1
Classify a paradox by its exit
Set an argument out as premises and a conclusion, check the steps, and name whether the best reply denies a premise, faults a step or accepts the conclusion.
Course · Philosophy
Paradoxes and Their Exits
A paradox is a valid-looking argument from plausible premises to a conclusion no one can accept. Here each one is computed: a sentence with no satisfying row, a series with an exact limit, a posterior that depends on a likelihood model, a modal sentence true at one world and believed at none. The labs find where the argument turns, and the choice of exit is left to you.
a paradox: valid, plausible, unacceptable
exit: deny a premise, fault a step, accept
p ↔ ¬p is false in every row
1/2 + 1/4 + 1/8 + ... has the limit 1
P(H | E) = P(E | H)·P(H) / P(E)
p ∧ ¬□p can be true, and cannot be known
What this course is about
- Subject
- Philosophy
- Lessons
- 9
- Format
- Interactive
Overview
What you will be able to do
By the end you can take a paradox apart into its premises, test what can be tested by computation, and say which exit a reply takes and what it costs.
2
Show a sentence has no satisfying row
Build the model or the truth table for the barber and the liar, and say why the failure of every row does not by itself say which premise to give up.
3
Compute an infinite sum exactly
Sum Zeno's halves and Achilles' gaps as geometric series, state the limit and the remainder after any number of steps, and say when there is no limit.
4
Separate what a series settles from what it leaves open
For Thomson's lamp, compute the time of every switch and show that the state after them has no limit.
5
Compute the posterior a paradox depends on
For the two envelopes, Sleeping Beauty and Monty Hall, write the likelihood model, compute the posterior and the best act, and say which assumption the disagreement is about.
6
Show a true sentence that cannot be believed
In a possible-worlds model, find the world where Moore's sentence is true and show that its believed form is true at no world of a reflexive frame.
Lessons
9 lessons
The method comes first, with the barber as the plainest case. The paradoxes of the infinite follow, because their arithmetic is exact, then those of probability, where the figures depend on a model, and the last paradox is about belief itself.
The Barber and the Anatomy of a ParadoxA paradox is a valid-looking argument from plausible premises to a conclusion no one can accept, and it has three exits.Not marked
The LiarThe liar sentence yields p ↔ ¬p, which has no satisfying row, and the exit that survives is to forbid the sentence.Not marked
Zeno's DichotomyThe halves of a unit distance form a geometric series whose partial sums are 1 − 1/2ⁿ and whose remainder shrinks below any bound.Not marked
Achilles and the TortoiseThe gaps in the race form a geometric series whose ratio is the speed ratio, and the meeting point is its limit.Not marked
Thomson's Lamp and SupertasksThe switching times of the lamp sum to a minute while its state alternates for ever, so the series fixes the time and not the state.Not marked
The Two EnvelopesThe argument that you should always switch needs the other envelope to be double with probability one half at every amount, and a bounded prior cannot supply that.Not marked
Sleeping BeautyThe halfer and the thirder compute the same Bayes' rule from two different likelihoods for “I am awake”, and the disagreement is about which likelihood is right.Not marked
Monty HallAfter the host opens a door the chance is not one half each, because what the host does depends on where the car is, and a different host gives a different answer.Not marked
Moore's Paradox and What Cannot Be BelievedA sentence of the form p and I do not know that p can be true, and the same sentence preceded by “I know that” is true at no world of a reflexive frame.Not marked
Recommended background
The rest of the Subject, since a paradox is where its tools are needed at once: the truth tables and counterexamples of Arguments and Validity, the credences and updating of Knowledge and Evidence, the expected values of Decision and Rationality, and the possible worlds of Identity, Modality and Freedom.
Practice suggestions
Work forward. “The Liar” uses the form that “The Barber and the Anatomy of a Paradox” ends with, and the lessons on the infinite assume the geometric series of “Zeno's Dichotomy”. The three exits are introduced once and used in every lesson afterwards.
Expect to meet old paradoxes again. The sorites was taught in “Vagueness and the Sorites” and the slippery slope in “Slippery Slopes and Small Differences”; the lottery paradox in “The Lottery Paradox”; Newcomb's problem in “Newcomb's Problem”; Simpson's paradox in “Correlation, Confounding and Simpson's Paradox”; the ravens in “The Raven Paradox”; grue in “Grue and the New Riddle”; Gettier cases in “Gettier Cases and the Fourth Condition”; the ship of Theseus in “The Ship of Theseus”; and the Condorcet paradox in “Majority Rule and the Condorcet Paradox”. Each was worked where its tools were introduced, and the question asked of each there, which premise to doubt and at what cost, is given its general form in the first lesson here and asked of the rest.
Compute before you argue. Every lab here gives a figure that decides something: a row that is absent, a limit, a remainder, a posterior, a world. Read the figure first, then ask which premise it leaves standing.
Hold the exits loosely. Each lesson states the argument at its strongest and each exit at its strongest, with what it costs. Which exit to take is left to you, on purpose, and the labs cannot choose it.
What it does not cover
The surprise examination and the unexpected hanging. They need an epistemic logic with announcements that the Subject does not build, and a treatment that leaves them as stories would misrepresent them.
Curry's paradox and the logics built to avoid it. It needs a consequence relation weaker than the classical one, and every lab here uses the classical table.
The Ross–Littlewood paradox, which needs a limit of sets and not of numbers, and Pascal's mugging, which needs an unbounded utility. Both are named here and not worked.
The full theory of truth. The liar is taught through the table and Tarski's hierarchy; the fixed-point constructions, revision theories and the arguments about revenge are not built.
Continuous probability and infinite outcome spaces. Every posterior is over finitely many hypotheses, and the two envelopes are treated under a bounded prior for that reason.