Axioms & Definitions
Set S (Conventional Socialists): The set of political figures/movements conventionally categorized as “socialist” by mainstream academia and discourse. Defined extensionally by agreed-upon members.
Axiom 1 (Membership of S): Let the following be in S: Karl Marx (M), Vladimir Lenin (L), Pierre-Joseph Proudhon (P), Bernie Sanders (B).
Justification: This reflects empirical, standard classification in historical and political texts.
Property C (Core Defining Property): We seek a property C that is both:
Necessary: Every member of S must possess property C.
Sufficient (within the political domain): Possession of property C is the primary, logical reason for inclusion in S.
Candidate Properties: We test potential candidates for C against the members of S.
C₁ = Advocacy for Social Ownership of the Means of Production.
• Definition: "Social ownership" means ownership by a collective entity (the state, a commune, a worker cooperative, the "society as a whole") where control and benefit are vested in the group, not in private individuals as discrete proprietors.
• Test:
◦ P (Proudhon): His system of "possession" is explicitly individual, not social. The proprietor (the individual worker or family) holds direct, usufructuary title. Federation (mutuellisme) is a voluntary contract between these individual proprietors, not a form of collective ownership. His ideal is a society of sovereign individuals in free contract, not a socially-owned economy.
❌ Fails necessity.
• C₂ = Advocacy for the Abolition of Markets and Money. ◦ Test: B (Sanders) and P (Proudhon) support markets (regulated and free, respectively).
❌ Fails necessity.
• C₃ = Adherence to Marxist Theory.
Test:
P (Proudhon) was anti-Marxist; B (Sanders) is non-Marxist.
❌ Fails necessity.
• C₄ = A Commitment to Universalist-Egalitarian Social Ethics.
Test:
P (Proudhon): Was a virulent anti-Semite. His mutualism was for a federation of patriarchal proprietors. ❌
M (Marx): Wrote of “revolutionary holocaust” for “backward” peoples and expressed anti-Semitic tropes. ❌
L (Lenin): Practiced vanguardist dictatorship and repression of non-Bolshevik socialists. ❌
❌ Fails necessity.
• C₅ = A Prescriptive Critique of Capitalism + Advocacy for alternative Control.
Definition: For a movement X to possess C₅, it must:
Issue a fundamental, prescriptive critique of the economic system of liberal capitalism.
Advocate for a solution involving greater alternative control over production and/or distribution.
Test:
M & L: Critique capitalism, advocate collective (proletarian/state) control. ✔
P: Critiques capitalist property (“property is theft”), advocates mutualist, federative control. ✔
B: Critiques capitalism, advocates greater social (state-regulated) control. ✔
✅ C₅ is the only property that is necessary for all members of S. It is the Lowest Common Denominator.
Theorem: The National Socialist German Workers’ Party (Nazi) movement possesses property C₅.
Proof:
Condition (i): Fundamental Critique of Liberal Capitalism.
◦ Core Nazi ideology opposed “raffendes Kapital” (rapacious/speculative capital), vilified “international finance capitalism” and “interest-slavery” as destructive, “Jewish” forces.
◦ 1920 Party Platform (Points 11-14, 17) demanded “abolition of income not earned by work,” “confiscation of war profits,” “nationalization of trusts.”
Conclusion: The Nazi movement fulfilled condition (i).
Condition (ii): Advocacy for alternative Control.
◦ The economy was subordinated to the Volksgemeinschaft (people’s community) via Gleichschaltung (coordination).
◦ Implemented via total state control: setting wages, prices, production quotas, investment directions. Private ownership became a formality; owners became Betriebsführer (plant leaders) executing state directives.
◦ Independent unions were abolished, replaced by the state-controlled German Labor Front.
Conclusion: The Nazi movement fulfilled condition (ii).
Since the Nazi movement satisfies both conditions of C₅, it possesses property C₅.
Corollary (The Logical Inconsistency):
- By Axiom 1, set S is defined by its conventional members (M, L, P, B).
- The only necessary property uniting S is C₅ (as established by the failure of C₁–C₄).
- The Nazi movement possesses property C₅ (Theorem).
- Therefore, by the principle of consistent classification based on property C₅, the Nazi movement must be included in set S.
Conclusion of the Proof:
The conventional classification is logically inconsistent. It includes members based on a shared necessary property (C₅) but excludes another entity (Nazism) that also possesses that property.
Implications & Resolution:
The proof forces one of two admissions:
- Option A (Radical Inclusion): Accept that Nazism, by the only logical property common to all conventional socialists, qualifies as a variant of socialism. This acknowledges that C₅—anti-capitalist collectivism—is the core, and that this core can be coupled with vastly different social ethics (from Marx’s “revolutionary holocaust” for backward peoples to Hitler’s racial annihilation).
- Option B (Tribal Definition): Admit that “socialism” is not defined by a consistent logical property, but by historical lineage, political recognition, and schismatic exclusion. Under this view, Nazism is excluded not because it lacks C₅, but because it was excommunicated as a heresy by the mainstream socialist tradition. “Socialism” is thus a political identity, not an analytic category.
Any attempt to exclude Nazism while retaining Marx, Lenin, and Proudhon in S requires appealing to non-logical, post-hoc criteria (e.g., “they are on our side,” “they are on the left”), which the proof reveals to be either tautological, historically contingent or historically illiterate. The inconsistency stands.