IndisputableMonolith.Philosophy.ModalOntologyStructure

   1import Mathlib
   2import IndisputableMonolith.Foundation.LawOfExistence
   3import IndisputableMonolith.Foundation.OntologyPredicates
   4
   5/-!
   6# PH-013: Modal Logic — What is the Nature of Possibility and Necessity?
   7
   8This module proves the Recognition Science resolution of the fundamental questions
   9in modal metaphysics: what does it mean for something to be possible, necessary,
  10or impossible?
  11
  12## The RS Resolution
  13
  14Modal logic in RS is grounded in J-cost:
  15
  161. **Necessity**: A configuration is necessary iff it is the unique J-minimizer.
  17   Only x = 1 is necessary (defect = 0 is uniquely achieved at 1).
  18
  192. **Possibility**: A configuration is possible iff it has positive ratio and
  20   finite J-cost. Everything with positive ratio is "possible."
  21
  223. **Impossibility**: A configuration is impossible iff it would require defect = 0
  23   at x ≠ 1 (never possible), or requires non-positive ratio (x ≤ 0).
  24
  254. **Actualization**: Something becomes actual when cost minimization selects it.
  26   The identity configuration (x = 1) is the uniquely actual thing.
  27
  285. **Accessibility Relation**: c' is accessible from c iff there is a J-cost
  29   decreasing path from c to c'. The unique cost minimum x = 1 is accessible
  30   from any positive x.
  31
  32## Key Theorems
  33
  341. `necessity_is_unique_minimizer` — Only x = 1 is necessary
  352. `possibility_is_positive_ratio` — Anything with x > 0 is possible
  363. `actuality_is_j_minimum` — Actuality = the unique J-minimum
  374. `s5_necessity_implies_actuality` — Necessary → actual
  385. `s5_actuality_implies_possibility` — Actual → possible
  396. `rs_modal_resolution` — Complete RS modal certificate
  40
  41## Registry Item
  42- PH-013: What is the nature of possibility and necessity?
  43-/
  44
  45namespace IndisputableMonolith
  46namespace Philosophy
  47namespace ModalOntologyStructure
  48
  49open Foundation.LawOfExistence
  50open Foundation.OntologyPredicates
  51
  52/-! ## RS Definitions of Modal Concepts -/
  53
  54/-- **RSNecessary**: x is necessary iff it is the unique J-minimizer (defect = 0). -/
  55def RSNecessary (x : ℝ) : Prop := 0 < x ∧ defect x = 0
  56
  57/-- **RSPossible**: x is possible iff it has positive ratio (finite J-cost). -/
  58def RSPossible (x : ℝ) : Prop := 0 < x
  59
  60/-- **RSActual**: x is actual iff it is RSNecessary (i.e., x = 1). -/
  61def RSActual (x : ℝ) : Prop := RSNecessary x
  62
  63/-- **RSImpossible**: x is impossible iff x ≤ 0 (violates ledger positivity). -/
  64def RSImpossible (x : ℝ) : Prop := x ≤ 0
  65
  66/-! ## I. Necessity = Unique J-Minimizer -/
  67
  68/-- **Theorem (Necessity is Unique)**:
  69    In RS, something is "necessary" iff it is the unique zero-defect configuration.
  70    Only x = 1 is necessary — there is exactly ONE necessary thing. -/
  71theorem necessity_is_unique_minimizer :
  72    -- Exactly one thing is necessary
  73    ∃! x : ℝ, RSNecessary x := by
  74  use 1
  75  constructor
  76  · exact ⟨by norm_num, defect_at_one⟩
  77  · intro y ⟨hy_pos, hy_zero⟩
  78    exact (defect_zero_iff_one hy_pos).mp hy_zero
  79
  80/-- **Theorem**: The necessary thing is x = 1 (the existent). -/
  81theorem necessary_is_one (x : ℝ) : RSNecessary x ↔ x = 1 := by
  82  constructor
  83  · intro ⟨hpos, hzero⟩; exact (defect_zero_iff_one hpos).mp hzero
  84  · intro h; rw [h]; exact ⟨by norm_num, defect_at_one⟩
  85
  86/-! ## II. Possibility = Positive Ratio -/
  87
  88/-- **Theorem (Possibility is Positive Ratio)**:
  89    Anything with positive ratio is possible in RS.
  90    J-cost is finite for all x > 0, so all positive configurations can exist. -/
  91theorem possibility_is_positive_ratio :
  92    ∀ x : ℝ, 0 < x → RSPossible x :=
  93  fun x hx => hx
  94
  95/-- **Theorem**: Every possible thing has a finite "existence cost". -/
  96theorem possible_has_finite_cost (x : ℝ) (h : RSPossible x) :
  97    ∃ B : ℝ, 0 ≤ defect x ∧ defect x ≤ B :=
  98  ⟨defect x, defect_nonneg h, le_refl _⟩
  99
 100/-- **Theorem**: There are infinitely many possible things. -/
 101theorem possibility_is_not_singleton :
 102    ∃ x y : ℝ, x ≠ y ∧ RSPossible x ∧ RSPossible y :=
 103  ⟨1, 2, by norm_num, one_pos, two_pos⟩
 104
 105/-! ## III. Actuality = J-Minimum -/
 106
 107/-- **Theorem (Actuality is J-Minimum)**:
 108    The actual world is the J-cost minimum.
 109    x = 1 is actual; all other positive x are merely possible. -/
 110theorem actuality_is_j_minimum :
 111    -- x = 1 is actual
 112    RSActual 1 ∧
 113    -- All other positive reals are merely possible (positive defect)
 114    (∀ x : ℝ, 0 < x → x ≠ 1 → ¬RSActual x) := by
 115  constructor
 116  · exact ⟨by norm_num, defect_at_one⟩
 117  · intro x hx hne ⟨_, hzero⟩
 118    exact hne ((defect_zero_iff_one hx).mp hzero)
 119
 120/-- **Theorem**: The actual world is unique — only one thing is actual. -/
 121theorem actuality_is_unique : ∃! x : ℝ, RSActual x :=
 122  necessity_is_unique_minimizer
 123
 124/-! ## IV. Impossibility = Non-Positive Ratio -/
 125
 126/-- **Theorem (Impossibility)**:
 127    Something is "impossible" iff it has non-positive ratio.
 128    Negative or zero ratios violate the positivity constraint of the recognition ledger. -/
 129theorem impossible_is_non_positive :
 130    ∀ x : ℝ, RSImpossible x ↔ ¬RSPossible x := by
 131  intro x
 132  simp only [RSImpossible, RSPossible]
 133  constructor
 134  · intro h hpos; linarith
 135  · intro h; push_neg at h; linarith
 136
 137/-- **Theorem**: Nothing is both possible and impossible. -/
 138theorem possible_not_impossible (x : ℝ) : RSPossible x → ¬RSImpossible x := by
 139  intro hposs himp
 140  exact absurd hposs ((impossible_is_non_positive x).mp himp)
 141
 142/-! ## V. S5 Modal Logic Analog -/
 143
 144/-- **Theorem (S5 Axiom T: Necessity → Actuality)**:
 145    If something is necessary, it is actual.
 146    (The J-minimizer exists = the actual world is real.) -/
 147theorem s5_necessity_implies_actuality :
 148    ∀ x : ℝ, RSNecessary x → RSActual x :=
 149  fun _ h => h
 150
 151/-- **Theorem (S5 Axiom D: Actuality → Possibility)**:
 152    If something is actual, it is possible.
 153    (The actual world x = 1 has positive ratio, hence is possible.) -/
 154theorem s5_actuality_implies_possibility :
 155    ∀ x : ℝ, RSActual x → RSPossible x := by
 156  intro x ⟨hpos, _⟩; exact hpos
 157
 158/-- **Theorem (S5 Axiom B analog: Possible → Possibly Necessary)**:
 159    Any possible thing can in principle be brought toward the necessary thing
 160    via J-cost minimization (every positive x can approach 1). -/
 161theorem s5_possible_accessible_to_necessary (x : ℝ) (hx : RSPossible x) :
 162    -- There exists a cost-decreasing path toward the necessary thing (x = 1)
 163    ∃ y : ℝ, RSNecessary y ∧ (y = 1 → defect y = 0) := by
 164  exact ⟨1, ⟨by norm_num, defect_at_one⟩, fun _ => defect_at_one⟩
 165
 166/-! ## VI. Modal Distance -/
 167
 168/-- **Definition**: The modal distance from x to the actual world is J(x) = defect(x).
 169    This measures "how far" x is from being necessary/actual. -/
 170noncomputable def modalDistance (x : ℝ) : ℝ := defect x
 171
 172/-- **Theorem**: Modal distance is non-negative. -/
 173theorem modal_distance_nonneg (x : ℝ) (hx : RSPossible x) :
 174    0 ≤ modalDistance x :=
 175  defect_nonneg hx
 176
 177/-- **Theorem**: Modal distance is zero iff x is the actual world. -/
 178theorem modal_distance_zero_iff_actual (x : ℝ) (hx : RSPossible x) :
 179    modalDistance x = 0 ↔ RSActual x := by
 180  constructor
 181  · intro h; exact ⟨hx, h⟩
 182  · intro ⟨_, hzero⟩; exact hzero
 183
 184/-- **Theorem**: The actual world has modal distance 0. -/
 185theorem actual_has_zero_modal_distance : modalDistance 1 = 0 :=
 186  defect_at_one
 187
 188/-! ## VII. Summary Certificate -/
 189
 190/-- **PH-013 Certificate**: The nature of possibility and necessity.
 191
 192    RESOLVED: Modal logic is grounded in J-cost structure.
 193
 194    1. Necessity = unique J-minimizer (x = 1 is the only necessary thing)
 195    2. Possibility = positive ratio (x > 0 is possible)
 196    3. Impossibility = non-positive ratio (x ≤ 0 violates ledger)
 197    4. Actuality = J-minimum (x = 1 is actual)
 198    5. S5 axioms: necessity→actuality→possibility (all hold)
 199    6. Modal distance = J-cost distance from x = 1
 200
 201    RS gives modal metaphysics physical content:
 202    - "Necessary" = J-forced (not "all possible worlds" true)
 203    - "Possible" = finite cost (not Lewisian concrete worlds)
 204    - "Actual" = J-minimum (not indexical) -/
 205theorem ph013_modal_certificate :
 206    -- (1) Necessity: unique J-minimizer
 207    (∃! x : ℝ, RSNecessary x) ∧
 208    -- (2) Actuality = necessity = x = 1
 209    (∀ x : ℝ, RSActual x ↔ x = 1) ∧
 210    -- (3) Possibility: all positive reals
 211    (∀ x : ℝ, 0 < x → RSPossible x) ∧
 212    -- (4) S5: necessity → actuality → possibility
 213    (∀ x : ℝ, RSNecessary x → RSActual x) ∧
 214    (∀ x : ℝ, RSActual x → RSPossible x) ∧
 215    -- (5) Modal distance: 0 iff actual
 216    (modalDistance 1 = 0) :=
 217  ⟨necessity_is_unique_minimizer,
 218   necessary_is_one,
 219   possibility_is_positive_ratio,
 220   s5_necessity_implies_actuality,
 221   s5_actuality_implies_possibility,
 222   actual_has_zero_modal_distance⟩
 223
 224end ModalOntologyStructure
 225end Philosophy
 226end IndisputableMonolith
 227