Against Intellectual Property (Formal Proof)
Purpose of the Formal Argument
I propose to question the justification of intellectual property based on a formal reasoning that establishes concepts of “good”, “use”, “authority”, “property” and “abundance”, among others. The formal system is structured by means of definitions and axioms that, together and with some premises, attempt to demonstrate solidly (solid deductive argument) that the assignment of exclusive property rights (especially in the context of abstracted ideas or properties) leads to contradictions when considering goods that are abundant (i.e., susceptible of being used simultaneously by different agents).
The strategy of the argument is to carry out a reduction to the absurd. It starts from an assumption, called the “statist assumption”, in which it is stated that there is at least one good (for example, an ideal or idea) that is abundant and, at the same time, exclusive property is assigned to some subject. It is then shown that, by assuming an abundant good, it follows that it is impossible -in modal terms- for exclusive ownership to exist, since exclusivity requires conditions (such as rivalry of use and normative authority) that cannot be met in shared goods. By generating this contradiction, the possibility of justifying intellectual property in such cases is formally rejected.
Detailed Analysis of Definitions and Axioms
Symbology and Notation
Before analyzing each axiom, it is important to explain the notation used:
≡: Used as a definition symbol, to be read as “is defined as.”
◊: Modal operator of possibility, used to express “it is possible that…”
The basic predicates and functions are:
Abundant(y): y is abundant; that is, it can be used or controlled by two or more agents simultaneously. (Axiom 7)
Good(x) = x is a good. (See Axiom 3)
Use(x, y, E) = x exercises control/use over y for the purpose E
ExclusiveControl(x, y) = x exercises exclusive control over y; that is, x has the capacity to prevent others from using y. (see Axiom 5)
Property(x, y) = x is the owner of y, and is defined in terms of exclusive control. (Axiom 6)
Authority(x, y, n): x possesses normative authority over y (that is, the capacity to decide, permit, or exclude the use of y by others) through norm n (or justified by norm n); it is understood as the capacity to decide, permit or exclude the use of y by others.
EthicalTheory(T): T is an ethical theory, understood as a consistent subset of the total set of norms which also has a model that validates its norms. (Axiom 8)
AvoidsConflicts(n, T): Norm n of theory T has the objective of avoiding conflicts.
Norm(n): n is a norm (or axiom) of theory T.
Conflicting(n, m): n conflicts with m.
Rival(y): y is rival; that is, it is defined as a good for which it is impossible (in the modal sense) for two different agents to use it simultaneously for given ends.
Essential Definitions/Axioms
1. Total Set of Norms N
N = { n : Norm(n) }
The set N is defined as the totality of norms. Here, each norm n is formally stated within the system and constitutes part of the normative corpus.
2. Consistent Subset of Norms
D = { T ⊂ N : ∀ n, m ∈ T, m ≠ ¬n }
D is defined as the set of subsets T of norms that are internally consistent, meaning that no pair of norms in T contradict each other (no norm is the negation of another). This consistency is fundamental for the normative system to be valid without incurring paradoxes (referring to any ethical theory insofar as it is a consistent subset of N).
3. Definition of Good
∀x, y, E Good(y) ≡ ◊Use(x, y, E)
This definition establishes that something is a good if it is possible for some subject x to use y for some end E.
4. Definition of Rival
∀x ≠ z, y, E₁, E₂ Rival(y) ≡ ¬◊(Use(x, y, E₁) ∧ Use(z, y, E₂))
A good y is defined as rival when it is impossible (in the modal sense) for two different agents to simultaneously use it for ends E₁ and E₂.
5. Definition of ExclusiveControl
∀x ≠ z, y, E₁, E₂ (ExclusiveControl(x, y) ≡ Rival(y) ∧ Authority(x, y, n))
This statement specifies that x exercises exclusive control over y if y is a rival good—meaning its use cannot be shared—and x possesses authority (normative power) over y through some norm n. Exclusivity implies both the impossibility of shared use (rivalry) and the consolidation of normative authority.
6. Definition of Property
∀x, y Property(x, y) ≡ ExclusiveControl(x, y)
Property is defined directly in terms of exclusive control: for x to be the owner of y, x must have exclusive authority and control over y. This definition equates appropriation with exclusivity in control and use.
7. Definition of Abundant Good
∀x ≠ z, y, E₁, E₂ Abundant(y) ≡ Good(A) ∧ ◊(Use(x, A, E₁) ∧ Use(z, A, E₂))
The concept of an “abundant good” is introduced as a good y that, unlike rival goods, can be used (modally) by at least two different subjects. Notice that the definition of abundance does not include normative authority; only simultaneous use by different agents is required.
8. Definition of EthicalTheory
∀T EthicalTheory(T) ≡ T ∈ D ∧ ∃M (∀n ∈ T [ M ⊨ n ])
A set T is considered an ethical theory if and only if T is a consistent subset of norms (that is, T ∈ D) and there exists a model M that validates or entails all the norms contained in T. This definition introduces a model-theoretic perspective in which the validity of norms is confirmed in a certain model.
9. Definition of Rivalry
∀x ≠ z, y, E₁, E₂ Rival(y) ≡ Good(y) ∧ (¬◊(Use(x, y, E) ∧ Use(z, y, E)))
This definition establishes that y is rival if, being a good, it is impossible (modally) for two distinct subjects x and z to simultaneously use the same good. That is, in a rival good, one subject’s exclusive use excludes the possibility of another’s use.
10. Objective of Conflict Avoidance in Ethical Theories
∀T (EthicalTheory(T) ∧ ∀n ∈ T AvoidsConflicts(n, T))
Every norm of an ethical theory T has the objective of avoiding conflicts.
11. Existence of Norms from Authority or Normative Power
∀x, y, n, T Authority(x, y, n) → ∃n ∃T (Norm(n) ∧ EthicalTheory(T) ∧ n ∈ T)
The intention is to establish that the exercise of authority over a good must be tied to the existence of norms within an ethical theory. That is, authority implies a normative basis in T.
12. Origin of Conflicts in Rival Goods
∀n ∈ T, x, y (AvoidsConflicts(n, T) → (Authority(x, y, n))) ↔ Rival(y))
If every norm n of an ethical theory T aims to avoid conflicts (as Axiom 10 states), then x has normative authority over y if and only if y is a rival good.
13. Incompatibility of Conflicting Norms within an Ethical Theory
∀n, T Norm(n) ∧ EthicalTheory(T) ∧ n ∈ T ∧ ¬(Conflicting(n, n))
This axiom imposes that no norm contained in an ethical theory can be in conflict with itself; that is, Conflicting(n, n) cannot occur. This condition is fundamental to guaranteeing the internal consistency of the normative system.
Statement of the Argument (Reductio ad Absurdum)
1. (Statist assumption)
∃y ∃x Abundant(y) ∧ Property(x, y)
2. ∀x, y Abundant(y) → ¬◊(Property(x, y))
(Justification)
2.1 (Statist assumption)
∃x ≠ z ∃y ∃E₁ ∃E₂ (Good(y) ∧ Use(x, y, E₁) ∧ Authority(x, y) ∧ Use(z, y, E₂))
(Assumption that there exists a good that can be used by different agents simultaneously and at least one subject has authority over it)
2.2 ∀T (EthicalTheory(T) ∧ ∀n ∈ T AvoidsConflicts(n, T))
(Axiom 10)
2.3 ∀n ∈ T, x, y (AvoidsConflicts(n, T) → (Authority(x, y, n))) ↔ Rival(y))
(Axiom 12)
2.4 ∴ ∀n ∈ T, x, y (Authority(x, y, n))) ↔ Rival(y))
(Modus Ponens from 2.2 with 2.3)
2.5 ∀n ∈ T, x ≠ z, y Good(y) ∧ (Use(x, y, E₁) ∧ Use(z, y, E₂)) → ¬◊(¬◊(Use(x, y, E) ∧ Use(z, y, E)))
(Principle of Non-Contradiction)
2.6 ∴ ¬◊(Rival(y))
(Application of Definition 8, Modus Ponens with 2.1 and 2.5)
2.7 ∴ ¬◊(Authority(x, y, n))
(Modus Ponens with 2.1, 2.2, 2.3 and 2.5)
2.8 ∴ ¬◊(Property(x, y))
(Application of Definition 5, which implies authority and rivalry, both impossible in this case)
3. ∴ Property(x, y) ∧ ¬◊(Property(x, y))
(Reductio ad absurdum from 1 and 2)
To avoid contradiction, the contradictory part of the assumption is denied, leaving the consistent conclusion:
3. ∴ ¬◊(Property(x, y))
Contextualization and Step-by-Step Development of the Argument
Initial Hypothesis and Statist Assumption
Fundamental Premise (Step 1):
∃y ∃x Abundant(y) ∧ Property(x, y)
Explanation:
The “statist assumption” presumes the existence of at least one abundant good y over which there exists a subject x who, according to the definition of property, has exclusive control over y. Since Property(x, y) is defined as ExclusiveControl(x, y), the assumption implies that x has both exclusive use and normative authority over y.
Denial of the Possibility of Property in Abundant Goods
Statement:
∀x, y Abundant(y) → ¬◊(Property(x, y))
Interpretation:
This statement, which we aim to derive, maintains that if a good y is abundant, then it is impossible (in the modal sense) for a subject x to have the right to exclusive property over y. The contradiction will be reached by showing that the “statist assumption” (from Step 1) and the real situation of an abundant good are incompatible.
Proof Strategy (Reductio ad Absurdum):
The opposite hypothesis to the desired conclusion will be assumed, and one will proceed to derive a contradiction so that the existence of a subject with property over an abundant good is rejected.
Development of the Reasoning Through Intermediate Steps (2.1 to 2.8)
The argument uses a series of inferences (numbered 2.1 through 2.8) to show how the hypothesis of an appropriable abundant good leads to a logical contradiction. Each of these steps is explained below:
- 2.1 Multiple-Use Hypothesis
∃x ≠ z ∃y ∃E₁ ∃E₂ (Good(y) ∧ Use(x, y, E₁) ∧ Authority(x, y) ∧ Use(z, y, E₂))
Explanation:
In this step (as part of the statist assumption), it is assumed that there exists a good y which, being a good (by the Definition of Good), is used by two different agents: x (who additionally has normative authority over y, using it for some end E₁) and z (for whom only use is assumed with some end E₂, without z having normative authority over y).
– 2.2 Assertion of the Normative Goal
∀T (EthicalTheory(T) ∧ ∀n ∈ T AvoidsConflicts(n, T))
Explanation:
This statement reiterates Axiom 10, which establishes that in any ethical theory the goal is to avoid conflicts. This premise is the basis upon which normative authority will be related to rivalry (see Axiom 12).
– 2.3 Relation Between Conflict Avoidance, Authority, and Rivalry
∀n ∈ T, x, y (AvoidsConflicts(n, T) → (Authority(x, y, n))) ↔ Rival(y)))
(Axiom 12)
Explanation:
According to Axiom 12, for any norm n in the ethical theory T that seeks to avoid conflicts, it follows that x has authority over y by means of n if and only if y is rival. That is, the condition for normative authority —and therefore for exclusive property— is that the good behaves as rival (i.e., it does not allow simultaneous use by different subjects).
– 2.4 Deduction of the Equivalence Between Normative Authority and Rivalry
Applying modus ponens to statements 2.2 and 2.3, we deduce:
∴ ∀n ∈ T, x, y (Authority(x, y, n))) ↔ Rival(y))
Inference Rule:
Here biconditionality is used and applied through modus ponens based on the hypothesis that in every ethical theory the goal is to avoid conflicts, so authority is linked precisely to rivalry. (It should be noted that strict application of modus ponens to a biconditional normally requires first proving “if A then B” and then “if B then A”; the way the axiom is presented already suggests an equivalence, from which the conclusion is drawn.)
– 2.5 Application of the Principle of Non-Contradiction to Simultaneous Use in Goods
∀n ∈ T, x ≠ z, y Good(y) ∧ (Use(x, y, E₁) ∧ Use(z, y, E₂)) → ¬◊(¬◊(Use(x, y, E) ∧ Use(z, y, E)))
Explanation:
This step is based on the principle of non-contradiction: if a good is used by two agents (which would classify it as abundant), then it is not possible for that good to also be rival, since rivalry requires that there cannot be simultaneous use that permits exclusivity. A conditional inference is used: given that simultaneous use contradicts exclusivity of use (and, by extension, exclusive control), the modal possibility of such exclusive use is denied.
– 2.6 Conclusion on the Impossibility of Rivalry
From the application of the definition of Rivalry (Definition 8) and the consequence of step 2.5, we deduce:
∴ ¬◊(Rival(y))
Inference Rule:
Modus ponens is used: since simultaneous use (presence of abundance) prevents rivalry (by the definition of Rival:
∀x ≠ z, y, E₁, E₂ Rival(y) ≡ Good(y) ∧ (¬◊(Use(x, y, E) ∧ Use(z, y, E)))),
we conclude that it is impossible for y to satisfy the condition of being rival in the sense required for normative authority.
– 2.7 Deduction of the Impossibility of Exercising Normative Authority
From the equivalence established in 2.4 (that Authority(x, y, n) is equivalent to Rival(y)) and the result of 2.6, we conclude:
∴ ¬◊(Authority(x, y, n))
Inference Rule:
Modus ponens is again applied to the equivalence: if exercising authority requires that y be rival, and it has been shown that it is impossible for y to be rival (in modal terms) when simultaneous uses are present, it follows that normative authority over y cannot exist.
– 2.8 Impossibility of Exclusive Property
Finally, using the Definition of Property:
Property(x, y) ≡ ExclusiveControl(x, y)
and recalling that exclusive control requires both rivalry and normative authority (see the Definition of ExclusiveControl in Definition 6), we deduce:
∴ ¬◊(Property(x, y))
Inference Rule:
The conclusion is obtained by modus ponens applied to the relationship between authority, rivalry, and property. Since the modal impossibility of having authority has already been demonstrated (and, consequently, of achieving exclusive control), it follows that it is impossible for any subject x to have property over y when y is abundant.
Conclusion by Reductio ad Absurdum (Step 3)
Final Statement:
The initial hypothesis of the statist assumption is:
∃y ∃x (Abundant(y) ∧ Property(x, y))
But from the development (Step 2), it is demonstrated that for all x and y the following holds:
Abundant(y) → ¬◊(Property(x, y))
The combination of both statements produces the contradiction:
Property(x, y) ∧ ¬◊(Property(x, y))
Explanation:
The contradiction is achieved through reductio ad absurdum: the existence of an abundant good with exclusive property is assumed, and from the chain of inferences (which link the definition of simultaneous use, the impossibility of rivalry and, therefore, the impossibility of normative authority and exclusive control), it is deduced that it is impossible (modally) for exclusive property to exist. Therefore, the initial hypothesis is rejected.
Inference Rule:
Reductio ad absurdum is applied by noting that the assumption of the existence of exclusive property in an abundant good leads to a logical contradiction, and thus we conclude:
¬◊(Property(x, y))
This is the final conclusion of the argument, which rejects the formal justification of intellectual property in the presence of abundant goods.
Conclusion of the Analysis
To summarize rigorously:
1. Introduction and Objective:
The argument aims to demonstrate that in a normative system where good, use, authority, exclusive control, and property are defined, the assignment of exclusive property is incompatible with the abundance of a good (that is, the possibility that two or more agents may make use of it).
2. Definitions and Axioms:
Each of the axioms and definitions is used to construct the theoretical framework linking the possibility of use with the exclusivity required for having property. In particular, property is equated with exclusive control, which requires that the good be rival (that it prevents simultaneous uses) and that normative authority be exercised.
3. Development of the Argument:
– The argument begins with the statist assumption asserting the existence of an abundant good over which some subject is the owner (Property(x, y)).
– It is assumed that there exists at least one simultaneous use by two agents (which defines the capacity of being an abundant good).
– Through the relationship established in the ethical theory (especially through the equivalence between normative authority and rivalry), it is deduced that if a good is simultaneously used by different agents, it cannot satisfy the condition of rivalry required for authority and therefore for exclusive property.
– The modal impossibility of property (¬◊(Property(x, y)) ) is obtained, which contrasts with the initial hypothesis, and through reductio ad absurdum, the statist assumption is rejected.
4. Clarified Inference Rules:
– Existential introduction and conjunction are used to formulate the use hypotheses.
– Modus ponens is employed in various steps to apply implications derived from the axioms (for example, the equivalence between normative authority and rivalry).
– Finally, reductio ad absurdum is used to conclude that the assumption leads to a contradiction.
The final conclusion is that, within this formal framework, it is not possible to justify exclusive property over a good that is inherently abundant, which challenges the formal justification of intellectual property when applied to goods that can be used by multiple agents.