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Possibility-Based Syllogism

Introduction

Possibility-Based Syllogisms test whether a stated relation can be true without contradicting the premises. Instead of asking what must follow, these questions ask what is possible - a subtle but common twist in competitive reasoning.

Mastering possibility checks is important because many problems present incomplete information; recognizing which conclusions remain logically possible (even if not certain) separates cautious solvers from excellent ones.

Pattern: Possibility-Based Syllogism

Pattern

The key idea: A conclusion is a possibility if it does not contradict any given premise; it is impossible if it directly contradicts a premise.

Rules to apply:

  • If a conclusion violates an explicit universal (“All” / “No”), it is impossible.
  • If nothing in the premises contradicts a conclusion, treat it as possible (unless the premises force the opposite).
  • Possibility questions often use words like “possible,” “can,” or ask whether a conclusion may be true.
  • Distinguish carefully between follows (must be true) and is possible (may be true).

Step-by-Step Example

Question

Statements:
1️⃣ All painters are artists.
2️⃣ Some artists are sculptors.

Which of the following is possible?
Options:
A. All painters are sculptors.
B. Some painters are sculptors.
C. No painter is a sculptor.
D. All sculptors are painters.
E. All of the above

Solution

  1. Step 1: Restate premises

    All Painters ⊂ Artists. Some Artists ↔ Sculptors (partial overlap between Artists and Sculptors).

  2. Step 2: Check Option A

    “All painters are sculptors” would mean Painters ⊂ Sculptors. No universal premise forbids this, so it is possible. ✅

  3. Step 3: Check Option B

    “Some painters are sculptors” is also possible (it would follow if the painters occupy the overlapping portion). ✅

  4. Step 4: Check Option C

    “No painter is a sculptor” is also not contradicted by the premises: painters might lie in the artist portion disjoint from sculptors. So this is possible. ✅

  5. Step 5: Check Option D

    “All sculptors are painters” (Sculptors ⊂ Painters) is not guaranteed but not explicitly contradicted by the given statements (we only know some artists are sculptors). So this is also possible. ✅

  6. Final Answer:

    All options A, B, C, and D are possible. → Option E

  7. Quick Check:

    None of the options contradicts a universal premise (there is no statement like “No artist is a sculptor” or “All sculptors are not artists”), so all four listed possibilities can logically hold in some scenario. ✅

Quick Variations

1. Possible vs Impossible: replace “follows” with “possible” to change answer patterns.

2. Mix of All/No/Some: an explicit “No A is B” immediately makes any “A can be B” impossible.

3. Existential constraints: if premises include “Some A are B”, then “No A is B” becomes impossible.

4. Multiple premises: check each premise for direct contradiction before declaring possibility.

Trick to Always Use

  • Step 1 → Scan for universal negatives (“No …”) first - they create impossibilities.
  • Step 2 → If no premise directly contradicts the candidate conclusion, treat it as possible.
  • Step 3 → Remember: “possible” ≠ “follows”; mark as possible unless explicitly ruled out.

Summary

Summary

  • Identify universal statements first - they create hard impossibilities.
  • If a conclusion doesn’t contradict any premise, it is logically possible.
  • “Some” premises provide existence but do not eliminate alternative possibilities unless contradicted.
  • Always re-check each premise against the candidate conclusion for direct contradiction.

Example to remember:
All A are B; Some B are C ⇒ It is possible that All A are C, or Some A are C, or No A are C - unless a premise forbids it. ✅

Practice

(1/5)
1. Statements: 1️⃣ All cats are mammals. 2️⃣ All mammals are animals. 3️⃣ Some animals are wild. Conclusions: I. Some cats are wild. II. All cats are animals.
easy
A. Only Conclusion I follows
B. Only Conclusion II follows
C. Both I and II follow
D. Neither I nor II follows

Solution

  1. Step 1: Link relations

    All Cats ⊂ Mammals; All Mammals ⊂ Animals ⇒ All Cats ⊂ Animals.

  2. Step 2: Evaluate Conclusion I

    ‘Some animals are wild’ is an existential about Animals (the superset). That existential does not guarantee that the particular wild animals include Cats (the subset). Therefore ‘Some cats are wild’ is only possible, not certain. ❌

  3. Step 3: Evaluate Conclusion II

    From the universals All Cats ⊂ Mammals and All Mammals ⊂ Animals we can deduce All Cats ⊂ Animals. Hence ‘All cats are animals’ is definitely true. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option B

  5. Quick Check:

    All → All gives a universal inclusion; an existential about the superset does not force overlap with a particular subset. ✅
Hint: An existential about the superset doesn't imply the subset is part of that existential.
Common Mistakes: Assuming 'Some animals are X' automatically includes every subset of animals.
2. Statements: 1️⃣ All roses are flowers. 2️⃣ No flower is a tree. Which of the following is impossible?
easy
A. Some roses are trees
B. No rose is a tree
C. All roses are flowers
D. Some trees are not roses

Solution

  1. Step 1: Restate premises

    All Roses ⊂ Flowers; Flowers ∩ Trees = ∅ (flowers and trees are disjoint).

  2. Step 2: Derive consequence

    Since Roses ⊂ Flowers and Flowers ∩ Trees = ∅, it follows that Roses ∩ Trees = ∅ (roses cannot overlap with trees).

  3. Step 3: Determine impossibility

    ‘Some roses are trees’ asserts Roses ∩ Trees ≠ ∅, which directly contradicts Roses ∩ Trees = ∅. Therefore this statement is impossible.

  4. Final Answer:

    Some roses are trees. → Option A

  5. Quick Check:

    Subset of a set disjoint with another set ⇒ the subset is also disjoint with that set → 'Some roses are trees' impossible. ✅
Hint: If A ⊂ B and B ∩ C = ∅ then A ∩ C = ∅ (so any 'Some A are C' is impossible).
Common Mistakes: Treating equivalent phrasings as distinct; ensure you map subsets first.
3. Statements: 1️⃣ All pens are instruments. 2️⃣ Some instruments are musical. 3️⃣ All musical things are enjoyable. Conclusions: I. Some pens are enjoyable. II. Some enjoyable things are instruments.
easy
A. Only Conclusion II follows
B. Both I and II follow
C. Only Conclusion I follows
D. Neither I nor II follows

Solution

  1. Step 1: Map the chain

    All Pens ⊂ Instruments; Some Instruments ↔ Musical; All Musical ⊂ Enjoyable.

  2. Step 2: Evaluate Conclusion I

    ‘Some pens are enjoyable’ would require that the particular instruments which are musical include pens. The premises only say some instruments are musical - they do not guarantee that those musical instruments include pens. So Conclusion I is not certain (only possible). ❌

  3. Step 3: Evaluate Conclusion II

    Some Instruments are Musical and Musical ⊂ Enjoyable ⇒ the musical instruments are enjoyable. Therefore there exist Enjoyable things that are Instruments - i.e., Some enjoyable things are instruments. This conclusion follows. ✅

  4. Final Answer:

    Only Conclusion II follows. → Option A

  5. Quick Check:

    All + Some + All produces an existential about the middle that transfers to the superset; the subset (pens) only becomes existential if explicitly overlapping the 'Some' portion. ✅
Hint: A 'Some' in the middle yields a guaranteed 'Some' for related supersets/subsets only when overlap is specified.
Common Mistakes: Assuming the middle 'Some' automatically includes all subsets of the superset.
4. Statements: 1️⃣ No poet is a banker. 2️⃣ Some bankers are writers. Which of the following is impossible?
medium
A. Some poets are writers
B. All writers are poets
C. Some writers are bankers
D. No writer is a banker

Solution

  1. Step 1: Restate premises

    No Poet ⟂ Banker; Some Bankers ↔ Writers (existential overlap between Bankers and Writers).

  2. Step 2: Test each option

    A (Some poets are writers) is possible because poets could be writers and remain non-bankers; B (All writers are poets) is not contradicted by premises (writers could all be poets who are not bankers) - it's extreme but possible; C (Some writers are bankers) follows from premise 2 and is therefore possible/true. D (No writer is a banker) directly contradicts premise 2 (Some bankers are writers implies some writers are bankers), so D is impossible.

  3. Final Answer:

    No writer is a banker. → Option D

  4. Quick Check:

    'Some bankers are writers' rules out any universal claim that 'No writer is a banker'. ✅
Hint: An existential ('Some ...') immediately invalidates the opposite universal ('No ...').
Common Mistakes: Confusing possibility with necessity: extreme universal claims may still be logically possible unless directly contradicted by an existential.
5. Statements: 1️⃣ Some engineers are designers. 2️⃣ All designers are creative. 3️⃣ Some creative people are artists. Conclusions: I. Some engineers are artists. II. Some artists are designers.
medium
A. Both I and II follow
B. Neither I nor II follows
C. Only Conclusion I follows
D. Only Conclusion II follows

Solution

  1. Step 1: Understand chain

    Some Engineers ↔ Designers; All Designers ⊂ Creative; Some Creative ↔ Artists.

  2. Step 2: Evaluate Conclusion I

    ‘Some engineers are artists’ would require the particular designers who are engineers to overlap with the particular creative people who are artists. The premises do not guarantee this overlap - it is only possible, not certain. ❌

  3. Step 3: Evaluate Conclusion II

    ‘Some artists are designers’ would require the particular creative people who are artists to overlap with the particular designers. The premises state Some Creative are Artists and All Designers ⊂ Creative, but that does not force overlap between Artists and Designers. Therefore Conclusion II is not guaranteed either. ❌

  4. Final Answer:

    Neither I nor II follows. → Option B

  5. Quick Check:

    Two separate existential facts about different parts of the same superset do not ensure those particular parts overlap. ✅
Hint: Avoid chaining multiple 'Some' facts into certainty unless an explicit overlap is given.
Common Mistakes: Assuming existence in the superset implies overlap across different existential parts.

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