1
Where the box sits is part of what is said
□(p ∨ ¬p) puts one box around the whole disjunction. □p ∨ □¬p puts a box on each side. They are different sentences, and the second is a much stronger claim.
Lesson · Identity, Modality and Freedom · Modality
Modal Fallacies and the Sea Battle
“Necessarily, either there will be a sea battle tomorrow or there will not” is true at every world of every model. “Either it is necessary that there will be one, or it is necessary that there will not” is false at a world that sees both kinds of future. The lab evaluates both, and the difference is where the box sits.
□(p ∨ ¬p) true at every world
□p ∨ □¬p false if both p, ¬p seen
the box does not split over ∨
open future: the seen worlds disagree
One box over a disjunction, or one on each side
Core ideas
What this lesson establishes
The argument that the future is fixed is an old one, and one version of it has a precise flaw that a small model shows.
2
A logical truth is true at every seen world
p ∨ ¬p is true at every world of every model, so it is true at every world a world sees, and □(p ∨ ¬p) is true wherever you evaluate it.
3
An open future is a disagreement among the seen worlds
□p ∨ □¬p requires the seen worlds to agree on p, either all true or all false. If one has p and another has ¬p, neither box holds.
The material
The fatalist's step
The argument at its strongest, the model that refutes the step, and what is left of fatalism.
Put the argument as its defender would. Tomorrow either there is a sea battle or there is not, and that is not up to anyone: it is a matter of logic, and logic is necessary. So either it is necessary that there will be a sea battle, or it is necessary that there will not. If the first, nothing anyone does can prevent it; if the second, nothing can produce it. In either case deliberating about it is idle.
Write p for “there will be a sea battle tomorrow”. The first premise is □(p ∨ ¬p). The conclusion the fatalist wants is □p ∨ □¬p. Whether the argument is valid is the question “Validity by Truth Table” teaches, with one change: the counterexample is a world in a model, not a row.
premise: □(p ∨ ¬p)
conclusion: □p ∨ □¬p
Build the model. World w1 is today. It sees w2, a world where the battle happens, and w3, a world where it does not. The letter p is true at w2 and false at w3. At w1 the premise asks whether p ∨ ¬p holds at w2 and at w3. It does at both, so the premise is true. The first disjunct of the conclusion asks whether p holds at both seen worlds, and it fails at w3. The second asks whether ¬p holds at both, and it fails at w2. Both disjuncts are false, so the conclusion is false. The premise holds and the conclusion fails, so the form is invalid.
Definition · Modal scope fallacy
The scope fallacy here is the move from a box over a disjunction to a disjunction of boxes. It treats the box as if it distributed over “or”. It distributes over “and”, and not over “or”.
A look at the third preset shows when the disjunction of boxes is true. If w1 sees only worlds where the battle happens, then □p is true, so □p ∨ □¬p is true. That is a model of a settled future, and the conclusion is true in it for a reason that has nothing to do with the premise: the seen worlds agree.
Two features of the lab need a word. Worlds w2 and w3 see nothing, so any box is vacuously true there, and that is why the worlds tile for the conclusion lists them. The value to read is the one at w1. And the model says nothing about whether the future has a truth value today; it is a point about a scope, not a theory of time.
What the fatalist can say instead
Take the repair at its strongest. The fatalist can drop the disjunction and assert that what is true is already necessary: p → □p. If p is true today and that makes it necessary, then there is no open future. This is a different argument. Its premise is false at a world that has p and sees a world without it, so the lab agrees with the fatalist only where the seen worlds already agree. Accepting the premise is accepting that the future is settled, and then the conclusion is not argued for but assumed.
Interactive
Possible worlds
Modal formulas at every world, and the axioms the frame validates
Method
Finding a scope shift
Four moves for any argument that moves a box.
1
Mark where each box sits
Write the premise and the conclusion with brackets. A box before an opening bracket governs everything inside it. A box before a letter governs the letter only.
2
Build a world that sees one world of each kind
Give the premise's subject matter two seen worlds that disagree. One with p and one with ¬p is the smallest.
3
Evaluate the premise and then the conclusion at that world
The premise is a logical truth and will hold. Ask the conclusion about each seen world separately.
4
Name the shift
If the premise holds and the conclusion fails, the box was moved across the connective. Say which connective, and say that the inference is invalid.
Where the box sits
Score 0 / 0Choose the answer you can justify.
The sea battle at w1
World w1 sees w2, where p is true, and w3, where p is false.
p or not p at w2: true, at w3: true
box of (p or not p) at w1: true
p at w3: false, so box of p at w1: false
not p at w2: false, so box of not p at w1: false
box of p or box of not p at w1: false
the premise holds and the conclusion fails
The disjunction of boxes is false because neither box holds, and neither holds because the seen worlds disagree. The inference from the first line to the last needs the seen worlds to agree, and the premise says nothing of the kind.
Common mistakes
1
Believing that “necessarily, either there will be a battle or not” entails that one of them is necessaryThe first is □(p ∨ ¬p) and the second is □p ∨ □¬p. In the lab's model the first is true at w1 and the second is false, because w2 has p and w3 lacks it. A premise true and a conclusion false at one world is a counterexample, and the form is invalid however natural the English sounds.
2
Thinking an open future requires denying that the battle either will or will not happenThe model keeps □(p ∨ ¬p) true. It is the claim that the outcome is settled that the model withholds. The two claims are different, and the open-future view needs only the second to be false.
3
Concluding that fatalism is refutedOne argument for it has been shown invalid. A fatalist who asserts p → □p has a different argument, and the lab agrees with it exactly where the seen worlds already agree. Whether truth today makes the future necessary is the premise to judge, and no model decides it.
Check your understanding
✓
Finish when you can show, with a model, that a box does not distribute over a disjunction.
Given an argument that passes from a necessary disjunction to a necessary disjunct, you should be able to draw a world that sees two worlds that disagree, evaluate the premise and the conclusion there, and name the connective the box was moved across. An answer that says only “it confuses scope” has not shown the world.
Note: Worlds w2 and w3 see nothing, so a box is vacuously true there, and the verdict to read is the one at w1.