Marxist Theory Formalization

𝒜: Set of all agents (individuals)
𝒞: Set of all social classes
ℳ: Set of all modes of production
ℛ: Set of all economic resources
𝒯: Ordered set of time periods, 𝒯 ⊂ ℤ⁺
m(t) ∈ ℳ: Mode of production at time t
Class(a, c, t): Agent a ∈ 𝒜 belongs to class c ∈ 𝒞 at time t
Owns(c, r, t): Class c ∈ 𝒞 owns resource r ∈ ℛ at time t

B(t): Economic base at time t
B(t) = (m(t), OWNERSHIP(t))
S(t): Superstructure at time t
S(t) = f(B(t))

Bourgeoisie(c, t) := (m(t) = Capitalism) ∧ (∀r ∈ ℛ_mp, Owns(c, r, t))

Proletariat(c, t) := (m(t) = Capitalism) ∧ (∀a ∈ 𝒜, Class(a, c, t) ⇒ ¬∃r ∈ ℛ_mp, Owns(a, r, t))

L(r, t) ∈ ℝ⁺: Socially necessary labor time
V(r, t) ∈ ℝ⁺: Value of resource r at time t
V(r, t) = L(r, t)

W(a, t) ∈ ℝ⁺: Wage paid to worker a
L_created(a, t) ∈ ℝ⁺: Value created by worker a
s(a, t) = L_created(a, t) - W(a, t)
S_t = ∑_{a ∈ 𝒜w} s(a, t)

K(c, t) ∈ ℝ⁺: Capital owned by class c at time t
∀t′ > t, K(c_b, t′) > K(c_b, t)

∀t′ > t, ∑_{a ∈ 𝒜cp} W(a, t′) ≤ ∑_{a ∈ 𝒜cp} W(a, t)

CC(c, t) ∈ {0, 1}: Class consciousness
CC(c_p, t) = F(B(t), S_t)

Crisis(t) ∈ {0, 1}: Economic crisis at time t
Rev(t) := (m(t) = Capitalism) ∧ CC(c_p, t) = 1 ∧ Crisis(t) = 1

Rev(t) ⇒ (m(t+1) = Socialism) ∧ (S(t+1) = f(B(t+1))) ∧ (B(t+1) ⇒ c_p)

(m(t) = Socialism) ∧ (ClassAntagonism(t) ≈ 0) ⇒ ∃Δt > 0, m(t+Δt) = Communism