1 Introduction 1

1.1 What Is a Paradox? 1

Paradoxes are robust, widespread intellectual illusions in which seemingly compelling reasoning generates an absurd or contradictory conclusion.

1.2 What Is a Solution? 3

A solution should dispel the illusion, so that the paradoxical reasoning no longer seems compelling.

1.3 How to Seek Solutions

We should not expect a common approach to apply to all paradoxes – but self-evident principles such as those of classical logic must always apply.

1.4 Why Paradoxes? 4

Paradoxes are charming, fun, and may reveal deep confusions about important philosophical matters.

1.5 Paradoxes Not Covered 5

I address only philosophical paradoxes that do not depend on controversial views and that I have not previously addressed.

2 The Liar 7

2.1 The Paradox 7

The liar sentence, “This sentence is false”, is apparently both true and false.

2.2 A Third Truth-Value 7

Some say the liar sentence is “indeterminate”. But what about the sentence, “This sentence is false or indeterminate”?

2.3 True Contradictions 8

The view that there are true contradictions is confused.

2.4 Meaninglessness 10

Perhaps the liar sentence is meaningless for one of the following reasons.

2.4.1 Self-Reference 10

Due to self-reference? But there are benign cases of self-reference.

2.4.2 False Presupposition 11

Due to containing a false presupposition? But we can easily remove the putative presupposition.

2.4.3 Lack of Communicative Use 12

Because it cannot be sincerely asserted? But other paradoxical sentences can be sincerely asserted. Because it cannot be used to convey information? But very similar sentences can be so used.

2.5 Putting the Blame on Truth 12

Some say that there is something wrong with the general concept of truth. But this approach is self-undermining and rules out too many innocent sentences.

2.6 A Solution 14

2.6.1 An Inconsistent Language 14

Our language contains inconsistent rules for how to interpret certain sentences, which results in sentences with no propositional content.

2.6.2 Meaning Deficiency 17

The liar sentence is not meaningless; it merely has a defective meaning that fails to pick out a proposition.

2.6.3 The Truth-Teller 17

“This sentence is true” also fails to express a proposition.

2.6.4 “The Liar Sentence Is Not True” Is True 17

It is permissible for a sentence other than L itself to say that L is not true.

2.6.5 This Sentence Is False or Meaning-Deficient 18

“This sentence is false or meaning-deficient” also fails to express a proposition.

2.6.6 Liar Cycles 19

In cases where a group of sentences generate a liar-like paradox, all the members of the group fail to express propositions.

2.6.7 Prohibiting Liars 20

There is no need to devise new syntactic rules for identifying liar-like sentences or ruling them “ungrammatical”.

2.7 Curry’s Paradox 21

Given a sentence, “This sentence is false, or p”, we can seemingly prove that p must be true. Again, the sentence fails to express a proposition.

2.8 The Paradox of Non-Self-Applicability 21

The property of “not applying to oneself” seemingly must apply to itself if and only if it does not apply to itself. Solution: there is no such property.

2.9 Russell’s Paradox 22

The set of all sets that don’t contain themselves must contain itself if and only if it doesn’t contain itself. Solution: there is no such set.

3 The Sorites 24

3.1 The Paradox 24

Removing a single grain from a heap of sand does not convert the heap to a non-heap. This principle entails that if a million grains of sand make a heap, then one grain of sand makes a heap.

3.2 Deviant Logic 25

Some respond with theories of indeterminacy or degrees of truth. These views have trouble explaining second-order vagueness. Degrees of truth introduce more precision than is plausibly present. Also, classical logic is self-evident.

3.3 Supervaluationism 27

Some say that a sentence is true provided that it would be true on any acceptable way of making the vague terms precise. This view has trouble with second-order vagueness. It also violates classical logic, conflicts with the T-schema, and implies that a statement of the theorist’s own view is false.

3.4 Epistemicism 29

Some say vague terms have precise boundaries that we merely fail to know. This is implausible since there is nothing that could make a particular boundary the correct one.

3.5 A Moderate Nihilist Solution 32

3.5.1 Fit Determines Content 32

Mental states can be satisfied to varying degrees by different possible states of the world. The idea of the “propositional content” of a mental state is only a rough description of a mental state’s meaning, as if the state were always fully satisfied or fully unsatisfied.

3.5.2 When Thoughts Are Vague 34

Thoughts are vague when they have intermediate degrees of satisfaction in some possible situations. There are degrees of vagueness.

3.5.3 Uncertainty About Vagueness 35

We can be introspectively uncertain whether a thought is vague.

3.5.4 Vague Thoughts Make for Vague Language, and Vice Versa 36

The vagueness of thought makes language vague, and vice versa.

3.5.5 An Argument that Vague Statements Do Not Express Propositions 37

Vague sentences and thoughts do not express precise propositions, and there are no vague propositions, so vague sentences and thoughts don’t express propositions at all.

3.5.6 Arguments by Analogy 39

In other cases of semantic indecision, we accept that sentences fail to express propositions.

3.5.7 Logic Is Classical 40

This view of vagueness preserves the law of excluded middle and bivalence, properly understood.

3.5.8 How to Almost Say Something 42

Vague sentences often come close to expressing propositions.

3.5.9 Almost Is Good Enough 43

It is common and appropriate to use language approximately.

3.5.10 Applying Logic to Vague Sentences 44

We can apply our logical faculties to sentences even when they fail to express propositions.

3.5.11 Interpreting “Truth”: Strict vs. Loose Truth 45

It is unsettled whether “is true”, as applied to sentences, means “expresses a true proposition” or instead means “expresses a thought that the world satisfies to a high degree”.

3.5.12 Why Is There Second-Order Vagueness? 47

First- and second-order vagueness both arise from the same features of mental states described above.

3.6 Conclusion 48

The sorites argument fails since none of its sentences express propositions. The premises almost express truths and the inference form is valid, but this does not guarantee a true or nearly true conclusion.

4 The Self-Torturer 50

4.1 The Paradox 50

The self-torturer repeatedly increases his torture level by undetectable increments, each time receiving a large financial reward. Seemingly rational individual choices lead to an intolerable end result.

4.2 Quinn’s Solution 51

Quinn holds that it is not always rational to choose the best option available at the time, and that rational choice is not always forward-looking.

4.3 An Orthodox Solution 52

4.3.1 In Defense of Undetectable Changes 52

The case actually shows that there can be unnoticeable changes in subjective experience.

4.3.2 Indeterminacy 53

It cannot be indeterminate how bad a pain is.

4.3.3 In Defense of an Optimal Setting 55

Since pain has constant marginal disutility, while money has diminishing marginal utility, there is an optimal point for the self-torturer to stop.

4.3.4 Detectable and Undetectable Values 57

It is not so strange that an undetectable bad can outweigh a detectable good. Undetectable quantities can often be larger than detectable ones.

4.3.5 Advantages of This Solution 58

My solution to the problem preserves classical logic and decision theory, without positing anything particularly strange.

5 Newcomb’s Problem 59

5.1 The Paradox 59

You are asked to choose between taking box A and taking both A and B, where B contains $1000, and A contains either $1 million (if a reliable predictor thought you would take only A) or $0 (if the predictor thought you would take both). Dominance reasoning supports taking both, but expected utility maximization seemingly supports taking only A.

5.2 Objections to the Scenario 60

The scenario can be made more realistic without eliminating the paradox.

5.3 The Right Expected Utility Principle 61

5.3.1 The Right Way to Make Good Things More Likely 62

Rational agents seek to “make more likely” the achievement of their goals in a causal sense, not an evidential sense.

5.3.2 Two-boxing Maximizes Expected Utility: Doing the Math 63

When we incorporate the above insight into the notion of expected utility, Dominance and Expected Utility principles agree.

5.3.3 Why This Is the Best Solution 65

The causal decision theorist’s solution preserves central intuitions about rational choice.

5.4 The Case of Perfect Reliability 66

What if the predictor is 100% reliable? This requires either backward causation (making one-boxing rational) or determinism (ruling out free choice).

5.5 Rationality and Long-Run Benefit 67

One-boxers say that rationality must be tied to how much one benefits in the long run. They then appeal to one of the following claims.

5.5.1 One-boxers as a Group Do Better 67

But the wellbeing of the group one belongs to is irrelevant in rational choice theory.

5.5.2 One-boxers Tend to Do Better in any Given Case 68

But this rests on a mistaken interpretation of the probabilities used in rational choice theory.

5.5.3 One-boxers Do Better in Repeated Games 69

But this changes the scenario in a way that may also change the causal decision theorist’s answer.

5.5.4 Being a One-Boxer Is Predictably Advantageous 69

But this only shows that there are scenarios that systematically reward irrationality. This applies to any paradigmatic form of irrationality.

5.6 Uncertainty About Decision Theory 71

In decision-making, it can be rational to give some weight to each of two conflicting theories of rational choice.

6 The Surprise Quiz 74

6.1 The Paradox 74

The teacher announces a surprise quiz next week. It can seemingly be shown that there is no day on which the quiz can occur.

6.2 Rejecting the Assumptions 74

The scenario uses idealized assumptions about the students’ reasoning. But these assumptions can be relaxed without detriment to the paradox.

6.3 What Is Surprise? 75

We may assume that there is a threshold level of antecedent credence in an event that renders the event non-surprising.

6.4 Quiz Comes if and only if Surprising 77

In one version of the story, we assume that the professor has no interest in giving a quiz unless it will be a surprise.

6.4.1 Self-Undermining Beliefs with a Vague Surprise Threshold 77

If the quiz has not come by Thursday, the students should adopt a credence that would make a Friday quiz a borderline case of a surprise.

6.4.2 Self-Undermining Beliefs with a Precise Threshold 78

Or they should adopt a credence such that they will merely not know whether the Friday quiz would count as a surprise.

6.4.3 The Rest of the Week 79

Thursday and earlier quizzes will then count as surprising.

6.5 Quiz Comes, with or without Surprise 79

In another version of the story, we assume the professor will give a quiz whether or not it will surprise.

6.5.1 No Friday Surprise 80

In this version, a Friday quiz would not surprise.

6.5.2 Borderline Thursday Surprise 80

A quiz on Thursday would be a borderline case of a surprise, or it would be unknown whether it counted as a surprise.

6.5.3 The Rest of the Week 80

A quiz on any earlier day would be a surprise.

6.6 Surprising as Not-Most-Expected 81

If a quiz counts as “surprising” only when it was not antecedently considered most likely to occur on that day, then a surprise quiz cannot be given.

7 The Two Envelopes 82

7.1 The Paradox 82

There are two indistinguishable envelopes containing money, one with twice as much as the other. It can be argued that each envelope has a higher expected value than the other.

7.2 The Use of Probability in the Paradox 83

7.2.1 An Objection 83

Some think the paradoxical reasoning misuses the concept of probability.

7.2.2 Three Interpretations of Probability 83

Probability can be interpreted epistemically, subjectively, or physically.

7.2.3 Rational Choice Uses Epistemic Probabilities 84

Epistemic probability is the correct interpretation for rational choice theory. The paradox thus does not misuse the concept of probability.

7.2.4 Probabilities in Causal Decision Theory 85

This does not conflict with our earlier defense of causal decision theory.

7.3 The Use of Variables in the Paradox 85

The paradoxical reasoning confuses variables with constants.

7.4 The Correct Analysis 86

A correct analysis would assign a coherent probability distribution to each possible way of distributing money across the two envelopes. This leads to both envelopes having the same expected value.

8 The Principle of Indifference 89

8.1 The Principle of Indifference 89

The PI holds that, given no reason for preferring any of a set of alternatives over any other, all are equally probable.

8.2 The Paradoxes of the Principle of Indifference 89

There are cases in which the PI can seemingly be used to justify incompatible probability assignments.

8.2.1 The Colored Book 89

Given that a book is red, green, or blue, what is the probability that it is red?

8.2.2 France and England 90

Given that England is a proper part of the U.K., what is the probability that England is smaller than France? That the U.K. is smaller than France?

8.2.3 The Car Ride 90

Given a 100-mile car trip with duration between 1 and 2 hours, what is the probability of a duration between 1 and 1.5 hours? What about a velocity between 66.7 and 100 mph?

8.2.4 The Cube Factory 90

Given that a cube is between 0 and 2 inches on a side, what is the probability of a side between 0 and 1 inch? What about a volume between 0 and 1 cubic inch?

8.2.5 The Circle and Chord 91

Given a circle with an equilateral triangle inscribed inside, what is the probability of a random chord being longer than a side of the triangle?

8.3 Wherefore Indifference? 92

8.3.1 Theories Rejecting the PI 92

Empiricists say we can have no probabilities prior to empirical evidence. Subjectivists say any coherent initial probabilities are permissible. These views reject the PI.

8.3.2 The PI Is Intuitive 93

The PI is extremely intuitive in many cases.

8.3.3 The PI Is an Analytic Truth 93

The PI seems to follow from the meaning of epistemic probability. Without a priori probabilities, there are no epistemic probabilities at all.

8.3.4 The PI Underlies the Least Controversial Probability Assessments 94

Probability assessments based on randomization or statistical evidence also depend on the PI.

8.4 Interpreting the Principle of Indifference: The Explanatory Priority Proviso 96

The PI should be applied to the most explanatorily basic set of possibilities.

8.5 Solutions 97

8.5.1 The Colored Book 97

A uniform probability density should be assigned over the color solid.

8.5.2 France and England 98

Equal probabilities should be assigned to each possible complete set of “larger than” relations involving England, France, and the U.K.

8.5.3 The Car Ride 98

Velocity is privileged over time since velocity causally determines duration, given a fixed distance.

8.5.4 The Cube Factory 99

Quantity of material is privileged over width or volume since quantity of material causally determines size.

8.5.5 The Circle and the Chord 99

The PI should be applied by averaging over all known methods of random selection of a chord.

8.6 A Philosophical Application: The Problem of Induction 101

8.6.1 The Traditional Problem 101

Skeptics say that there is no reason to believe the course of nature is uniform, and hence no justification for relying on induction.

8.6.2 A Probabilistic Formulation of the Problem 102

Skepticism can be defended by applying the PI to all possible sequences of observations.

8.6.3 A Solution 103

We should instead apply the PI to all possible values of the objective chance of a given event.

8.6.4 The Mathematics of the Inductivist Solution 104

This can be used to derive Laplace’s Rule of Succession.

8.7 Another Application: The Mystery of Entropy 105

8.7.1 Why Entropy Increases 105

There is a probabilistic argument for why, starting from a low-entropy state, the entropy of a physical system should spontaneously increase.

8.7.2 The Reverse Entropy Law 106

A time-reversed version of that argument can be used to conclude that any given low-entropy state was probably preceded by a higher-entropy state.

8.7.3 Reverse Entropy Is Crazy 107

This has all sorts of ridiculous implications.

8.7.4 The Reverse Argument Misuses the Principle of Indifference 109

The argument for Reverse Entropy seems to ignore that the past is explanatorily prior to the present.

8.7.5 The Isolated Box 111

The diagnosis of the preceding subsection may prove too much. It is only plausible in a certain range of cases.

9 The Ravens 113

9.1 A Paradox of Confirmation 113

It seems that, in general, the observation of an A that is B supports “All A’s are B.” Therefore, observation of a purple shoe supports “All non-black things are non-ravens.” This is logically equivalent to “All ravens are black.” So purple shoes provide evidence that all ravens are black.

9.2 Solution 113

Whether an observation of a purple shoe supports “All ravens are black” or not depends upon how the observation was gathered – e.g., whether it was gathered by selecting randomly from the class of non-ravens, or selecting randomly from the class of non-black things.

10 The Shooting Room 116

10.1 The Paradox 116

The Shooting Room is set up such that (i) it is guaranteed that at least 90% of people who ever enter it are shot, but (ii) for any given person, whether they are shot or not depends on the flip of a fair coin. Q: Given that V is called into the room, what is the probability that V is shot?

10.2 A Finitist Solution 117

The paradox depends on metaphysically impossible assumptions about an infinite population of potential victims, and infinite speed or elapsed time. Given any metaphysically possible (finitist) assumptions, the correct probability comes to 50%.

11 Self-Locating Beliefs 122

11.1 The Sleeping Beauty Paradox 122

Beauty is put to sleep and woken up either once or twice, depending on the flip of a coin; after each waking, she will fall asleep and forget having woken. Upon waking, what should be her credence that the coin came up heads? Some say 1/2; others say 1/3.

11.2 The Fine Tuning Argument 123

The apparent fine tuning of the universe’s physical parameters, which is required for life to exist, might be evidence that there are many universes.

11.3 The Doomsday Argument 124

The number of people who have existed before you might constitute evidence that not many more will exist after you.

11.4 The Multiverse: Pro and Con 126

11.4.1 The “This Universe” Objection 126

Some object that the fine tuning evidence does not support multiple universes, because the existence of other universes would not explain anything about this universe.

11.4.2 In Defense of the Multiverse 128

The “this universe” objection seems parallel to some incorrect objections in other cases.

11.4.3 Four Cases Resolved 129

Evidence supports a theory (for you) when your having that qualitative evidence would be more likely if the theory were true than if it were false. This view gives the right verdicts in four cases of interest.

11.4.4 Personal Identity and the Multiverse Theory 131

Conclusion: the fine tuning evidence supports multiple universes, if and only if it would be possible for you to exist in another universe.

11.5 Against Doomsday 132

The Doomsday argument fails since, given the impossibility of backward causation, no hypothesis about humanity’s future affects the probability of your now having the evidence that you have.

11.6 Sleeping Beauty: For a Third 134

Upon waking, Beauty should update on the evidence, “I am awake now”. This results in a credence of 1/3 that the coin came up heads.

12 Concluding Remarks 137

12.1 Seven Varieties of Error 137

The preceding paradoxes exhibit several kinds of problem that also beset human thinking in more ordinary cases. These include: hidden assumptions, neglect of the small, confusion, binary thinking, oversimplification, inappropriate idealization, and inference from partial data.

12.2 Against Radical Revision 140

We should not give up extremely obvious principles, such as those of classical logic, to avoid paradox. We should prefer to qualify principles rather than rejecting them outright. Our mistakes are likely to be subtle, not blatant.

12.3 Reality Is Intelligible, with Difficulty 141

The world is not inconsistent or incomprehensible. Human reason is highly fallible but correctable with effort.