truth flows down, falsity flows up)

⚡ HAT-UG Common Mistakes:

• Treating “Some A are B” as equivalent to “Some A are not B” — they are not negations of each other
• Forgetting that a valid syllogism with false premises can still have a false conclusion
• Not using Venn diagrams systematically — visual methods are more reliable than intuition

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for HAT-UG and NTS analytical reasoning preparation.

Conditional Syllogisms:

Modus ponens: If P → Q. P. ∴ Q. VALID. Modus tollens: If P → Q. ¬Q. ∴ ¬P. VALID. Affirming the consequent: If P → Q. Q. ∴ P. INVALID ( converse error). Denying the antecedent: If P → Q. ¬P. ∴ ¬Q. INVALID (inverse error).

Hypothetical Syllogisms: If P → Q. If Q → R. ∴ If P → R. VALID (chain argument). If P → Q. If P → R. ∴ P → (Q and R). VALID.

Disjunctive Syllogisms: P or Q. ¬P. ∴ Q. VALID (for exclusive or; for inclusive or, this is not certain). P or Q. P. ∴ ¬Q. VALID (for exclusive or only).

** Logical Deduction from Multiple Statements:**

When given several conditional statements:

1. Translate each into logical notation
2. Combine where possible: if A → B and B → C, then A → C
3. Use contrapositives: if A → B, then ¬B → ¬A
4. Look for chains that link all relevant terms

Example set:

• If it is hot, it is summer. (H → S)
• If it is summer, it is sunny. (S → U)
• If it is sunny, we go to the beach. (U → B)
• We did not go to the beach. (¬B)
• From U → B and ¬B: ¬U (modus tollens)
• From S → U and ¬U: ¬S (modus tollens)
• From H → S and ¬S: ¬H (modus tollens) ∴ It is not hot.

Sorites (Chain of Syllogisms): A chain of categorical statements with a shared middle term forming one long syllogism.

Example: All A are B. All B are C. All C are D. All D are E. ∴ All A are E. (Valid through repeated application of Barbara.)

Testing for Logical Validity — Checklist:

1. Is the middle term distributed in at least one premise? (If not: undistributed middle — invalid)
2. Does the conclusion distribute a term more broadly than its premise? (If yes: illicit process — invalid)
3. Are there any negative premises without a negative conclusion? (If yes: exclusive premises — invalid)
4. Are both premises negative? (If yes: no conclusion follows — invalid)

NTS/HAT-UG Patterns:

• Syllogisms appear regularly in NTS NAT-I and HAT-UG analytical reasoning
• Focus on identifying mood and figure to quickly eliminate invalid forms
• Practice translating everyday language into categorical form
• Use Venn diagrams for all categorical syllogisms — never rely on intuition

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