Causality

Causation is a form of event generation. To present an explicit definition of causation requires introducing some ontological concepts to formally characterize what is understood by ‘event’.

The concept of individual is the basic primitive concept of any ontological theory. Individuals associate themselves with other individuals to yield new individuals. It follows that they satisfy a calculus, and that they are rigorously characterized only through the laws of such a calculus. These laws are set with the aim of reproducing the way real things associate. Specifically, it is postulated that every individual is an element of a set s in such a way that the structure S = ⟨s, ◦, ◻⟩ is a commutative monoid of idempotents. This is a simple additive semi-group with neutral element.

In the structure S, s is the set of all individuals, the element ◻ ∈ s is a fiction called the null individual, and the binary operation ◦ is the association of individuals. Although S is a mathematical entity, the elements of s are not, with the only exception of ◻, which is a fiction introduced to form a calculus. The association of any element of s with ◻ yields the same element. The following definitions characterize the composition of individuals.

1. x ∈ s is composed ⇔ (∃ y, z) s (x = y ◦ z)
2. x ∈ s is simple ⇔ ∼ (∃ y, z) s (x = y ◦ z)
3. x ⊂ y ⇔ x ◦ y = y (x is part of y ⇔ x ◦ y = y)
4. Comp(x) ≡ {y ∈ s|y ⊂ x} is the composition of x.

Real things are distinguished from abstract individuals because they have a number of properties in addition to their capability of association. These properties can be intrinsic (P_i) or relational (P_r). The intrinsic properties are inherent and they are represented by predicates or unary applications, whereas relational properties depend upon more than a single thing and are represented by n-ary predicates, with n ≥ 1. Examples of intrinsic properties are electric charge and rest mass, whereas velocity of macroscopic bodies and volume are relational properties.

An individual with its properties make up a thing X : X =< x, P(x) >

Here P(x) is the collection of properties of the individual x. A material thing is an individual with concrete properties, i.e. properties that can change in some respect.

The state of a thing X is a set of functions S(X) from a domain of reference M (a set that can be enumerable or nondenumerable) to the set of properties P_X. Every function in S(X) represents a property in P_X. The set of the physically accessible states of a thing X is the lawful state space of X : S_L(X). The state of a thing is represented by a point in S_L(X). A change of a thing is an ordered pair of states. Only changing things can be material. Abstract things cannot change since they have only one state (their properties are fixed by definition).

A legal statement is a restriction upon the state functions of a given class of things. A natural law is a property of a class of material things represented by an empirically corroborated legal statement.

The ontological history h(X) of a thing X is a subset of S_L(X) defined by h(X) = {⟨t, F(t)⟩|t ∈ M}

where t is an element of some auxiliary set M, and F are the functions that represent the properties of X.

If a thing is affected by other things we can introduce the following definition:

h(Y/X ) : “history of the thing Y in presence of the thing X”.

Let h(X) and h(Y) be the histories of the things X and Y, respectively. Then

h(Y/X) = {⟨t,H(t)⟩|t ∈ M},

where H≠ F is the total state function of Y as affected by the existence of X, and F is the total state function of X in the absence of Y. The history of Y in presence of X is different from the history of Y without X .

We can now introduce the notion of action:

X ▷ Y : “X acts on Y”

X ▷ Y =^def h(Y/X) ≠ h(Y)

An event is a change of a thing X, i.e. an ordered pair of states:

(s₁, s₂) ∈ E_L(X) = S_L(X) × S_L(X)

The space E_L(X) is called the event space of X.

Causality is a relation between events, i.e. a relation between changes of states of concrete things. It is not a relation between things. Only the related concept of ‘action’ is a relation between things. Specifically,

C'(x): “an event in a thing x is caused by some unspecified event e^x_xi“.

C'(x) =^def (∃ e^x_xi) [e^x_xi ∈ E_L(X) ⇔ x_i ▷ x.

C(x, y): “an event in a thing x is caused by an event in a thing y”.

C(x, y) =^def (∃ e^x_y) [e^x_y ∈ E_L(x) ⇔ y ▷ x

In the above definitions, the notation e^x_y indicates in the superscript the thing x to whose event space belongs the event e, whereas the subscript denotes the thing that acted triggering the event. The implicit arguments of both C’ and C are events, not things. Causation is a form of event generation. The crucial point is that a given event in the lawful event space E_L(x) is caused by an action of a thing y iff the event happens only conditionally to the action, i.e., it would not be the case of e^x_y without an action of y upon x. Time does not appear in this definition, allowing causal relations in space-time without a global time orientability or even instantaneous and non-local causation. If causation is non-local under some circumstances, e.g. when a quantum system is prepared in a specific state of polarization or spin, quantum entanglement poses no problem to realism and determinism. The quantum theory describes an aspect of a reality that is ontologically determined and with non-local relations. Under any circumstances the postulates of Special Relativity are violated, since no physical system ever crosses the barrier of the speed of light.