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intermediate20 min

Syllogisms and Logical Deduction

Determine whether a conclusion validly follows from two categorical statements.

What you'll learn

  • Represent categorical statements (All/Some/No) using set membership
  • Determine whether two 'All' statements validly chain into an 'All' conclusion
  • Recognize when a 'Some' premise cannot support a universal ('All' or 'No') conclusion

Prerequisites

Explanation

A syllogism gives you two statements (premises) and asks whether a conclusion necessarily follows -- not whether it sounds plausible, but whether it's logically guaranteed by the premises alone.

Categorical statements come in a few forms: All A are B (every member of A is also in B), No A are B (A and B share no members), and Some A are B (at least one member is in both). Thinking of A and B as sets makes the rules concrete: "All roses are flowers" means the set of roses is entirely contained within the set of flowers.

The classic valid chain is two "All" statements sharing a middle term: "All roses are flowers" + "All flowers need water" -> "All roses need water" is valid, because the middle term (flowers) links them: roses are a subset of flowers, flowers are a subset of things-that-need-water, so roses are a subset of things-that-need-water too.

Two traps show up constantly. First, the converse is not implied: "All A are B" does not mean "All B are A" -- all roses are flowers, but not all flowers are roses. Second, "Some" premises can never yield a universal conclusion. "Some dolphins are mammals" only tells you about some dolphins, not all of them -- so a conclusion like "No dolphins are fish" (a universal claim) overreaches what the premise actually supports, even if it happens to be true in the real world. The premises given, not outside knowledge, are what determine validity.

The reliable check for a two-"All"-premises chain: do the premises share a middle term, with the first premise's predicate matching the second premise's subject? If yes, the chain is valid. If either premise is "Some" instead of "All," a universal conclusion never validly follows.

Example

Checking whether one group is entirely contained within another.

function allAreSubsetOf(groupA, groupB) {
  return groupA.every((member) => groupB.includes(member));
}
// allAreSubsetOf(['cat', 'dog'], ['cat', 'dog', 'bird']) -> true

Guided exercise

Guided exercise

Write allAreSubsetOf(groupA, groupB) that returns true if every member of groupA also appears in groupB.

Checks: Confirms a genuine subset relationship · Rejects a group with a member missing from the second · plus 1 hidden check

Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.

Loading editor…

Stuck? Get a hint.

Independent exercise

Independent exercise

Write followsAllAAreC(premise1, premise2) where each premise is { type: 'all'|'some', subject, predicate }. Return true only if both premises have type 'all' AND premise1's predicate matches premise2's subject (a valid middle-term chain).

Checks: Confirms a valid two-'All'-premise chain · Rejects premises with mismatched middle terms · plus 1 hidden check

Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.

Loading editor…

Stuck? Get a hint.

Common mistakes

  • Assuming the converse of 'All A are B' (that 'All B are A') is also true.
  • Treating a 'Some' premise as if it supported a universal ('All' or 'No') conclusion.
  • Judging a conclusion by whether it sounds true in real life instead of whether it's guaranteed by the given premises.

Knowledge check

Knowledge check

1. Statement: All roses are flowers. All flowers need water. Conclusion: All roses need water. Is this conclusion valid?
2. Statement: No fish are mammals. Some dolphins are mammals. Conclusion: No dolphins are fish. Is this valid?
3. Why can't a 'Some A are B' premise ever validly support an 'All' or 'No' conclusion?

Takeaway

A syllogism's conclusion is only as strong as its weakest premise -- a single 'Some' premise caps the conclusion at 'Some' too.

Summary

Categorical statements can be reasoned about as set membership. Two 'All' premises sharing a middle term validly chain together, but a 'Some' premise can never support a universal conclusion, and a statement's converse is never automatically implied.

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