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Reverse Syllogism

Introduction

Reverse Syllogism asks you to work backwards: conclusions are given first and you must decide which set(s) of premises could validly support them. This skill trains you to recognise which premises are sufficient, which are incompatible, and which are merely possible - a frequent exam-style twist.

Practising reverse syllogisms is important because many tests assess whether a candidate can construct or identify valid premises for a target conclusion, not just evaluate forward inference.

Pattern: Reverse Syllogism

Pattern

The key idea: Given one or more conclusions, identify premises (or select from options) that together make the conclusions logically valid.

Rules & checklist:

  • Match quantifiers carefully - a conclusion that is universal (All / A) requires premises that distribute the middle term appropriately.
  • For a particular conclusion (Some / I), at least one existential premise or chain ending in an existential is required.
  • Watch for undistributed middle: premises must link subject and predicate through a term that is distributed correctly in at least one premise.
  • A negative conclusion (No / E or Some ... not / O) demands at least one negative premise or a valid chain producing exclusion.

Step-by-Step Example

Question

Conclusions:
I. Some P are Q.
II. No R is Q.

Which set of premises (choose the valid set) would make both conclusions true?
Options:
A. All P are S; Some S are Q; Some R are Q.
B. Some P are S; Some S are Q; No R is Q.
C. All P are S; Some S are Q; No R is Q.
D. Some P are Q; All Q are T; No R is T.

Solution

  1. Step 1: Identify requirements from conclusions

    Conclusion I (Some P are Q) requires an existential overlap between P and Q - either directly (Some P are Q) or via a chain that guarantees at least one P maps into Q (e.g., Some P are S and Some S are Q, with overlapping S portion that includes the P-element).
    Conclusion II (No R is Q) requires an exclusion: a premise that places R disjoint from Q (a direct No R is Q) or a chain that ensures R cannot be Q.

  2. Step 2: Evaluate Option A

    Option A: All P are S; Some S are Q; Some R are Q.
    - All P are S + Some S are Q ⇒ Some P could be Q (possible) - satisfies I only if the same S-element is both in P and Q; but "Some S are Q" and "All P are S" do not guarantee that the particular S which is Q overlaps the P-subset (existential uncertainty).
    - Some R are Q contradicts Conclusion II (No R is Q). So A fails to support II. ❌

  3. Step 3: Evaluate Option B

    Option B: Some P are S; Some S are Q; No R is Q.
    - Some P are S and Some S are Q still do not guarantee that the same S-element connects P and Q (two particular premises do not ensure overlap), so I is not guaranteed (only possible).
    - No R is Q directly gives II. So B secures II but fails to guarantee I. ❌

  4. Step 4: Evaluate Option C

    Option C: All P are S; Some S are Q; No R is Q.
    - All P are S plus Some S are Q still leaves existential uncertainty about whether the particular S that is Q is inside the P-subset, so Some P are Q is not guaranteed from these two alone. However, among the options this is the strongest chain that can make I true if we interpret the existential S as overlapping P - but strict syllogistic logic requires an explicit existential bridge (i.e., Some P are S AND Some S are Q with a common S element) to definitely conclude I. Thus C gives II (No R is Q) but does not strictly guarantee I. ❌

  5. Step 5: Evaluate Option D

    Option D: Some P are Q; All Q are T; No R is T.
    - Some P are Q directly satisfies Conclusion I. ✅
    - All Q are T and No R is T together imply No R is Q (if all Q are inside T but R is disjoint from T then R is disjoint from Q). More precisely: No R is T ⇒ R ∩ T = ∅; All Q ⊂ T ⇒ Q ⊂ T; therefore R ∩ Q = ∅ ⇒ No R is Q. Thus D also secures Conclusion II. ✅

  6. Final Answer:

    Option D - the premises in D make both conclusions valid.

  7. Quick Check:

    Pick premises that directly provide the existential for I and a negative/exclusion chain for II. Option D supplies both explicitly (Some P are Q; and All Q are T + No R is T → No R is Q). ✅

Quick Variations

1. Conclusions given as universals (All / No) usually require premises with universal links and correct distribution of the middle term.

2. When conclusions mix existential and universal claims, one premise must supply existence (Some...) and another must supply distribution/exclusion as needed.

3. If multiple premise-sets are offered, prefer the set that explicitly provides the existential and the necessary universal/negative - do not rely on two particulars to force overlap.

4. Reverse problems may ask you to construct premises - ensure the middle term is properly distributed if you want a universal conclusion.

Trick to Always Use

  • Step 1 → Look for an explicit existential (Some) if any conclusion is particular; without it you cannot be certain.
  • Step 2 → For a negative conclusion (No / Some not), look for an explicit negative premise or a universal that, combined with a negative, yields exclusion.
  • Step 3 → Avoid assuming two Some premises guarantee overlap; they don't unless a common element is stated.

Summary

Summary

  • Reverse Syllogism requires checking whether candidate premises supply the necessary existential or universal/distributive conditions for the conclusions.
  • Two particular premises (Some + Some) do not guarantee a particular conclusion unless a shared element is given.
  • Negative conclusions demand at least one negative premise or a universal + negative chain that enforces exclusion.
  • When multiple premise-sets are offered, choose the set that explicitly supplies the existentials and distributions the conclusions need.

Example to remember:
Conclusions: Some A are B; No C is B.
Premises that work: Some A are B; All B are D; No C is D ⇒ No C is B. ✅

Practice

(1/5)
1. Conclusions: I. Some X are Y. II. No Z is Y.<br>Which set of premises would make both conclusions definitely true?
easy
A. Some X are Y; All Y are W; No Z is W
B. All X are W; Some W are Y; Some Z are Y
C. Some X are W; Some W are Y; No Z is Y
D. All X are Y; Some Z are Y; No Y is W

Solution

  1. Step 1: Understand the conclusions

    We need 'Some X are Y' (overlap between X and Y) and 'No Z is Y' (Z and Y must be separate).

  2. Step 2: Check Option A

    Option A gives 'Some X are Y' directly - this satisfies the first conclusion. It also says 'All Y are W' and 'No Z is W'. Since Y is fully inside W and Z has no connection with W, it means Z and Y can never overlap. So, 'No Z is Y' is also true. ✅

  3. Final Answer:

    Option A → Option A

  4. Quick Check:

    'Some' ensures overlap; 'All + No' ensures separation. Perfect match for both conclusions. ✅
Hint: For 'Some' use explicit overlap; for 'No', use a universal + negative link.
Common Mistakes: Using two 'Some' statements together and assuming overlap automatically exists between unrelated groups.
2. Conclusions: I. No A is B. II. Some C are B.<br>Choose the premise set that makes both conclusions true.
easy
A. All A are D; No D is B; Some C are B
B. Some A are B; All C are B; No A is C
C. No A is B; Some C are D; All D are B
D. All A are B; Some B are C; No C is D

Solution

  1. Step 1: Understand what is required

    'No A is B' means A and B must be totally separate. 'Some C are B' means there should be an overlap between C and B.

  2. Step 2: Check Option A

    ‘All A are D’ and ‘No D is B’ → A is inside D, and D is totally separate from B, so A and B are also separate (first conclusion true). It also has ‘Some C are B’, which gives us the overlap needed for the second conclusion. ✅

  3. Final Answer:

    Option A → Option A

  4. Quick Check:

    Universal + Negative gives separation; explicit 'Some' gives overlap. ✅
Hint: Use a universal + negative chain for 'No', and add an explicit 'Some' for overlap.
Common Mistakes: Picking premises that only suggest a possibility but don’t guarantee the exclusion.
3. Conclusions: I. Some M are N. II. Some N are not P. Which premises guarantee both conclusions?
easy
A. All M are Q; Some Q are N; Some N are P
B. Some M are N; No N is R; Some R are P
C. Some M are N; Some N are not P; All R are N
D. All M are N; All N are P; Some P are Q

Solution

  1. Step 1: Understand what is required

    Conclusion I (Some M are N) needs at least one M that is also N. Conclusion II (Some N are not P) needs at least one N that lies outside P.

  2. Step 2: Check Option C

    Option C explicitly states Some M are N - this satisfies Conclusion I directly. It also explicitly states Some N are not P - this satisfies Conclusion II directly. Both conclusions are therefore guaranteed by these premises.

  3. Final Answer:

    Some M are N; Some N are not P. → Option C

  4. Quick Check:

    Option C restates both required 'Some' relations exactly, so it unambiguously guarantees both conclusions. ✅

Hint: Prefer options that explicitly restate the 'Some' or 'Some not' conclusions rather than implying them indirectly.
Common Mistakes: Picking choices with only universals ('All') which cannot prove partial overlaps or partial exclusions.
4. Conclusions: I. All S are T. II. Some U are not T.<br>Select the premise set that makes both conclusions valid.
medium
A. Some S are T; All U are V; No V is T
B. All S are R; All R are T; Some U are not T
C. All T are S; Some U are S; No U is R
D. All S are T; No U is T; Some U are V

Solution

  1. Step 1: Understand the requirements

    'All S are T' means S is fully inside T. 'Some U are not T' means U must have some members outside T.

  2. Step 2: Check Option B

    'All S are R' and 'All R are T' → by chain rule, All S are T (first conclusion true). The premise 'Some U are not T' is also directly mentioned (second conclusion true). ✅

  3. Final Answer:

    Option B → Option B

  4. Quick Check:

    Two universals give 'All', while explicit 'Some not' completes the second part. ✅
Hint: For 'All' use two universal links; for 'Some not' use an explicit negative.
Common Mistakes: Mixing up directions like 'All T are S' instead of 'All S are T'.
5. Conclusions: I. Some P are Q. II. Some R are not Q.<br>Which of the following premise-sets guarantees both conclusions?
medium
A. All P are S; Some S are Q; All R are Q
B. Some P are Q; No R is Q; Some R are T
C. Some P are S; Some S are Q; Some R are not Q
D. All P are Q; Some R are Q; No R is S

Solution

  1. Step 1: Understand the need

    'Some P are Q' means there’s overlap between P and Q. 'Some R are not Q' means some R are outside Q.

  2. Step 2: Check Option B

    ‘Some P are Q’ directly satisfies the first conclusion. 'No R is Q' ensures all R are outside Q, which means at least some R are not Q. The line 'Some R are T' guarantees that R exists, satisfying both conclusions. ✅

  3. Final Answer:

    Option B → Option B

  4. Quick Check:

    Explicit 'Some' confirms existence; 'No' ensures separation. ✅
Hint: Look for one 'Some' statement (for overlap) and one 'No' (for separation).
Common Mistakes: Ignoring that 'No' already includes 'Some not', making it more than sufficient.

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