Introspection allows me to analyze the structure of an experience—its phenomenal structure—by dissecting it into coarse components, let’s call them compound distinctions, and the way these distinctions are bound, corresponding to compound relations. However, introspection alone is hardly adequate to the task of dissecting an experience down to what I might consider to be elementary distinctions and relations. For example, I can distinguish the book and my left hand, and see they are related—the hand lies on the book. I can also distinguish the book’s cover, its top and bottom edges, and its title, just as I can distinguish my thumb and index finger. I can also see that the two fingers are related to a different region of the book. But I cannot possibly hope to dissect my experience all the way down, or to count the innumerable ways in which the contents of my experience are related [2].

Composition is immediate because I do not need to infer the structure of experience, the way the experience feels. It is already there, the way it is, "in front of my eyes"—the book bound to its shape, color, and position in space; my body lying over my bed within the room; and so on. Composition is also irrefutable, in the sense that its negation is self-contradictory or absurd. Try to think of an experience that had no components at all, not even a single distinction. The structure of the experience would be composed of nothing, which is the same as saying that there would be no experience [3]. And while I can conceive of an experience structured in a different way, it would remain true that when I experience it, it would be structured in its particular way—it would feel the way it feels—reaffirming the validity of the axiom. Moreover, composition is true not just of the experience I am having now, but it must be true of every conceivable experience, which makes it an essential property of phenomenal existence.

How can we formulate the phenomenal property of composition in physical terms? Phenomenally, composition means that every experience is structured, being composed of phenomenal distinctions bound by phenomenal relations to form a phenomenal structure that is the way it is. Therefore, the substrate of consciousness must have cause–effect power that is structured in a particular way. We can reveal this internal structure by considering, first, whether and how subsets of units take and make a difference over subsets of units within a complex, specifying their own cause–effect states and yielding causal distinctions. Second, we must consider how these distinctions’ causes and effects overlap over subsets of units, yielding causal relations. Altogether, causal distinctions and relations form a cause–effect structure, also called a Φ-structure.

We saw in the previous postulates that the complex as a whole must specify both a cause and an effect over itself. Likewise, a subset of its units can only exert cause–effect power within the complex if it specifies a cause–effect (a cause and an effect) within the complex, called a causal distinction. A distinction comprises a mechanism, a cause purview, and an effect purview: a mechanism is a subset of units within a complex that can take and make a difference within it, its cause purview is the subset of units from which the mechanism takes a difference, and its effect purview is the subset of units to which it makes a difference.  The link between cause and effect provided by a mechanism can be roughly expressed as “given this cause, then this effect.” [In the figures], distinctions are illustrated by orange lines connecting a mechanism and its cause and effect purviews. 

For a substrate constituted of n units, there are 2n–1 combinations of units (the powerset of the n units, excluding the empty set [1]), so the substrate could specify up to 2n–1 distinctions. In a given mechanism or purview, the term order refers to the number of units involved. In the example here, the mechanisms and effect purviews are both second order, while the cause purviews are first order.

As components of the cause–effect power of the complex, distinctions—like the complex as a whole—must be characterized in accordance with the postulates of physical existence (save for composition because, being components themselves, they do not have components). Recall that the complex as a whole must be able to take and make a difference (existence) and do so within itself (intrinsicality). It must also select a specific cause and effect state (information). Similarly, a distinction can only exist for the complex if it can make and take a difference. For this, it needs a mechanism that links a cause with an effect. Distinctions must also comply with the intrinsicality postulate: their cause–effect power must be exerted within a system, and from the intrinsic perspective of the mechanism. Likewise, they must comply with the information postulate: a mechanism in its specific current state must select a specific cause–effect state. As for the system, this is the state for which intrinsic information is maximal (indicated by the red and green squares on the TPM in [the slides above]) [2].

Finally, distinctions must comply with the integration and exclusion postulates. A distinction’s integrated information is measured by φd (“phi” with subscript d for distinction), which quantifies how much specific cause–effect power a mechanism has over its purviews as one mechanism. As with the system, and in line with the principle of minimal existence, we assess integration by considering the partition for which the difference is minimal, relative to the maximum possible integrated information for that partition. We also take the minimum between the cause and the effect sides [3]. To comply with the exclusion postulate, we follow the principle of maximal existence and establish the border of a distinction by determining the cause–effect purview over which a mechanism achieves maximal integrated information, excluding any purview with lower φd [4].

Note that when assessing integrated information of higher-order mechanisms, φd is not a measure of how much additional effects (or causes) a higher-order mechanism (say, Abcd) specifies over first-order mechanisms (say, A, b, c, d). Additional effects, say, due to non-linear interactions, may be relevant from an extrinsic perspective. Yet from the intrinsic perspective of an entity, what matters with respect to structure is whether the mechanism Abcd has an effect (and cause) in a way that is maximally irreducible. In other words, does the complex have at its disposal the integrated "pulley" or "handle" Abcd to specify a certain effect (and cause) within itself? [For more, see FAQ: Why do we assess the causal power of all orders of mechanisms? Why not simply assess the causal power of the individual units alone?]

In principle, a complex constituted of n units can specify a very large number of relations, up to n*22^n. A relation is composed of one or more distinctions (the relation’s degree) [1]. Each relation has one or more faces (the face’s degree), each of which corresponds to a distinct combination of causes and/or effects of the participating distinctions. Each face has a corresponding face purview—the maximal congruent overlap shared by two or more causes and/or effects. For example, a relation between distinctions b and cd is indicated [in the slides above] by a dark line “binding” two labels in the mechanism powerset; the respective relation faces are shown as blue lines (for second degree) and a plane (for third degree) connecting the specified causes and effects in the cause and effect powersets. 

The union of all face purviews is called the relation purview, or joint purview. [In the slides above], for example, the relation purview of distinctions b and cd comprises purview elements a and d. The particular combination of faces that constitute a relation characterize its type. For example, a self-relation is reflexive, a relation where all causes and effects are maximally congruent with one another is a complete relation (which is necessarily reflexive), a relation where the effects of one set of distinctions overlap congruently with the causes of the remaining distinctions (but not the other way around) is a directed relation, and so on. The type of relations will become important when accounting for the quality of experience [...].

Like distinctions, relations must be characterized according to the postulates of physical existence. Relations inherit existence from the cause–effect power of the distinctions that compose them. They inherit intrinsicality because the causes and effects that compose their faces are specified within the substrate. Moreover, relations are specific because their face purviews must be congruent for all causes and effects.

Relations satisfy integration because they capture the irreducibility of cause–effect power due to the overlap among distinctions [2]. The irreducibility of a relation (φr) is measured by “unbinding” distinctions from their joint purviews, one by one. By the principle of minimal existence, a relation can only be as irreducible as the minimal amount of irreducible information specified by any one distinction in the relation. Therefore, the relation integrated information (φr) is given by the minimum φd value scaled by the number of unique units over the face purviews of all faces in the relation [3].

Finally, a relation complies with exclusion in that its integrated information is naturally maximized over the maximally congruent overlap for each relation face (taking subsets of these overlaps could only reduce the integrated information of the relation).

In sum, for both distinctions and relations, φ is again a quantifier of integrated existence: a distinction or relation only exists as one distinction or relation to the extent that it is causally irreducible.

Relations also serve to “relate” the identity and state of the units of a complex from its intrinsic perspective. Needless to say, intrinsically, neither the units in a purview nor their states come with a label (such as unit A or B, state ON or OFF). It is relations that specify which purview units in one distinction correspond to which purview units in another, and which states correspond to which states. In this way, relations can uniquely specify (or “lock in”) the identity of both the units and of the states from the intrinsic perspective of the complex [4].

To make fully explicit the internal structure of the cause–effect power specified by a complex, we need to unfold all the distinctions it supports, of any order, and all the relations that bind them together, of any degree. However, in line with intrinsicality and information, from the intrinsic perspective of the complex, the only distinctions and relations that truly exists are those that are congruent with its own cause–effect state. A distinction’s mechanism must be constituted by units of the complex, whose current state is a subset of the state of the complex as a whole. Likewise, the cause purview of any distinction must be in the same state as the cause state of the complex as a whole, and similarly for the effect purviews and the joint purviews of relations. This can be seen in [the slides above], where the three powersets [...] are always congruent with the cause–effect state of the complex. The complete set of distinctions and the associated relations specified by the complex compose a cause–effect structure—or, for brevity, a Φ-structure [1]. A Φ-structure thus captures the fully ‘unfolded’ cause–effect power of a complex in its current state [2].

In the Φ-structure depicted [in the slides], mechanisms are represented by black letters, cause purviews by red letters, and effect purviews by green letters. The order of the mechanism or purview is the number of units that constitute it, which corresponds to the z-axis “level” in the figure. Distinctions are depicted by orange lines linking a cause purview (left side) and an effect purview (right side) through a mechanism (middle), and the thickness of the line reflects the φd value. Relation faces are depicted as lines (for second degree) and planes (for third degree), whose color intensity reflects the value of φr (the large number of relation faces higher than third degree cannot be visualized). Note that the Φ-structure is “planted” on the complex through its first-order mechanisms and, like the complex, it is depicted in a blue palette. This is to indicate that a Φ-structure is not a mathematical abstraction, disconnected from the substrate; it is rather the fully unfolded cause–effect power of that substrate.

Together, a complex and the Φ-structure unfolded from it comprise an intrinsic entity. The distinctions and relations that compose the Φ-structure characterize what exists, and the way they are organized characterize which way it exists—what it is like. The φd and φr values measure how much the distinctions and relations exist within the structure. The sum of their φd and φr values is the Φ value (“big phi”) of the Φ-structure, which measures its total structure integrated information—how much the entity exists. 

What is meant by the term axiom in IIT?

If the axioms are “immediate” and “irrefutably true,” shouldn’t they also be self-evident?

Why do we assess the causal power of all orders of mechanisms? Why not simply assess the causal power of the individual units alone?

Does a Φ-structure "emerge" from a complex?

Does an experience "emerge" from its corresponding Φ-structure?