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A paradox is an argument
It has premises, steps and a conclusion. A sentence cannot be a paradox by itself, only the argument that leads to it, and that is why a paradox can be worked on: each premise and each step can be examined.
Lesson · Paradoxes and Their Exits · Method
The Barber and the Anatomy of a Paradox
A paradox is an argument that looks valid, starts from premises that each look plausible, and ends where no one can stay. That definition says where the work lies: deny a premise, fault a step, or accept the conclusion. The barber's defining sentence is the cleanest case, and in every model the lab builds for it the sentence fails at the barber himself.
a paradox: valid, plausible, unacceptable
exit 1: deny a premise
exit 2: fault a step that only looks valid
exit 3: accept the conclusion
barber: b shaves x iff x does not
shave x -- fails at x = b
Three parts to a paradox, three exits from it
Core ideas
What this lesson establishes
Three ideas fix what the word means in this course. The barber is the first argument to be tested against them.
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Three features, all at once
The argument looks valid, so that no case makes the premises true and the conclusion false. Each premise looks plausible when taken alone. The conclusion cannot be accepted. Remove any one feature and what is left is an ordinary refutation or an ordinary surprise.
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A contradiction is the usual bad conclusion, not the only one
A contradiction is a sentence false in every row of its truth table. Many paradoxes end in one, which is the clearest way for a conclusion to be unacceptable. Some end only in something absurd, and the later lessons meet those too.
The material
Anatomy of a paradox, and the village barber
The definition first, then the exits it implies, then the barber as the case that tests them.
The word paradox is used for anything surprising. Here it has a narrower meaning, and the narrowness is what makes it useful. A paradox is an argument with three features. It looks valid: from “Validity and Soundness”, no case makes every premise true and the conclusion false. Each premise looks plausible when taken by itself. And the conclusion is one that cannot be accepted. An argument that is plainly invalid has nothing more to say for itself, and one whose conclusion is merely unfamiliar has nothing to explain.
Definition · Paradox
A paradox is an argument that appears valid, from premises that each appear plausible, to a conclusion that cannot be accepted. It is the combination that creates the problem: the argument cannot be waved away, and its conclusion cannot be kept.
Because the argument appears valid and the conclusion has to go, there are three things to do. The first exit is to deny a premise: one of them was not as plausible as it looked, and the reader must say which, and why it deceived. The second exit is to fault a step: the argument looked valid and is not, because a step trades on an ambiguity or on a rule that fails in this setting. The third exit is to accept the conclusion, on the ground that it was unacceptable only because it was unfamiliar. Each lesson of this course ends by asking which exit the best answer takes and what it costs.
The oldest tidy example is a village with a barber, and a rule about him: the barber shaves all and only those who do not shave themselves. Write b for the barber. The rule is one sentence:
∀x (Shaves(b, x) ↔ ¬Shaves(x, x))
Read it as saying that for every individual x, the barber shaves x exactly when x does not shave x. It is a sentence of the kind that “Quantifiers and Their Order” taught you to read and “Compositional Truth Conditions” taught you to evaluate in a finite model, and the lab below does that.
Now apply the rule to the barber himself, which is allowed because the rule is about every individual. Put x = b. The sentence says that b shaves b exactly when b does not shave b. Write s for “b shaves b”. The instance has the form s ↔ ¬s, and that form is false whichever truth value s has. If s is true, the left side is true and the right side is false. If s is false, the left side is false and the right side is true. Either way the biconditional fails.
Example · The whole argument
Premise one: there is a barber b of whom the village rule is true. Premise two: what a rule says of every individual it says of b. Conclusion: s ↔ ¬s. The steps are valid, the second premise is the logic of “every”, and the conclusion is a contradiction. So the first premise goes.
That is exit one, and it is the cheapest exit this course will offer. Nothing about shaving or logic is overturned. The village rule can be written down, and it can be understood, and there is no barber who meets it. The rule is like “the largest whole number”: a description that no one satisfies, however clearly it reads.
The lab builds a model: three individuals a, b and c, with the name barber given to b, and a list of who shaves whom. It evaluates the sentence and names the first individual in the domain at which it fails. In the first preset a shaves himself, b shaves c, and c shaves no one, which satisfies the rule for a and for c and leaves b as the failure. Edit the list and try to repair it. Adding the pair for b shaving b never helps, because the instance at b has the form s ↔ ¬s either way. Removing the pair for a shaving a makes the lab name a first, since the tile reports the earliest failure; b is still among the failures.
The lab's limit should be stated plainly. It evaluates the sentence in the models you build, and it says False in each of them. It does not search all models. That the sentence is false in every model, of any size, is the paragraph above: the instance at b has the form s ↔ ¬s and no truth value for s makes that true. The lab shows the instance; the argument generalises it.
A change of wording changes the verdict, and the third preset shows it. If the rule only ranges over the villagers, and the barber is not a villager, the sentence becomes ∀x (Villager(x) → (Shaves(b, x) ↔ ¬Shaves(x, x))). Nothing now demands anything of b, there is no instance at b, and the lab reports True. So the barber can exist if he lives outside the rule. The paradoxes that follow are harder because the premise that has to go is not so easily spared: it will be a sentence that we can plainly write down and can plainly understand.
Interactive
Truth in a model
A first-order sentence evaluated over a domain you can edit
Method
Taking a paradox apart
Four moves, in this order, for any argument that seems to prove what cannot be true.
1
Write it in standard form
List the premises and the conclusion, as in “Premises, Conclusions and Standard Form”. A paradox hides in prose, and the premises are rarely all stated.
2
Check the steps
Ask whether the conclusion follows from the premises, by a table or by an instance. If the argument is valid, the work is in the premises. If a step fails, that is exit two and the paradox was only an appearance.
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Test each premise alone
Ask of each premise whether it is as plausible as it looked. Mark the one that would be cheapest to give up, and write what it costs.
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Name the exit and its price
Say which of the three exits the best reply takes. For the barber, exit one, and the price is nothing: no one satisfied the rule.
Parts and exits
Score 0 / 0Choose the answer you can justify.
The village rule, instantiated at the barber
The rule is true of the barber, and the barber is one of the individuals it is about. Set the individual to the barber and read.
rule: for every x, b shaves x iff x does not shave x
take x to be b itself
instance: b shaves b iff b does not shave b
let s stand for: b shaves b
the instance has the form: s iff not s
s true: true iff false, so the instance is false
s false: false iff true, so the instance is false
no truth value for s makes the rule true of b
so no b meets the rule: exit one, deny the premise
The argument is valid at every step, and its second premise is only the logic of “every”. The only premise left to deny is the one that said there was such a barber, and denying it costs nothing.
Common mistakes
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Treating a paradox as a contradictionA contradiction is a sentence false in every row. A paradox is an argument that ends in one, or in something equally unacceptable, from premises that looked fine. The barber sentence is not itself the paradox. The argument from “there is a barber who meets the rule” to s ↔ ¬s is, and it is worth studying because it shows that one of the premises has to go.
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Hunting for a flaw in the logic of the barberThe steps are valid. Instantiating a rule that is about everyone at the barber is the logic of “every”, and s ↔ ¬s is false in both rows. People who look for the trick in the word “shaves” are looking at the wrong place. The exit is in the premise that such a barber exists.
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Reading the lab's False as a proof for all modelsThe lab says False in the models you have built, and a model has the individuals you gave it. The general claim comes from the argument about the instance at b, not from running the lab more often. Say which of the two you are relying on.
Check your understanding
✓
Finish when you can classify an argument by its exit and refute the barber in a model.
Given a short argument, you should be able to set it out as premises and a conclusion, say whether each of the three features of a paradox is present, name the exit the best reply takes and what it costs. For the barber you should be able to build a model, read the individual at which the sentence fails, and say why no model removes the failure.
Note: The barber is a popular form of Russell's paradox, which is about the collection of all collections that are not members of themselves; the same instance with the form s ↔ ¬s is the heart of it. The next lesson takes the same form, p ↔ ¬p, and finds that its exits are much dearer, because the sentence in question is not a barber that can be quietly left out of the village.