Integration

3rd axiom & postulate of IIT

Summary 

Integration is an essential property of experience (an axiom) and, by inference, also an essential property of the substrate of consciousness (a postulate). 

The axiom states that experience is unitary: it is a whole, irreducible to separate experiences. Formulated in physical terms, the postulate states that the cause–effect power of the substrate of consciousness must be unitary: it must specify its cause–effect state as a whole set of units, irreducible to separate subsets. 

Integration Axiom

Experience is unitary: it is a whole, irreducible to separate experiences. 

[This excerpt is taken from Tononi, G. (forthcoming). On Being.
Ch. 2: The axioms of phenomenal existence.] 

For example, I experience a whole scene, which cannot be reduced to seeing the left side of space separately from seeing the right side of space (as indicated in fig. 2.5.1). Said otherwise, the experience of space cannot be subdivided into separate parts, such that, say, the left side of space existed without the right side, or the other way around. The experience is always whole. Likewise, the scene I see is colored, and the experience cannot be reduced to seeing the scene without the color, or the color without the scene.

Integration is immediate because I do not have to infer the unity of the left and right side of space within the experience—their unity is right “in front of my eyes.” Integration is irrefutable, in the sense that its negation is self-contradictory or absurd. Consider an experience of a sight (say, a flash from a camera) and a sound (say, a bang from a gun) that have little to do with each other except for their co-occurrence in the experience. Would this experience falsify integration? No, because even in this case, I would have precisely a single experience of sight and sound, a flash–bang. This unitary experience of a flash–bang cannot be reduced to two separate experiences, one of the flash and one of the bang [1]. Thus, trying to conceive of an experience that were not unitary leads to conceiving of two separate experiences, each of which is unitary, which reaffirms the validity of the axiom. Finally, integration is true not only of my current experience, but it must be true of every conceivable experience; hence it is an essential property of phenomenal existence [2].

Footnotes

[1] Or think of two different consciousnesses, one experiencing the flash and the other the bang. In this case, there is no one who experiences both, which is quite different from there being one consciousness (me) experiencing both contents. [...][2] Both Descartes and Kant recognized unity as a fundamental property of consciousness. Descartes emphasized that mind—the “res cogitans”—was indivisible, whereas matter—the “res extensa”—was divisible. As he wrote in the Sixth Meditation, “the mind is utterly indivisible. For when I consider the mind, that is, myself insofar as I am only a thinking thing, […] I understand myself to be manifestly one complete thing.” In the synopsis of the Meditations, he also wrote, “we cannot understand the mind to be anything but indivisible. For we cannot conceive of half of a mind […]” (Descartes, R. (1984). The Philosophical Writings of Descartes. Trans. by J. Cottingham, R. Stoothoff, and D. Murdoch. Cambridge: Cambridge University Press. P. 89).In response to Hume’s skepticism about unity, Kant emphasized what he called “the transcendental unity of apperception.” In the Critique of Pure Reason, he wrote, “The synthetic unity of consciousness is, therefore, an objective condition of all knowledge. It is not merely a condition that I myself require in knowing an object, but is a condition under which every intuition must stand in order to become an object for me. For otherwise, in the absence of this synthesis, the manifold would not be united in one consciousness. […] this proposition makes synthetic unity a condition of all thought […]." (Kant, I. (2007). Critique of pure reason, second edition (2nd ed.; N. K. Smith, Ed.). New York: Palgrave Macmillan. p. 156.) 

Integration Postulate

The cause–effect power of the substrate of consciousness must be unitary: it must specify its cause–effect state as a whole set of units, irreducible to separate subsets of units.

[This excerpt is taken from Tononi, G. (forthcoming). On Being.
Ch. 3: The postulates of physical existence I.] 

Phenomenally, integration means that experience is unitary—it is irreducible to separate experiences. This essential property of experience can be formulated as a physical property of the substrate of consciousness by requiring that its cause–effect power must also be unitary: the cause–effect state specified by the substrate as a whole must be irreducible to that specified by separate subsets of units. This can be determined by assessing whether a partition of the candidate substrate affects the intrinsic information of that substrate—that is, its specific cause–effect power over itself. If so, a candidate substrate is in fact one substrate.

To establish integration in a principled manner, we follow another ontological principle of IIT discussed at the end of the chapter—the principle of minimal existence. The principle states that, when it comes to a requirement for existence, something cannot exist more than the least it exists. In the case of integration, a substrate cannot exist as one substrate more than it exists over its minimum partition. The minimum partition can be thought of as the "weakest link" or "fault line" of the system. As a simple analogy, consider an old rope that is heavily frayed in one section. That frayed section is the rope’s weakest point, and thus the rope as a whole cannot exist more than what that frayed section allows.

For a candidate substrate, the irreducibility of its specific, intrinsic cause–effect power is measured by integrated information over the minimum partition, yielding φs ("phi" with subscript s for "system"). As illustrated in the figure [below], a partition divides the substrate into two or more "parts" by severing connections among them. The partition among Ab, c, and d here (depicted by the dashed orange line and scissors) is the minimum because it makes the least difference to the intrinsic information specified by Abcd (other partitions, not shown, would all yield a φs greater than 1.32). Because φs for the minimal partition is greater than zero, the candidate substrate Abcd satisfies the integration postulate. It would not, however, if there were a partition for which φs were zero: a substrate cannot be one substrate if it can be subdivided into subsets of units that exist separately from one another [1].

In general, system partitions are made up of various unidirectional cuts, which sever all causal connections to or from one or more parts, as illustrated by the dashed connection arrows from part c to d [2]. The cuts can be unidirectional because, from the intrinsic perspective of a system, a subset of units must be able to interact with the rest of the system in both directions (cause and effect) to be truly a part of it. As a metaphor, think of a group having an audio conference call to arrive at a collective decision. If one of the participants cannot hear the others—so he cannot be affected by them—there will be no “group” decision of which he is part (as a true “participant”). The same is true if he can hear them but his line is muted, so he cannot affect them.

Note that the integrated information of a system is not only the minimum φs of all possible partitions, but also the minimum between the cause and effect sides of that partition. This is because, by the 0th postulate, existence requires both cause and effect power; hence a substrate cannot exist as one substrate more than the least it is integrated on the cause or effect side. In the example, therefore, φs = 1.32 corresponds to the minimum partition on the cause side. 

Integration Postulate - 08/2023

The integrated information of the system (φs) is thus a quantifier of integrated existence: a system only exists as one system to the extent that it cannot be partitioned (unidirectionally) without loss. [...] Integrated information is sensitive to a tension between expansion and dissolution: it can increase for a larger system as long as the system specifies higher integrated information across larger partitions (expansion), but it decreases if "fault lines" appear—partitions across which the system specifies low integrated information (dissolution).

[To see how the integration postulate is operationalized mathematically, see Computing Φ, Step 4: Compute integrated information (φs).]

Footnotes

[1] The idea that unity is necessary for existence can be found in some scholastic philosophers, and especially in Leibniz: “I consider as an axiom this self-identical proposition, diversified by emphasis only: that which is not truly one being is not truly a being at all” (Leibniz, G. W. [1988]. Discours de métaphysique et Correspondance avec Arnaud. Paris, Vrin. p. 165).[2] Bidirectional cuts can be considered as a special case of multiple unidirectional cuts. Note, however, that a cut must sever all connections to or from a given part. In the example here, all connections to part c are severed; it would not be possible to only sever, say, connection b to c while leaving A to c and d to c intact. 

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For the integration axiom, please cite Tononi, Giulio. Forthcoming. On Being. Chapter 2: The axioms of phenomenal existence.For the integration postulate, please cite Tononi, Giulio. Forthcoming. On Being. Chapter 3: The postulates of physical existence I.