Questions: Higher-Order Knowledge and Iteration

Question 1

In an epistemic model where the accessibility relation is not transitive, which of the following holds?

• A. Both positive and negative introspection hold, as introspection is independent of accessibility structure
• B. Positive introspection (Kₐp → KₐKₐp) fails — an agent may know p without knowing that she knows p
• C. Negative introspection fails but positive introspection still holds
• D. The agent cannot know anything, because knowledge requires transitivity

Answer: (B)

Positive introspection holds if and only if the accessibility relation is transitive (the S4 condition). Without transitivity, there can be a world w where Kₐp is true (p holds in all worlds accessible from w), yet the agent at w cannot 'see' that she knows — because the worlds accessible from her accessible worlds may not themselves be accessible from w. So Kₐp does not imply KₐKₐp. Option A is the common misconception that introspection is automatic.

Question 2

What distinguishes S5 from S4 as an epistemic logic system?

• A. S5 adds positive introspection (Kₐp → KₐKₐp) to S4's axioms
• B. S5 adds negative introspection (¬Kₐp → Kₐ¬Kₐp), so agents always know what they don't know
• C. S5 replaces transitivity with reflexivity as the core accessibility condition
• D. S5 limits knowledge to finitely many iterations, preventing infinite regress

Answer: (B)

S4 corresponds to reflexive and transitive accessibility (capturing positive introspection: knowing implies knowing-you-know). S5 adds symmetry, which yields negative introspection: if you don't know p, you know that you don't know p. This collapses all iterated knowledge operators — KₐKₐp and Kₐp become equivalent. S5 is the strongest standard epistemic system, but its negative introspection axiom is epistemically demanding and widely disputed.

Question 3

Negative introspection is the principle that if an agent knows p, she knows that she knows p.

• T. True
• F. False

Answer: False

False — that is the definition of *positive* introspection (Kₐp → KₐKₐp). Negative introspection runs in the opposite direction: if an agent does NOT know p, she knows that she does not know p (¬Kₐp → Kₐ¬Kₐp). Positive introspection says knowledge is self-revealing; negative introspection says ignorance is self-revealing. Both are substantive assumptions about self-knowledge, and neither follows from the basic definition of the knowledge operator.

Question 4

In the S5 epistemic system, KₐKₐp and Kₐp are logically equivalent — knowing p and knowing that you know p are the same thing.

• T. True
• F. False

Answer: True

True. S5 includes both positive and negative introspection, and as a consequence all iterated knowledge operators collapse: KₐKₐp ↔ Kₐp, KₐKₐKₐp ↔ Kₐp, and so on. You can 'see all the way down' without information loss. This follows from the fact that in S5, the accessibility relation is an equivalence relation (reflexive, transitive, symmetric), and within an equivalence class, every world agrees on everything an agent knows.

Question 5

Explain why the transitivity of the accessibility relation corresponds to positive introspection. Walk through the possible-worlds reasoning.

Model answer: Suppose Kₐp is true at world w: p is true at every world accessible from w. For KₐKₐp to hold at w, Kₐp must be true at every world w' accessible from w. Kₐp is true at w' if p holds at every world accessible from w'. If accessibility is transitive, then any world w'' accessible from w' is also accessible from w, and we already know p is true there. So yes, Kₐp holds at every w' accessible from w, meaning the agent at w knows she knows p. Without transitivity, w'' might be accessible from w' but not from w, so we can't guarantee p holds at w'' — and positive introspection fails.

Transitivity is exactly the 'chain-following' property: if you can reach w'' by going w → w' → w'', transitivity says you can also reach it directly as w → w''. This is what lets an agent at w 'see' not just what's true at the first step of her accessibility, but also what's true at the second step. Without it, there are blind spots about her own knowledge.