“It is raining but I do not know it” sounds absurd to say, yet it could be true. In a possible-worlds model the lab finds a world where it is true. Prefix the box, so that it says the speaker knows it, and the lab finds no world of the frame where that holds. A sentence can be true and still be one that no one in a world like that could know.
Core ideas
Three ideas separate what is absurd to say from what is false.
The sentence says that p is true and the speaker does not know it. Nothing forbids that: many truths are not known by anyone. There is a world, in the lab a model with two worlds, where the sentence is true. So it is not a contradiction.
The box says that the speaker knows what follows. □(p ∧ ¬□p) says that the speaker knows both that p and that they do not know it. The added claim is stronger, and it is the one that fails.
In a reflexive frame every world sees itself, so whatever is known is true there. If the speaker knew ¬□p it would be true, so they would not know p, while also knowing p. No world of such a frame survives.
The material
The sentence, the model where it is true, the model where it cannot be known, and what that costs.
G. E. Moore noticed that some sentences are odd to assert although nothing about their content is odd. “It is raining, but I do not know that it is” is one. Each half may be true, and the whole may be true, as when it is raining in a room where no one has looked. Yet no one can sincerely say it. The puzzle is why a sentence that could be true cannot be sincerely stated, and set out as this course sets out a paradox it is three premises that cannot all hold.
Write p for “it is raining” and read the box □ as “the speaker knows that”, the reading the lab names and “Leibniz's Law and the Masked Man” used before. Moore's sentence is p ∧ ¬□p. Its stated form is the sentence itself, and what would have to hold of the speaker for the assertion to be sincere, that they know what they assert, is the sentence with the box in front:
The first is true at a world where it rains and the speaker does not know it. The lab models that with two worlds. At the first, w1, it rains. The speaker at w1 sees w1 and w2, and at w2 it does not rain, so the speaker does not know it is raining. The lab evaluates p ∧ ¬□p at w1 and reports True. It also lists the worlds where the sentence holds, and the list is w1, since at w2 it is not raining. So the sentence is satisfiable, and Moore's paradox is not that it is contradictory.
Moore's paradox is that a sentence of the form “p, but I do not know that p” can be true, and yet cannot be known by the person it is about, so that it cannot be sincerely asserted, nor believed by a believer who knows their own mind.
The second sentence is a different matter. Suppose a world where □(p ∧ ¬□p) is true. The box distributes over the conjunction, so at that world the speaker knows p and knows ¬□p. Now use the frame. If every world sees itself, whatever the speaker knows is true there: that is the axiom □p → p of “Frames, Axioms and What Necessity Obeys”, which a reflexive frame validates and the lab's axioms tile lists as T. So ¬□p is true at the world: the speaker does not know p. But the speaker knows p. The two cannot both hold, so there is no such world.
The lab checks this. In the second preset the frame is reflexive: w1 sees w1 and w2, and w2 sees itself. The formula □(p ∧ ¬□p) is evaluated at every world and the lab's list of worlds where it holds is empty. The argument above shows that the same would hold in any reflexive frame, whatever the number of worlds. The lab shows it for this one, and the argument generalises it.
A world that sees no world satisfies every boxed sentence vacuously, because there is nothing for the box to range over. A reflexive frame has no dead ends, which is part of why it gives no world for the boxed Moore sentence. The third preset drops reflexivity, and its list of worlds includes a dead end for exactly this reason.
So Moore's sentence is a truth that no one in a reflexive frame can know. That is exit one, and the premise denied is the second, that whatever can be true can be known. The premise is natural: it treats the limits of knowledge as the limits of fact. The price of denying it is that some truths are unknowable by their own subject, not through ignorance of the world but through the form of the sentence. Nothing is wrong with the logic. The sentence says something about the speaker's knowledge that knowing it would falsify.
The third preset shows what the argument needed. It has three worlds, w1 sees w2 and w2 sees w3, and no world sees itself. At w1 the speaker knows that p and that they do not know it, because at the one world w1 sees, w2, it rains and the speaker there sees a dry world. The lab reports the boxed sentence True at w1. Without reflexivity, what is known need not be true, and this is a model of mistaken belief. Belief does not have to be reflexive, which is why the paradox is put in terms of knowledge here.
A believer who can look into their own beliefs recovers the result without reflexivity. The fourth preset is a frame in which each world sees only worlds that see the same worlds, as an agent does who knows their own mind, and the lab again finds no world that satisfies the boxed sentence. What the lab shows in each case is a property of the frame and not a finding about people. It says what a speaker whose frame has the property cannot coherently hold.
Interactive
Modal formulas at every world, and the axioms the frame validates
Method
Five steps, for Moore's sentence and for any sentence that says something about the speaker's own knowledge.
Give the sentence, then put the box in front of it. For Moore's sentence these are p ∧ ¬□p and □(p ∧ ¬□p).
Build a small model. If one exists the sentence is satisfiable, and the paradox is not a contradiction.
Read the list of worlds where it holds. In a reflexive frame the list is empty.
Ask what the frame property did. Without reflexivity, or the introspective condition of the fourth preset, the known form can hold.
Deny that whatever can be true can be known. The price is that some truths are unknowable by the person they are about, because of what they say.
The model has two worlds. The speaker at w1 sees w1 and w2, and at w2 sees w2. It is raining at w1 only.
At w1 the sentence is true; prefix the box and no world of the frame satisfies it. A reflexive frame cannot have one at all.
Given a sentence about the speaker's own knowledge, you should be able to build a model where it holds, evaluate its boxed form at every world, say what the frame property did, and name the premise the best reply denies and what it costs.
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