Lesson · Identity, Modality and Freedom · Modality

Leibniz's Law and the Masked Man

I know that my father is here and I do not know that the masked man is here, yet the masked man is my father. In a model whose box is read as “it is known that”, the lab shows both sentences true at the actual world and the two boxes with different values. The substitution fails inside the box, and outside it the law holds.

Open the interactive lesson Practice

if a is b, a and b share every property f ↔ m true, but □f ↔ □m false swap freely outside a box, not inside one world: nothing to disagree about

Where substitution holds and where it fails

Core ideas

What this lesson establishes

The lab reads the box epistemically here, as “it is known that”. The computation is the same as in “Necessity, Possibility and Possible Worlds”. The subject matter has entered through the arcs.

Leibniz's law

If a and b are the same thing, then whatever is true of a is true of b. It is a principle about identity and about properties of the thing.

A box looks at other worlds

□f is true at w1 only if f is true at every world compatible with what is known there. Two sentences that agree at w1 can disagree at another such world, and then the boxes differ.

Being known is not a property of the man

“Is known by me to be here” depends on how the man is picked out, as the father or as the masked man. It is a property of the man under a description, and the law concerns the properties of the man himself.

The material

A true equivalence that does not survive a box

The masked man as a model, the substitution that fails, and the one that cannot.

The old argument runs as follows. I know who my father is, so I know that my father is here. I do not know who the masked man is, so I do not know that the masked man is here. But the masked man is my father. So by Leibniz’s law my father has a property that the masked man lacks, being known by me to be here, and they are not the same man. Since they are the same man, something has gone wrong.

The lab makes the failure exact. Let f say that my father is here and m that the masked man is here. At the actual world, w1, both are true: the man in the mask is my father. Read the box as “it is known that” and draw the arcs from w1 to the worlds compatible with what I know. There are two: w1 itself and w2. At w2 my father is here but the man in the mask is not, since for all I know the masked man is a stranger.

world f m w1 T T w2 T F

Now evaluate. □f is true at w1, because f holds at both worlds w1 sees. □m is false at w1, because m fails at w2. And f ↔ m is true at w1, since both are true there. So at one world two sentences are equivalent and their boxed versions differ in value. That is the whole of the masked man: substitution of equivalents inside a box fails, because a box consults worlds at which the equivalence need not hold.

Outside the box nothing fails. From f ↔ m at w1 you may swap f and m in any context that looks only at w1: in a conjunction, a negation, a conditional. The extensional preset shows the opposite end. With a single world the box has only that one world to look at, f and m agree there, and □f ↔ □m is true. Substitution inside the box is harmless when the box has nothing to disagree about.

Definition · Opaque context

A context is opaque when swapping two sentences, or two names for one thing, that agree in value at the actual world can change the value of the whole. A box is opaque whenever the worlds it looks at can disagree.

The same shape appears with necessity. The number of planets is eight, and eight is necessarily even, but it does not follow that the number of planets is necessarily even, since there are worlds where the planets are seven. The model is the same, with the arcs read alethically.

What survives for Leibniz is a point about what the law covers. It speaks of properties of the man, and being known to be here is not one of them, because it changes with the description. The reply is not free: it commits the reader to a distinction between properties of things and properties of things under descriptions, and some philosophers resist that. The lab computes the failure of substitution; it does not decide how to describe it. It also contains no individuals and no identity sign, so it models the sentences and not the man.

Interactive

Possible worlds

Modal formulas at every world, and the axioms the frame validates

Substitute inside the box and outside it

The box is read as it is known that. World w1 sees itself and w2. At w1 both sentences are true, at w2 only f. The first three presets are the three sentences of the argument. Type []f <-> []m yourself on the first model to see the boxed versions disagree.

Model

Worlds2

Accessibility: 1-2 means w1 sees w2

Valuation: p: 1 2; q: 3

Formula

Evaluate at

Read the box as

Value—

Worlds where it holds—

Frame—

Axioms valid on the frame—

Type [] for the box and <> for the diamond, or use the symbols, with ~ & | -> <-> as usual.

Method

Telling a failed substitution from a counterexample to the law

Four moves for any puzzle that seems to refute Leibniz's law.

Find the sentences that are true together

Evaluate both at the actual world. The masked man gives f and m both true.

Find the box in the context that goes wrong

Look for knowing, believing, necessity, or any operator that looks at other worlds. The property that seems to differ is built from it.

Evaluate the boxed sentences separately

Type each into the lab. If the boxed versions differ while the unboxed ones agree, the substitution failed inside an opaque context.

Ask whether the changing property belongs to the thing

If it depends on how the thing is described, it is not a property of the thing, and the law was never applied to it.

Substituting inside a box

Score 0 / 0

Choose the answer you can justify.

The masked man at w1

World w1 sees w1 and w2. At w1, f and m are both true. At w2, f is true and m is false.

box of f at w1: f at w1 true, at w2 true: true box of m at w1: m at w1 true, at w2 false: false f iff m at w1: true iff true: true box of f iff box of m at w1: true iff false: false swap f for m outside any box: still true swap f for m inside the box: the value changes

The two sentences agree where they are evaluated and differ in the worlds the box consults. That is all the failure is, and it needs no property of the man.

Common mistakes

Thinking the masked man refutes Leibniz's lawThe lab shows f ↔ m true at w1, □f true and □m false. The pair that differ are the boxed sentences, which are about the worlds compatible with what I know, and not about the man. The law was never applied to anything it covers, so nothing has refuted it.

Thinking substitution fails in every contextOutside a box it holds. From f ↔ m at w1 you may replace one by the other inside a negation, a conjunction or a conditional and keep the value at w1. The failure belongs to contexts that look at other worlds.

Treating the failure as a quirk of talk about knowledgeThe same arcs read as necessity give the same failure: eight is necessarily even, the number of planets is eight, and the number of planets is not necessarily even. Any operator that quantifies over worlds is opaque in this way. Reading the box as knowledge changed the words in the status line and left the computation alone.

Check your understanding

✓

Finish when you can show with a model that two true sentences differ under a box, and say why that leaves the law untouched.

You should be able to give a model in which two sentences agree at the actual world and their boxed versions do not, say which worlds make the difference, and state what Leibniz’s law covers that the boxed sentences are not. A claim that substitution “fails” without the world at which the sentences come apart has not been shown.

Note: World w2 sees nothing, so any box is vacuously true there. The tiles that matter in this lesson are the values at w1.