Pareto Principles in Infinite Ethics by Amanda Askell A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy New York University May 2018 Professor Cian Dorr Abstract It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms { the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the `at least as good as' relation holds between two worlds then it holds between qualitative duplicates of this world pair { then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. iii Contents Abstract iii List of Figuresv Introduction1 Chapter 1: The Foundations of Infinite Ethics3 1.1: The Possibility of an Infinite World ....................... 4 1.2: Basic Locations of Value ............................. 7 1.3: Sensitivity, Equity, and completeness....................... 14 1.4: Pareto Principles in Infinite Ethics........................ 34 1.5: Infinite Aggregation Principles.......................... 45 Chapter 2: Pareto, the < Relation, and the Permutation Principle 70 2.1: Pareto and Expansionism............................. 71 2.2: The < Relation .................................. 87 2.3: The Permutation Principle ............................ 99 Chapter 3: The Incomparability Results 113 3.1: Incomparability in Disjoint Population Pairs .................. 114 3.2: Incomparability in Identical Population Pairs.................. 121 3.3: The General Four World Result ......................... 130 3.4: The Cyclic Result................................. 135 Chapter 4: Extending the Incomparability Results 147 4.1: Weak Catching-Up................................. 148 4.2: Addition Principles and the Weak People Criterion . 156 4.3: Against Accumulation Principles......................... 172 Chapter 5: The Implications for Ethics 178 5.1: The Incomparability Result as an Impossibility Result . 179 5.2: Transitivity, the Permutation Principle, and Qualitativeness . 181 5.3: Rejecting Pareto.................................. 196 5.4: Embracing Incomparability............................ 216 Conclusion 256 Bibliography 263 iv List of Figures 1 Act outcomes for generations .......................... 12 2 Act outcomes for agents within generations................... 13 3 An infinite temporal permutation........................ 23 4 An infinite permutation of agent utilities.................... 24 5 Difference at even-indexed and odd-indexed utilities.............. 26 6 A partial order of worlds............................. 29 7 A finite utility permutation with disjoint populations............. 41 8 A world worse than w10 by Pareto........................ 41 9 A finite utility permutation of w2 ........................ 41 10 Comparing worlds w1 and w20 .......................... 42 11 World w2 is better at a single location ..................... 47 12 Applying Ordered Catching-Up to w1 and w2 . 48 13 Lower limit inferior with infinitely many agents better off........... 50 14 A single location makes a difference to Ordered Catching-Up......... 50 15 A world pair that Catching-Up and Overtaking do not rank ......... 52 16 Natural density of two utility streams...................... 56 17 Utility sequences with distinct medial limits .................. 58 18 The surreal tree.................................. 63 19 Clement and Stormy ............................... 72 20 Mansion and Shack................................ 73 21 Optimistic Cubeland and Pessimistic Cubeland ................ 74 22 Segments of Optimistic Cubeland........................ 75 23 An allowable expansion.............................. 76 24 Allowable expansions of Clement and Stormy.................. 77 25 Utility difference of Cubeland regions...................... 78 26 Squares of suffering across time ......................... 79 27 Utilities in the regions of the sphere of suffering................ 80 28 Utilities in the regions of the sphere of happiness ............... 80 29 Utility differences between sphere of suffering and sphere of happiness . 81 30 Utility differences between the spheres under a different allowable expansion 81 31 Balmy and Blustery................................ 83 32 Agent happiness in Balmy and Blustery..................... 84 33 Utility streams of Dan and Daniela....................... 85 34 Balmy and Duplicate Balmy........................... 90 35 Sets of agents in Clement and Stormy...................... 92 36 Sets of agents in Duplicate Clement and Duplicate Stormy.......... 93 37 Happiness levels of Clement, Stormy, and duplicates.............. 95 38 Transitivity violation showing Clement <6 Stormy ............... 96 39 Alternative groupings of the agents of Clement and Stormy.......... 96 40 New Duplicate Clement and New Duplicate Stormy.............. 97 v 41 Happiness levels of Clement, Stormy, and new duplicates........... 98 42 Transitivity violation showing Stormy <6 Clement ............... 98 43 Permutation of worlds each containing two agents . 102 44 Switching the utilities of two agents....................... 104 45 Permuting the agents whose utilities have been switched . 105 46 An identical population world pair with variable step permuted utilities . 107 47 A disjoint population world pair with variable step permuted utilities . 108 48 Disjoint worlds with utilities outside the [0,1] interval . 114 49 A strict upgrade from w1 to w2 . 115 50 Transitivity violation in disjoint world pairs with a strict upgrade . 116 51 A world pair with distinct `inner' utility levels . 119 52 Permutations showing that world pairs with distinct inner utility levels are incomparable ................................... 120 53 An identical population world pair with distinct inner utility levels . 121 54 Permutations showing that identical population pairs with distinct inner util- ity levels are incomparable............................ 122 55 Identical population world pair with identical utilities . 123 56 Permuting the populations of an identical population, identical utility world pair ........................................ 124 57 Transitivity violation in identical population world pair with a bidirectional upgrade ...................................... 126 58 A further example of an identical population world pair . 127 59 A permutation showing that w1 <6 w2 . 128 60 A world pair with overlapping populations................... 130 61 Permutations showing that the overlapping population world pair is incom- parable....................................... 131 62 Transitivity violation in a world pair with a bidirectional upgrade . 133 63 Overlapping world pair with distinct utilities at shared population . 135 64 Permutation showing that w2 <6 w2 in pair with distinct utilities at shared population..................................... 136 65 Permutations f(hw1; w2i) and f(f(hw1; w2i)) .................. 139 66 Agent utilities in f(hw1; w2i) and f(f(hw1; w2i)) . 139 67 Transitivity violation in a pair with distinct utilities at shared population . 140 68 Cyclic transitivity violation ........................... 142 69 World pair requiring a 3-cycle permutation................... 143 70 A 3-cycle upgrade permutation ......................... 144 71 Agent utilities under the 3-cycle upgrade permutation . 144 72 A 3-cycle downgrade permutation........................ 145 73 Agent utilities under the 3-cycle downgrade permutation . 145 74 Identical population world pair with distinct upper utility levels . 148 75 A pair with finitely many agents with utilities outside the accumulation interval151 76 A permutation showing w1 <6 w2 by Weak Catching-Up . 152 vi 77 A permutation showing w2 <6 w1 by Weak Catching-Up . 153 78 Examples of Benevolent Addition and Malevolent Addition . 158 79 Worlds accumulation intervals [0,3] and [3]................... 159 80 Adding infinitely many utility 0 lives ...................... 159 81 Worlds with accumulation intervals [-1, 2] and [-3, 4] . 166 82 The neutral expansion of this world pair .................... 167 83 A subpopulation transformation of the neutral expansion pair . 167 84 A neutral contraction of the subpopulation transformation . 167 85 A counterexample to unrestricted transfer principles . 168 86 Two problematic cases for the Accumulation principle . 175 87 A problematic case for the weakened Accumulation principle . 176 88 Clement, Stormy, and duplicates again ..................... 183 89 A world pair strictly ranked by Expansionism . 210 90 Curing a