Questions: Introduction to Modal Logic

Question 1

In a Kripke frame with worlds {w₁, w₂, w₃} where w₁ accesses w₂ and w₁ accesses w₃, proposition p is true at w₂ but false at w₃. What is the truth value of ◇p at w₁?

• A. False, because p is false at w₃ which w₁ accesses
• B. True, because p is true at w₂ which w₁ accesses
• C. True, because □p holds at w₁
• D. Undefined, because p has different truth values at accessible worlds

Answer: (B)

◇p ('possibly p') is true at a world if p is true in at least one accessible world. Since w₁ accesses w₂ and p is true at w₂, ◇p is true at w₁ — regardless of what happens at w₃. Only one witness world is required for possibility. This contrasts with □p ('necessarily p'), which requires p to be true at ALL accessible worlds — which would fail here because p is false at w₃.

Question 2

In modal logic, if □p is true at a world w, then p is expected to be a logical tautology.

• T. True
• F. False

Answer: False

□p means p is true in all worlds accessible from w — not that p is true in all possible worlds of all frames, which is what logical validity (⊨ p) requires. In a frame where only a few worlds are accessible from w, □p can hold even if p is false elsewhere in the same frame or in other frames entirely. Necessity is frame-relative and world-relative; tautology is an absolute semantic property.

Question 3

What distinguishes modal system T from system K, and what feature of the accessibility relation accounts for this difference?

Model answer: System T adds the axiom □φ → φ (whatever is necessary is actually true), which corresponds to making the accessibility relation reflexive — every world accesses itself. System K imposes no conditions on accessibility.

In system K, a world might not access itself, so □φ could be true at w (φ holds in all worlds w sees) while φ is false at w (w doesn't see itself). Adding reflexivity ensures every world is among its own accessible worlds, which forces □φ → φ: if φ holds everywhere accessible, and w is accessible from itself, then φ holds at w. This axiom is intuitively plausible for metaphysical necessity — if something is necessarily true, it should be actually true.