Quadratures: Person, Subject, Agent and Object

In these  pages I frequently refer to and build upon a structure with four “modes” or “terms” – namely those of Person, Subject, Agent and Object. At its core, this model is not original, and stems from the work of many disparate authors, in different areas of knowledge.

For the sake of rigour and completeness, it is important to at least mention the key references and comment some aspects which will help understand the meaning of this formulation. This review will also help the reader to see what elements of my approach are derivative and which ones may be original, potentially also showing where my interpretation of the subject may have gone astray.

Initially my work was focused on the study of oppositional structures within organisations, and more specifically in the context of systems design and decision making. This lead to a model of Information Security which I called the “Security Perspectives” sometime in 2005.  A full account of this approach can be found here: http://carlos-trigoso.com/fundamental-conceptions-of-information/ .

Repeated use and refinement of this model allowed me to see more underlying structure in what seemed first only a permanent characteristic of business organisations. The “geometry of logic” –as studied by Alessio Moretti—stimulated more research beyond the strictly organisational or sociological context.

Moretti studies how opposition is not limited to negation, but also to entailment (implication) and contradiction. His work, as well as that of other logicians and philosophers: Fabien Schang,  Régis Pellissier  and Hans Smessaert is my primary source in this respect.  Moretti has pointed to the fact that this logical structure may have an even deeper model, fundamentally equivalent to Jean Piaget’s “logical capacities” square  and Walter Helbig Gottschalk’s “theory of quaternality.”  In other words, a statement is inevitably and logically linked with its negation, its contradiction and its implications. The oppositions occur in the unconscious, in a “zone” of the mind that could be described as a “verbal unconscious” for lack of a better term. Although this is not orthodox Lacanian theory, I think the French psychoanalyst probably was pointing to such a relation when he said the unconscious is “structured like a language.”

Moretti and the N-Opposition theoretical circle (led by the researchers quoted above ), start from the foundations of modal logic, in particular the so-called Square of Oppositions. This “square” representing negation, contradiction and entailment of logical statements, originated with Aristotle and took its definitive classical form with Apuleius. I won’t summarise here the vast research that exists on this subject, but it is important to know that the immense interest that this has in philosophy, logic, mathematics and other sciences cannot be exaggerated. The N-Opposition theory website has many references to researchers working on the implications of the Logic Square in their own areas.

Comparing Moretti’s approach with those of other scholars, I was able to recognise that there were at least three types of formulations regarding the Square of Opposition: those based on numerical speculation, those based on logic and those starting from descriptive classifications. I believe that—while they may be useful as study materials—both numerical and descriptive structures of “four factors” or “four terms” are inconsistent and arbitrary.

Numerically-based structures tend to give a special meaning to the number “four” perhaps underlining traditional sources and associations with mythology. In this “tradition” we can find some interesting but a-critical lists of quaternities. These tend to equate very different sources only because of the fact that all these seem to contain a four-based group of terms.

The key point to understand in this respect is that there is nothing special about the number four (or any other number), and if there is any relevance in the Quadrature it is not because of the number but because of the geometry it supports (i.e. the fact that we can observe the symmetries pertaining to the Klein four-group).  According to Rudolf Kaehr’s interpretation of Gotthard Gunther, “Each single value and each single logical function is entitled to have a logical meaning.” Gunther –and Kaehr after him– proposed that it is absurd to chase for the “meaning” of logical values and functions for arbitrary many-valued systems. Hence, a method, an arithmetic position system which is able to determine arbitrary numbers on a finite base system, has to be invented. This was Gunther’s approach to many-valued place-value systems (Stellenwertlogik).

In a similar way, as some researchers tend to become enthralled with the number four, others focus on the number three with similar reasons, mainly attributing some metaphysical value to the trinity or the triangle. But numerically-based speculation of this kind ignores the fact that—in geometric terms—structures of four and three terms are deeply interrelated, to the point that it can be said that quaternities entail trinities and vice-versa. In some cases apparent trinities can be shown to contain or point to a quaternity, as in the case of Franz von Baader: “Baader’s use of the term quaternity does not in any way involve an idolization of mathematics. He uses the symbol of a triangle with a point in the middle. That is the Quadrat. In one way, he seems to use it to show the dynamism of the Trinity. The “fourth” is the underlying inner unity of the Trinity. The complete triad is a quaternity. the fourth element is not on the same level; rather, a ground for the other 3, source of its unity (Franz von Baader, Werke 2,243).” (See: Glenn Friesen, “Quadrat,” – http://www.members.shaw.ca/jgfriesen/Definitions/Quadrat.html )

This is a good place to suggest, following the work by Professor Stephen Palmquist,  that while numbers are not privileged in a specific way, odd numbers, in particular 3 and 5, can be associated with “synthetic” models, while even numbers, like 2, 4 and 8 can be shown to be “analytic.”  In fact, even numbers just show clearly in their geometrical structure the underlying distinction mechanism (dichotomic logic), while odd numbers and the figures they represent (triangles and “stars”) hide the underlying dichotomies and present an image of unity. For example, the logics of Georg W. Hegel  and Charles S. Peirce  are synthetic in the sense that they propose a “mediation,” a third entity bridging the gap between the initial dichotomy (e.g. in Peirce “firstness” and “secondness”). In comparison with these, the logics of Immanuel Kant and Heidegger are “analytical” as they keep the dichotomic structure in view for example in Kant’s “four judgements”  and in Heidegger’s “Geviert.”

Understanding how human cognition is based on distinctions is not new and in fact has been useful in various areas of research. Notable examples of these applications are Chris Lofting’s work on neuro-cognitive roots of logic  and Anthony Judge’s discussion on conceptual polarities,  but before these authors we had also the work of Will McWhinney  on organisational change and the “personal construct” hypothesis of the American psychologist George A. Kelly.  Other relevant authors are listed in the  bibliography at the end of this post.

Classification-based structures show a different problem: the lack of sufficient structure. In many cases researchers using this approach manage to present very interesting material, but there is scant justification either for the subject represented or the way the material is structures. The number four or thematic concept maps based on four-symmetry are presented as frameworks or models for different areas of knowledge. This is very frequently the case in Management Consulting disciplines where specialists are fond to use “quadrants” as a mechanism for topic classifications.

The models may differ and the terminology is diverse, but we are familiar with this type of graphic classification, where we correlate some features against others.  Obviously, we take as a reference (sometimes unconsciously) the system of so-called Cartesian coordinates (x and y dimensions arranged on the geometrical plane), by which we can associate values on one dimension (e.g. “x”) against another (e.g. “y”). More ambitious models will add other dimensions: three, six or eight (for example in the “radar chart”).  What is noteworthy though is the underlying mechanism of such models. This is rarely discussed in technological milieus, but in fact every quadrant-based model and their variations are based on a “logic of distinctions.” So, for example, two dimensions will always determine four quadrants. The choice of dimensions used for the comparison is arbitrary, but obviously the consultant will try to make sense by choosing some close to the subject under discussion.

In this case the logic structure or the symmetry are not essential, and the model seeks instead to be “exhaustive” in representing the possible sub-topics of a particular area of interest.

More decisive for my own approach, although strongly characterised by this exhaustive and descriptive approach here portrayed, are the models developed by Stephen C. Pepper, Richard Jung and Derek Cabrera.

Pepper’s model of evidence and judgement –comprising the categories of Formism, Organicism, Mechanism and Contextualism- and Richard Jung’s hermeneutic metaphors of Mind, Organism, Mechanism and Template, can be neatly aligned to show surprising conceptual parallels.  My own “Four Security Perspectives” were initially based on these authors’ theories. Following Pepper’s model, I formulated the perspectives as follows:

•    Formism: Direction, i.e. definition of Trust, Risk-taking
•    Organicism: Selection, i.e. allocation of Trust, Risk sharing
•    Mechanism: Protection, i.e. enforcement of Trust, Risk avoidance
•    Contextualism: Verification, i.e. verification of Trust, Risk monitoring

Later –learning from Richard Jung’s approach–I translated the initial Four Perspectives to what I called “systemic action metaphors” i,e. interrelated “modes of action”:

•    Systemic action as distinction (value function or Mind in Jung’s terminology)
•    Systemic action as membership (relationship function – Organism)
•    Systemic action as object (material flow – Machine)
•    Systemic action as context (process flow – Template)

In this context it is also important to mention the work of Derek Cabrera. While also part of the descriptive-exhaustive class of models, Cabrera manages to build a system of interrelated categories which is quite satisfactory. The only limitation being that there doesn’t seem to be a logic underpinning this construct, or at least the logic is not manifest.

By aligning the concepts of Identity and Difference, Part and Whole, Affect and Effect, and Objective and Subjective, Cabrera is able to show a powerful oppositional structure which then allows him to develop a wide-ranging discourse on subjects as diverse as education, research and systems design. (See: Derek Cabrera, “Remedial Genius,” 2005)

Many more authors –some of which are listed in the references below—have similar approaches. It becomes important then to avoid the enumerative and descriptive limitations so to show where many intuitions are effectively rooted in an implicit or hidden logic. This is where I think that we can establish a connection between the logic-based and the classificatory schemes mentioned above. To do so it is only necessary to align the multiple classifications and intuitive suggestions found in the literature with the much more powerful framework of modal logic and the Square of Opposition. The following Diagram summarises my own approach to this kind of mapping:



The essence of this depiction is the logical structure generated by the superposition or “re-entry” of two classes of negation. First “classical” negation ( A and ~ A, shown in the horizontal axis), and second “negation as complement” (the complement of A and the negation of this complement, shown in the vertical axis).  The diagram also shows how to align the concepts of Identification and Differentiation with the horizontal axis, and the concepts of intensive and extensive abstraction with the vertical axis.

In the whole, the aim is to give a formal, logical infrastructure to any possible categorization or combination of categories as those shown here, for example Abstract/Concrete or Particular/Universal. Cabrera’s categories definitely align with (for example) the concepts of objective and subjective, as well as those of Identification and Differentiation in the arrangement shown above.

While the logic of the Square of Opposition and the work of Moretti and others underpins my research, I must say that the idea of overlapping two kinds of negation was due to my study of the works of Sri Aurobindo, in particular some parts of his “Letters on Yoga” and  “The Life Divine,” from where I present the following extracts:

“It all depends upon where the consciousness places itself and concentrates itself. If the consciousness places or concentrates itself within the ego, you are identified with the ego—if in the mind, it is identified with the mind and its activities and so on. If the consciousness puts its stress outside, it is said to live in the external being and becomes oblivious of its inner mind and vital and inmost psychic; if it goes inside, puts its centralising stress there, then it knows itself as the inner being or, still deeper, as the psychic being; if it ascends out of the body to the planes where self is naturally conscious of its wideness and freedom it knows itself as the Self and not the Mind, life or body. It is this stress of consciousness that makes all the difference. That is why one has to concentrate the consciousness in heart or mind in order to go within or go above. It is the disposition of the consciousness that determines everything, makes one predominantly mental, vital physical of psychic, bound or free, separate in the Purusha or involved in the Prakriti” – “Letters on Yoga,” pages 235 and 236.


“But what is this strongly separative self-experience that we call ego? It is nothing fundamentally real in itself but only a practical constitution of our consciousness devised to centralise the activities of Nature in us. We perceive a formation of mental, physical, vital experience which distinguishes itself from the rest of being, and that is what we think of as ourselves in nature—this individualisation of being in becoming. We then proceed to conceive of ourselves as something which has thus individualised itself and only exists so long as it is individualised, –a temporary or at least a temporal becoming; or else we conceive of ourselves as someone who supports or causes the individualisation, an immortal being perhaps but limited by its individuality. This perception and this conception constitute our ego-sense, we go no farther in our knowledge of our individual existence.” — “The Life Divine,” page 367.

According to Aurobindo’s philosophy, it is possible to distinguish then four methods or paths of consciousness:

-Knowledge by identity

-Knowledge by intimate direct contact

-Knowledge by separative direct contact

-Knowledge by indirect contact

While the first class of knowledge arises from complete identification with the object known, the second is a form of knowledge where the sense of identification is greater than the sense of differentiation. The third implies that the sense of identification is lesser than the sense of differentiation, while the fourth depends on a complete sense of differentiation. (See: Sri Aurobindo, “Life Divine, Knowledge by Identity and Separative Knowledge”)

It can be noted that these four forms represent a double entry or superposition of the same differentiation-identification dimension. I believe that this “re-entry” of the identification-differentiation dimension can be fully represented with the two modes of negation used in the schema above.

More labour is necessary to explain how the various quaternities and categorisations mentioned in this page (e.g. Pepper’s and Jung’s ) match this quadrangular model, but this will be the subject of successive articles to be published in these pages. This work should also cover the different contributions of the authors listed in the references and other sources.

It remains to say here that the terms Person, Subject, Agent and Model are the labels I use to refer to the four aspects generated by the logical model. These labels owe a lot to Jacques Lacan’s “four discourses” –known as those of the Barred Subject, the Agent, the Other and the Product—, but do not represent any reformulation of the Lacanian theories.


Aurobindo, Sri “Letters on Yoga,” Collected Works of Sri Aurobindo, 1972

Blanché, Robert “Structures intellectuelles – Essai sur l’organisation systématique des concepts,” ed. J. Vrin, 1966

Burrell, G. and G. Morgan, “Sociological Paradigms Organizational Analysis,” Heinemann Press, London, 1979

Cabrera, D. ”Systems Thinking: Four Universal Patterns of Thinking,” Ithaca, NY: Cornell University, 2006

Canterbury, Anselm of “Complete Philosophical and Theological Treatises of Anselm of Canterbury,” translated by Jasper Hopkins and Herbert Richardson, The Arthur J. Banning Press, Minneapolis

Dhillon, G. A. “Interpreting the Management of Information Systems Security,” University of London, 1995

Fiske, Alan. P., and Haslam, N. “The Four Basic Social Bonds: Structures For Coordinating Interaction,” in Interpersonal Cognition, 2005

Gottschalk, Walter Helbig “Theory of Quaternality,” 1953

Hegel, G.W.F, “Wissenschaft der Logik,” 1812,

Heidegger, Martin “The Thing,” 1950

Hirschheim, Rudy and Heinz K. Klein, “Four Paradigms Of Information System Development,” Communications Of The ACM, October 1989

Hohfeld, Wesley Newcomb “Fundamental Legal Conceptions,” 1978

Jung, Richard “A Quaternion of Metaphors for the Hermeneutics of Life,” 1985

Kant, I., “Critique of Pure Reason,” 1781

Kaehr, Rudolf – http://www.thinkartlab.com/pkl/lola/Transjunctional%20Semiotics/Transjunctional%20Semiotics.html

Lacan, Jacques, “Encore,” Séminaire Livre XX, Paris 1975

Lofting, Chris “The Neurocognitive Roots of Logic,” 2003

Maruyama, Magoroh “Individual Types: Subcultural Or Transcultural,” The General Psychologist, 2001

McFadzean, Elspeth, Ezingeard, Jean-Noël and Birchall, David, “Anchoring Information Security Governance Research: Sociological Groundings and Future Directions,” Henley Management College, 2004

McWhinney, W. “Paths of Change: Strategic Choices for Organizations and Society,” 1992

Moretti, Alessio “The Geometry of Logical Opposition,” 2009

Morgan, Gareth and Burrell, Gibson, “Sociological Paradigms and Organisational Analysis,” 1979

Palmquist, Steven “The Combination Of Analysis and Synthesis In Numerical Symbolism,” Kant on the Web and The Geometry of Logic http://staffweb.hkbu.edu.hk/ppp/

Peirce, C. S. “Collected Papers,” Harvard University Press, 1948

Pepper, Stephen C. “World Hypotheses – A Study in Evidence,” 1942

Piaget, Jean “Traité de logique. Essai de logistique opératoire,” 1972

Rotman, Brian “Mathematics as a Sign,” Stanford University Press, 2000.

Trigoso, Carlos, “Security Perspectives” – http://carlos-trigoso.com/public/security-perspectives/

Trigoso, Carlos, “Four Perspectives on Risk and Trust” – http://carlos-trigoso.com/public/four-perspectives-on-risk-and-trust/

Uckelman, Sara L. “Anselm’s logic of agency,” Institute for Logic, Language, and Computation, 2009

Wallace, A.F.C. “On Being Just Complicated Enough,” 1961

Walton, Douglas “St. Anselm and The Logical Syntax of Agency,” Franciscan Studies, 1976

White, Hayden “Metahistory: The Historical Imagination in Nineteenth-Century Europe,” The Johns Hopkins University Press, 1973