Syllogism Part 2: Three-Premise Chains and Negative-Statement Traps

Three-premise syllogism chains, named moods, and negative-statement edge cases appear in the harder sections of TCS NQT, AMCAT, and Infosys InfyTQ aptitude tests. This article picks up where the four statement types and six distribution rules in Part 1 stop. It adds: sorites chains, the Celarent-Darii-Ferio moods as fast pattern matchers, the Baroco form, and complementary pairs, all verified through ten worked examples.

Extending to Three Premises: Sorites Chains

A sorites is a chain where the conclusion from one two-premise step becomes a premise for the next. Three premises produce two valid mood applications in sequence; four premises produce three.

The method is the same regardless of chain length:

• Step 1: Apply a valid mood to the first two premises and record the intermediate conclusion.
• Step 2: Pair that intermediate conclusion with the third premise and apply a valid mood again.
• Step 3: The output from Step 2 is the final conclusion.

If any single step in the chain violates the six distribution rules from Part 1, no valid overall conclusion follows, even when the chain looks directionally plausible.

Named Moods as Pattern Matchers

Aristotle’s syllogistic identifies 24 valid moods across four figures. Placement tests draw from four Figure-1 moods repeatedly. Memorise the pattern; use the distribution rules to verify when uncertain.

Mood	Premise 1	Premise 2	Conclusion	Code
Barbara	All M are P	All S are M	All S are P	AAA
Celarent	No M are P	All S are M	No S are P	EAE
Darii	All M are P	Some S are M	Some S are P	AII
Ferio	No M are P	Some S are M	Some S are not P	EIO

Baroco (Figure 2) is the fifth worth knowing: All P are M, Some S are not M, therefore Some S are not P. The fastest path through a Baroco problem is by contrapositive: if all P have property M, then anything lacking M is not P. The particular-negative premise confirms some S lack M, so those S are not P.

Negative-Statement Edge Cases

Three patterns cause consistent mark losses in placement aptitude tests.

Ferio is not uncertain. When you see No M are P paired with Some S are M, the conclusion Some S are not P is definitively valid. The S-items that are M are blocked from P by the first premise. There is no ambiguity here. Older prep materials sometimes mark this as uncertain. That label is wrong.

“No X is Y” does not chain to “No X is Z” when the only bridge is “All Y are Z.” That chain tells you the Y-items are Z but not X, so some members of Z are not X. Whether other Z-items overlap with X is entirely undetermined by the premises. The valid output from those two premises is “Some Z are not X” (see Example 8 for the full verification).

Baroco outputs particular negative only. When one premise is particular negative and the other is universal affirmative, the valid conclusion is particular negative. Never escalate to a universal negative conclusion, even when the pattern looks clean.

Complementary Pairs: The Either-Or Shortcut

When two conclusions C1 and C2 are subcontrary (“Some A are B” and “Some A are not B”), they cannot both be false simultaneously, provided A is a non-empty set.

Placement-test answer choices sometimes include “either C1 or C2 follows.” This option is correct when:

• Neither C1 nor C2 independently follows from the given premises (typically because Rule 1 is violated).
• C1 and C2 are subcontrary: one is the particular affirmative, the other is the particular negative, with the same subject and predicate.
• The subject set is confirmed non-empty by at least one premise.

The trap version: applying the complementary pair shortcut to contradictories (All A are B vs Some A are not B) or to conclusions with mismatched subjects. The shortcut applies only to the specific subcontrary pair.

Ten Worked Examples

• Example 1 — Sorites Chain (Barbara followed by Barbara)

  • Premises: All programmers know data structures. All who know data structures understand algorithms. All who understand algorithms can design efficient code.
  • Conclusion tested: All programmers can design efficient code.
  • Verdict: Valid.
  • Verification: Step 1 (Barbara): All programmers know data structures plus All who know DS understand algorithms gives All programmers understand algorithms. Step 2 (Barbara): All programmers understand algorithms plus All who understand algorithms can design efficient code gives All programmers can design efficient code. Both links valid; the chain is a sorites.

• Example 2 — Celarent

  • Premises: No fish are warm-blooded. All sharks are fish.
  • Conclusion tested: No sharks are warm-blooded.
  • Verdict: Valid.
  • Verification: Middle term is “fish,” distributed in P1 as subject of No. All sharks are inside the fish group (P2). No fish are warm-blooded (P1), so no sharks are warm-blooded. Matches Celarent: No M are P, All S are M, therefore No S are P.

• Example 3 — Darii

  • Premises: All engineers pass the aptitude screening. Some freshers are engineers.
  • Conclusion tested: Some freshers pass the aptitude screening.
  • Verdict: Valid.
  • Verification: Middle “engineers” distributed in P1 as subject of All. The freshers who are engineers (P2) also pass the aptitude screening (P1). Conclusion is particular, matching the particular premise. Rule 6 from Part 1 satisfied.

• Example 4 — Ferio (the Canary Case)

  • Premises: No papers are pens. Some books are papers.
  • Conclusion tested: Some books are not pens.
  • Verdict: Valid. A prior version of this article labelled this uncertain. That was incorrect.
  • Verification: Middle “papers” distributed in P1 as subject of No. The books that are papers (P2) are blocked from being pens (P1). Some books are therefore definitively not pens. Matches Ferio: No M are P, Some S are M, therefore Some S are not P.

• Example 5 — Undistributed Middle Trap

  • Premises: All graduates are educated. Some educated people are employed.
  • Conclusion tested: Some graduates are employed.
  • Verdict: Invalid.
  • Verification: Middle “educated” is the predicate of a universal affirmative in P1 (not distributed) and the subject of a particular affirmative in P2 (not distributed). Rule 1 from Part 1 violated. The employed-educated people in P2 may be entirely outside the graduate subset. The conclusion cannot be drawn.

• Example 6 — Baroco

  • Premises: All placement-test toppers finish questions on time. Some candidates do not finish questions on time.
  • Conclusion tested: Some candidates are not placement-test toppers.
  • Verdict: Valid.
  • Verification: Contrapositive of P1: any candidate who does not finish on time is not a topper. P2 confirms some candidates do not finish on time. Those candidates are not toppers. Rule check: middle “finish on time” is the predicate of a particular negative in P2 (distributed). Rule 1 satisfied via P2.

• Example 7 — Complementary Pair

  • Premises: All teachers are graduates. Some graduates are employed.
  • Conclusion 1: Some teachers are employed.
  • Conclusion 2: Some teachers are not employed.
  • Verdict: Neither C1 nor C2 individually follows. Either C1 or C2 must follow (complementary pair).
  • Verification: Middle “graduates” is the predicate of a universal affirmative in P1 (not distributed) and the subject of a particular affirmative in P2 (not distributed). Rule 1 violated for both C1 and C2. But C1 and C2 are subcontrary: since teachers are confirmed non-empty (from P1) and are graduates, they must either include some who are employed or some who are not. One of the pair must be true.

• Example 8 — No X is Y Chain Trap

  • Premises: No cats are dogs. All dogs are mammals.
  • Conclusion tested: No cats are mammals.
  • Verdict: Invalid.
  • Valid conclusion from these premises: Some mammals are not cats.
  • Verification: P1 says cats and dogs are disjoint. P2 places dogs inside mammals. Dogs are both mammals (P2) and not cats (P1). So some mammals, namely dogs, are not cats. Whether cats are or are not mammals through an independent route is not determined by these premises. “No cats are mammals” extends beyond what the premises support.

• Example 9 — Three-Premise Negative Chain

  • Premises: All software engineers write code regularly. No regular code writer makes systematic errors. Some interns are software engineers.
  • Conclusion 1: All software engineers are free from systematic errors.
  • Conclusion 2: Some interns are free from systematic errors.
  • Verdict: Both valid.
  • Verification: Step 1 (Celarent): No regular code writer makes systematic errors plus All software engineers write code regularly gives No software engineer makes systematic errors (C1). Step 2 (Ferio): Some interns are software engineers plus No software engineer makes systematic errors gives Some interns do not make systematic errors (C2). Two valid moods chain together.

• Example 10 — Four-Premise Complex

  • Premises: All coders know algorithms. All who know algorithms can debug code. Some students are coders. No one who can debug code fails technical interviews.
  • Conclusion 1: Some students do not fail technical interviews.
  • Conclusion 2: All coders can debug code.
  • Verdict: Both valid.
  • Verification: Step 1 (Barbara): All coders know algorithms plus All who know algorithms can debug gives All coders can debug code (C2). Step 2 (Darii): All coders can debug plus Some students are coders gives Some students can debug code. Step 3 (Ferio): Some students can debug code plus No debugger fails technical interviews gives Some students do not fail technical interviews (C1). Three valid steps, both conclusions confirmed.

Logical reasoning tests in placement aptitude rounds combine syllogism with other deduction-based sections. The same habit of tracing term distribution through premises applies to number analogy questions and to the if-then structure in blood relation problems.

The four-premise chain in Example 10 builds a habit that transfers directly to evaluating AI-generated text: an LLM’s conclusion may flow fluently through a chain of plausible steps, but each link still needs to hold under the same distribution checks applied here. TinkerLLM at ₹299 lets you run those chain-of-thought experiments live rather than study them from a textbook.