Information Integration Theory: Unified Psychology based on three mathematical laws*

Psychology is unique among the sciences in its joint concern with dual worlds: (a) internal world of feeling and thought, and (b) external world of stimulus and response. The first major movement in psychology, the introspectionist movement of the late 1800s, took the inviting direct attack of studying the internal world with conscious report. Introspective methods, however, led to disarray. They were replaced by an intolerant behaviorism that allowed only observables of the external world. Many important results were thus obtained as with animal conditioning and rote learning.


Mathematical laws of thought and action
Mathematical psychology is a solid reality.ree simple mathematical laws-adding, averaging, multiplying-have been demonstrated in experimental studies in most areas of human psychology, from psychophysics and learning, to social attitudes and moral judgment.ese three laws allow for personal values of each individual person.ey hold generally with young children and adults, and they have been demonstrated in nations around the globe.
Dreams of mathematical laws have haunted the imagination of many psychologists.Some have presented hopeful equations as early as Aristotle's equation for fair division between two persons, A and B, working on some mutual project: share for A ÷ A's Contribution = share for B ÷ B's Contribution.A simpler conjecture applies to the judgment of blame, ubiquitous in society, from family to politics: Modern judgmentdecision theory is grounded on the SEV law: Subjective Expected Value Subjective Probability ´ Subjective Value.
Psychological measurement is the critical obstacle towards testing such conjectures.All three terms in the blame equation, for example, are personal values in the blamer's head.us, Responsibility may be imputed by the blamer to the one being blamed on the account of carelessness or lack of forethought even though no actual harm was caused.Psychological measurement is thus necessary to test this blame hypothesis.We must get inside the blamer's head to measure his/her personal values of all three terms.e same applies to several other hopeful equations of thought and action that have been proposed.

Integration diagram
e essential ideas of Information Integration eory (IIT) are given in the Integration Diagram of Figure 1 and in the Parallelism eorem for adding-type integrations.In Figure 1, external stimuli, denoted by S, impinge on the organism and are transmuted into goal-directed psychological values, y. ese internal values are integrated into an internal response, r, which is externalized to become an observable response, R.

Figure 1 Information integration diagram
Chain of three operators, V -I -A, leads from observable stimulus field, {S}, to observable response, R. Valuation operator , V, transmutes stimuli, S, into subjective representations, y.Integration operator , I, transforms subjective field, {y}, into internal response, r.Action operator , A, transforms internal response, r, into observable response, R (aer Anderson, 1981).
Source: own work e effectiveness of this approach depended on two things.First, on the development of the method of functional rating response (see Appendix A), which eliminates well-known biases of ordinary rating methods and validates Premise 2 of the parallelism theorem below; and Second, the miraculous beneficence of Nature which endowed the mind with three simple algebraic laws of stimulus integration.

Parallelism eorem
Analysis of adding-type integration models, such as the foregoing blame equation, is given by the parallelism theorem.Two stimulus informers are presented in an ordinary row ´ column factorial design.e subject responds to each cell in this design as illustrated in Figure 2 below.e parallelism theorem requires two premises.Premise 1: e internal integration is additive: r = y A + y B .Premise 2: e action operator is linear: R = c 0 r + c 1 r.(Here c0 and c1 are inessential zero and unit constants.)ese two premises imply that the row ´ column graph of R will show parallel curves (see following figures).
Observed parallelism thus provides a cornucopia of benefits.Benefit 1. Support for an adding-type model (Premise 1).Benefit 2. Support in favor of R as a true linear measure of internal response, r (Premise 2).Benefit 3. Support that the mean response in row j (column k) of the integration design is a true measure of y Aj (y Bk ).Benefit 2 solves the long-standing obstacle of true measurement of response .Benefit 3 solves the obstacle of true measurement of stimulus .is is called functional measurement , because it measures the values that functioned in the integration process.

Empirical applications
e three mathematical laws of Information Integration eory have done well in almost every field of human psychology, from judgmentdecision and learning to social attitude theory, person science, and moral judgment.A few examples of adding-type laws are noted here.
Blame .Blaming follows an adding-type law even in young children as shown by the parallelism in Figure 2. Subjects judged an amount of blame for a child who threw a rock with specified intent (malice, displacement, accident) that caused four degrees of harm to another child (horizontal axis).e parallelism supports the additive law: Blame = Harm + Responsibility.
is blame law has been confirmed and extended by several investigators.It holds in legal judgment (Anderson, 2015, Chapter 4).A similar integration law holds with fair shares in adults and children as young as 4 years of age (Anderson, 2015, Chapter 5).Source: own work Piaget's stage theory of child development, as well as that of Kohlberg, had thus been shown to be seriously invalid.Stage theories claim that development falls into more-or-less discrete stages, claims that can be extremely seductive.But Piaget and Kohlberg both relied on verbal rationalization of choices in moral dilemmas, a fatal mistake that was revealed as soon as the methods of Information Integration eory were applied.
Functional memory versus Reproductive memory .An essentially new conception of memory emerged from the integration laws.
Traditional memory research took for granted that memory was remembering -accurate reproduction or recognition of given material to be memorized.
is traditional view was universally accepted.In social attitude theory, for example, it was long an "article of faith" that the attitude produced by a message was determined by what of the message remained in memory.
Instead, a dissociation between attitude and memory was found in a 1963 experience on IIT (see Figure 3).e recalled data showed the standard recency effect : the later items in the message were better remembered.In sharpest contrast, the earlier items had the greater effect on the attitude produced by the message items-a primacy effect .is dissociation has been widely supported.Person cognition .Interpersonal interaction is ubiquitous in everyday life: family, work, politics, and TV.Person cognition has been studied in numerous studies of experiments on information integration.In its simplest form, a hypothetical person is described by a set of trait objectives; the subject judges, for example, the likeability of the person.But the parallelism implies that each adjective had a constant valuerelative to the operative goal, regardless of which other adjective it was paired with.is meaning invariance was verified in other experiments in which subjects wrote a paragraph describing the person in their own words before rating the person.is would destroy the parallelism if the adjectives interacted to change one another's meanings.
But parallelism was still obtained, as shown also by other investigators.is result also shows how integration theory can go below consciousness in the analysis of cognition.Indeed, nonconscious influences can be exactly measured (Figure 6).
Science of Phenomenology .Phenomenology can be made scientific with the integration laws.e idea that the mind can be analyzed with verbal report has remained popular despite the failure of the early introspection approach.Indeed, verbal report is the medium of everyday communication.
However, the failure of verbal report, even in the simple experiment of Figure 4, shows that it can be quite untrustworthy.is crux can be resolved with functional measurement theory; the successes of the integration laws validate the method of functional rating (Benefit 2 of the parallelism theorem).ese laws make it possible to develop the science of phenomenology.
Cognition unitization and marital interaction .Complex stimuli can be treated as cognitive units, exactly measureable, by the integration laws.e valuation operation may be quite complicated, but its end result is a single number that can be exactly measured with an integration law.
One illustration of cognitive unitization comes from a study of wifehusband discussion.In the first phase, both spouses received a common scenario in which a child had performed a harmful action with a certain intent.Each spouse made a private judgment about the badness of this action.Next, husband and wife received separate, private information, slightly negative for one spouse, moderately extenuating for the other.ey then discussed their own opinion and their added information with each other.Finally, they made private, revised judgments of badness.
Results are shown in Figure 5. e le side shows the private initial blame judgments, separately for wives and husbands.e parallelism of the three curves supports an adding-type law: Blame = Damage + Intent.e right side shows the private revised judgments, somewhat lower because the main information added in the wife-husband discussion was extenuating.Both spouses again show parallel curves, further support for the blame law.Lo, Med -, Med + , and Hi represent graded levels of damage (aer Anderson, 2015).
Source: own work e wife-husband interaction thus acted as a cognitive unit for each separate spouse in the judgment of blame even though this verbal interaction was very complicated, beyond detailed analysis.Hence, it could be numerically measured separately for each spouse (see Studies of Marriage in Anderson, 2008).
Cognitive unitization has also been demonstrated in other experiments, for example, with judged statesmanship of U.S. presidents described by biographical paragraphs, and with witness testimony in a jury trial.Cognitive unitization is invaluable because it allows exact analysis of complex stimulus fields and complicated mental processing.
Measuring the Nonconscious .Most cognitive processing is nonconscious or semiconscious.It can be exactly measured, however, by using integration experiments.A simple example from psychophysics is the size-weight illusion of Figure 6.But people are hardly aware that their conscious experience is influenced by the irrelevant size, much less that this nonconscious effect can be exactly measured.e integration laws thus provide a foundation for science of nonconscious cognition.

Multiplying Laws: Linear Fan eorem
Multiplying laws follow a linear fan theorem , similar to the parallelism theorem.is application of functional measurement successfully solved the long-standing conjecture: Subjective expected value = Subjective probability ´ Subjective value.
Multiplying laws have also been found in many areas: poker betting, snake phobias, and psycholinguistics (see Figures 1. 13-1.19 in the 1981 Foundations volume).Multiplicative integration appears to be an innate mode of stimulus integration.

Figure 2
Figure 2 Parallelism supports the averaging law: Blame = Harm + Responsibility Graph plots rated naughtiness of a story child who threw a rock with one of three intents (curve parameter) producing one of four levels of harm (horizontal axis).Le panel shows third-, fih-, and seventh-graders; right panel shows college students.No age trends except perhaps the apparent increase in main effect of Intent (aer Anderson, 2008).Source: own work

Figure 3
Figure 3 Functional memory differs conceptually and empirically from traditional verbal memoryRecall curve for the adjectives in person description shows strong recency over last six serial positions, a standard result.Judgment curve for the effect of these same adjectives in person cognition shows uniform primacy, with lesser effects at later serial positions.Contrast between recall recency and judgment primacy implies a basic difference between person memory and verbal memory (aerAnderson, 2015).Source: own workWhat is important in cognition are the values of the stimuli, as constructed by the goal-directed valuation operation in the Integration Diagram.ese values function in the response.Such functional memory should be a primary concern of memory theory.Person cognition .Interpersonal interaction is ubiquitous in everyday life: family, work, politics, and TV.Person cognition has been studied in numerous studies of experiments on information integration.In its simplest form, a hypothetical person is described by a set of trait objectives; the subject judges, for example, the likeability of the person.Figure4summarizes judgments of social desirability of hypothetical persons described by two of their traits, listed in the figure.e parallelism reveals simple adding-type integration.

Figure 4
summarizes judgments of social desirability of hypothetical persons described by two of their traits, listed in the figure.e parallelism reveals simple adding-type integration.

Figure 4
Figure 4 Parallelism pattern supports adding-type rule in person perception Subjects judge the likability of hypothetical persons described by two trait adjectives from an indicated Row ´ Column design.Each of these 3 ´ 3 = 9 person descriptions corresponds to one data point.Data averaged over third adjective for simplicity (aer Anderson, 1981).Source: own work Consciousness gives a very different picture.People strongly feel that trait adjectives interact to change one another's meaning.us, preoccupied might seem to have a different meaning in an earnest person than in an unproductive person.Not a few researchers have strongly agreed.But the parallelism implies that each adjective had a constant valuerelative to the operative goal, regardless of which other adjective it was paired with.is meaning invariance was verified in other experiments in which subjects wrote a paragraph describing the person in their own words before rating the person.is would destroy the parallelism if the adjectives interacted to change one another's meanings.But parallelism was still obtained, as shown also by other investigators.is result also shows how integration theory can go below consciousness in the analysis of cognition.Indeed, nonconscious influences can be exactly measured (Figure6).Science of Phenomenology .Phenomenology can be made scientific with the integration laws.e idea that the mind can be analyzed with verbal report has remained popular despite the failure of the early introspection approach.Indeed, verbal report is the medium of everyday communication.However, the failure of verbal report, even in the simple experiment of Figure4, shows that it can be quite untrustworthy.is crux can be resolved with functional measurement theory; the successes of the integration laws validate the method of functional rating (Benefit 2 of the parallelism theorem).ese laws make it possible to develop the science of phenomenology.

Figure 5
Figure 5 Independent judgments of blame by husbands and wives Initial judgments based on information about intent (curve parameter) and damage (horizontal axis).Revised judgments based on additional information presented by spouse.

Figure 6
Figure 6 Parallelism supports adding-type rule for size-weight illusion Subjects li and judge heaviness of cubical blocks in 3 ´ 5, Gram Weight ´ Block Size design.Verbal rating in le panel, graphic rating in right panel.e slope of the curves provides a true linear measure of the nonconscious heaviness effect of visual size.(aer Anderson, 1981).Source: own work e top curve in each panel shows the judged heaviness of a lied 250-gram cubical block of five different sizes.e upward slope of the curve shows that the same 250-gram weight feels substantially heavier in a smaller size.(Use an ounce of white feathers and an ounce of lead for a sure-fire class demonstration.)e parallelism of the three curves supports an adding-type model: Heaviness = Size + Weight.But people are hardly aware that their conscious experience is influenced by the irrelevant size, much less that this nonconscious effect can be exactly measured.e integration laws thus provide a foundation for science of nonconscious cognition.