{
  "one_box_argument": {
    "premises": [
      "P1: the prediction matches your choice with chance p, whichever box you take",
      "P2: one box is expected to pay 1,000,000 p dollars; two boxes 1,000 + 1,000,000 (1 - p)",
      "P3: p is above 1001/2000 (50.05%)",
      "Rule: take the act with the higher expected value"
    ],
    "conclusion": "C1: take one box",
    "valid": true,
    "grid_points_checked": 10004,
    "counterexamples": [],
    "without_P3": {
      "valid": false,
      "example_counterexample_p": "1/2",
      "largest_p_where_one_box_is_not_higher": "1001/2000"
    }
  },
  "two_box_argument": {
    "premises": [
      "P1: the boxes were filled before you choose, and your choice does not change what is in them",
      "P2: whatever the opaque box holds, two boxes pay 1,000 dollars more",
      "Rule: if one act pays more in every state and the states do not depend on the act, take it"
    ],
    "conclusion": "C2: take both boxes",
    "valid": true,
    "two_boxes_ahead_by_dollars": {
      "opaque box full": 1000,
      "opaque box empty": 1000
    },
    "with_any_fixed_chance_of_full_box": "two boxes ahead by exactly 1,000 dollars (checked for q = 0, 0.01, ..., 1)",
    "without_P1": {
      "valid": false,
      "example_p": "9/10",
      "expected_one_box_dollars": 900000,
      "expected_two_boxes_dollars": 101000,
      "note": "two boxes still pay 1,000 more in each state, but the state now goes with the choice"
    }
  },
  "clash": {
    "same_payoff_table": true,
    "p_range_where_the_rules_disagree": "1001/2000 < p <= 1",
    "grid_points_in_clash": 4996
  },
  "note": "Validity only: if the premises were true, the conclusion would follow. Which premises are true is the dispute."
}