Intransitivity

Understanding Intransitivity

Intransitivity is a property of binary relations. A relation R is intransitive if, for any elements A, B, and C, whenever A R B and B R C hold, then A R C does not hold.

Key Concepts

Unlike transitive relations (e.g., ‘greater than’), intransitive relations break the chain of inference. Consider rock-paper-scissors:

• Rock beats Scissors
• Scissors beats Paper
• But Rock does NOT beat Paper (Paper beats Rock)

This cyclic nature is a hallmark of intransitivity.

Deep Dive: Examples and Logic

Common examples include:

• Preference relations: If you prefer A over B, and B over C, you might still prefer C over A.
• Voting systems: Condorcet cycles demonstrate intransitivity where no candidate wins against all others in pairwise comparisons.
• Games: Many games involve intransitive outcomes.

In logic, intransitivity means you cannot deduce the relationship between the first and third elements based solely on the relationships between the first and second, and the second and third.

Applications of Intransitivity

Understanding intransitivity is crucial in:

• Game theory: Designing fair games and analyzing strategies.
• Social choice theory: Examining fairness and paradoxes in voting.
• Decision making: Recognizing potential pitfalls in sequential choices.

Challenges and Misconceptions

A common misconception is confusing intransitivity with simple asymmetry. While many intransitive relations are asymmetric, asymmetry doesn’t guarantee intransitivity. For example, ‘is not equal to’ is asymmetric but transitive.

FAQs

Is ‘loves’ an intransitive relation?

Not necessarily. If Alice loves Bob, and Bob loves Carol, it’s possible Alice also loves Carol, making it transitive in that instance. However, it’s not inherently transitive or intransitive; context matters.

What’s the opposite of intransitive?

The opposite is transitive. A relation is transitive if A R B and B R C implies A R C.