A Formula For Discovering & Deconstructing conceptual/Semantic hierarchies In the Quran

In the Quran, the broadness of linguistic import presents a problem—and a fruitful one at that. The acknowledgment of which constitutes a first step towards its solution, towards understanding the corpus and the weltanschauung that underlies its construction. To take an example: ask, why would the word 'Fulk,' by a slight diacritical modification, be applicable in as many different settings as it is—settings so different from one another that the unity of the word is indeed called into doubt?

Surah 7:64: "فَكَذَّبُوهُ فَأَنْجَيْنَاهُ وَالَّذِينَ مَعَهُ فِي الْفُلْكِ"(But they denied him, so We saved him and those who were with him in the ship)

Surah 21:33: "وَهُوَ الَّذِي خَلَقَ اللَّيْلَ وَالنَّهَارَ وَالشَّمْسَ وَالْقَمَرَ ۖ كُلٌّ فِي فَلَكٍ يَسْبَحُونَ"(And it is He who created the night and the day and the sun and the moon; all [heavenly bodies] in an orbit are swimming)

Surah 31:31: "أَلَمْ تَرَ أَنَّ الْفُلْكَ تَجْرِي فِي الْبَحْرِ بِنِعْمَتِ اللَّهِ لِيُرِيَكُم مِّنْ آيَاتِهِ ۚ إِنَّ فِي ذَٰلِكَ لَآيَاتٍ لِّكُلِّ صَبَّارٍ شَكُورٍ"(Do you not see that ships sail through the sea by Allah's Grace that He may show you some of His Signs? Verily, in this are signs for every patient, grateful person.)

Words with this broadness of import, and cross-domain application are generally known as polysemous.¹ Atomic words are polysemous, if by polysemous we mean a word that changes depending on its context. For example, the word "paper" can refer to the substance or a printed publication made of that substance, or how the word "bank" can mean a financial institution or the side of a river—both meanings are connected to the idea of "storage" or "edge."

Atomic words, though polysemous, extend beyond this point into polysynthesis, or polythetic language structures—a feature of language characterized by complex words consisting of several morphemes, in which a single word may function as a whole sentence.² Thus atomicity is a more technical, modificational, syntactic framework that links polysemy with polysynthesis. Quranic words are atomic, employing a polysynthetic strategy, in that the modifications applied to its terms (Fulk → Falak) are morphological operations that polysemously generate context-specific meanings from a single root.

Generality = Identity

Reading through these examples, the question naturally arises: after 'why the same word in different senses' and settings, what do these have in common such that the same word would suffice, by a slight modification?

Fulk may be understood as an orbit or course which a planet or celestial object follows, a ship or vessel, or in Noah's case, a vessel built to protect from a flood. The importance of understanding the atomic concept really shows here. We must look for that function that's general enough to be applicable in all three contexts.

The concept that's general is a concept that's more malleable relative to a concept that's less general. In mathematics, we look for averages because averages allow us to see patterns, compare and predict—i.e., to apply a more detached and transcendental mode of thought to otherwise rigid and opaque quantities. They allow us to perceive in our object an order of relations that does not involve knowing every detail, but rather—and this is the borderline miraculous thing about atomic concepts—to reconstruct details a priori once the idea we call atomic is grasped.

Word-analysis and finding the Phi-Point: Weaponizing Non-Commutativity

We can formalize this search for generality through non-commutativity testing—a method apply to formalize our approach to word analysis. If the relatively more general 'sense' of a concept contains the relatively more specific, and if we're dealing with the same sense of a word in different contexts - polysemously - how do we determine which of these multiple contextual 'meanings' of the same word is more general, and which, more specific?.

There's clearly a conceptual hierarchy in the order of polysemous word-use, what the non-commutativity formula does is reveal dependency asymmetry.¹ The test is straightforward:

• Does concept A depend on concept B?

• Does concept B depend on concept A?

If A→B but B↛A, the relation is non-commutative (asymmetric). This asymmetry reveals hierarchy.

Take a simple example: birds and wings. Does the concept of "bird" depend on the concept of "wing"? Yes—we can't fully conceptualize what a bird is without reference to wings (or their analogues). But does the concept of "wing" depend on "bird"? No—wings exist as a broader aerodynamic principle that applies to insects, aircraft, seeds, and countless other instances.

The result: wing is broader, more fundamental than bird.

Why this matters: In domain analysis, non-commutativity identifies the phi-point—the concept with maximum asymmetric dependencies pointing toward it. Everything else depends on the phi-point, but the phi-point doesn't depend on them. This is the organizing principle we're searching for, the atomic concept itself.

We can express this formally: For a concept C to be the phi-point of its domain:

∀x(x→C) ∧ ¬∃y(C→y)

Everything points to it; it points to nothing. Or more precisely: all variants in the domain depend on C, ∀x(x→C), but C depends on no particular variant ¬∃y(C→y).

Applied to Fulk: Does "ship" depend on the concept of protected trajectory through adverse medium? Yes. Does "orbit" depend on this same concept (being that the adverse medium is disorder or amorphousness)? Yes. Does "Noah's ark" depend on it? Yes. But does 'protected trajectory through adverse medium' (in a word 'Vehicle') depend on any of these specific instances? No. It needs to be 'independent' if it's to qualify them in that way. We know that all three cases are contextually unique and different, but we're simultaneously aware of the 'common denominator', 'defined trajectory through adverse medium (geodesic path in physics, or entelechy in Aristotelian metaphysics 'if we 'physicalize it': the path of least resistance).

The fact that 'protected trajectory through adverse medium' does not depend on any of these scenarios is precisely what makes it applicable or generalizable over, or quantifiable over all of them, hence our use of the notations ∀x and ∃y. To put it differently "It's what all the instances require, but it requires none of them. This is not just statistical correlation (which is symmetric)—it's logical priority. Non-commutativity reveals causation where correlation only shows association."

Note: We mean something specific by 'denominator', something relevant to the notion of 'averages' -> you add all the elements and divide them by their sum. The sum is a denominator in that formula, and tells you what those would be quantities have in common. This idea will resurface when we get to Aristotle's Koinon (common sense).

Suddenly what can be obtained from so little, so simple an idea is incredibly vast. But as we noted before, to obtain that idea we must think inductively,² to investigate some word, persistently, in different contexts, preferably in contexts that are thematically different from each other, until we can see in the divergences a common strand. The true import of atomic words - (those words that satisfy the formula (∀x(x→C) ∧ ¬∃y(C→y)) - as we call them, seems to lie in its cross-domain presece. This cross-domain presence is a feature of fundamental logical structures that requires 'lateral thinking' skills -> precisely what Moses was had the Israelites do (Baqara, 67-71).

Moses' challenge => Here's this calf, I'll give you only minimum information, very general information, and I want you to go fetch it. You'll do the obvious, except the obvious is the most rational, inquire, and inquire, until you've reached a specific-enough definition or picture of what you seek.

What we're seeing in those verse of chapter 2 is the 'socratization' of society. All of it is contained in this tight little formula.

Notice that Abraham went through the same thing (Al-An'am 6:75-79), 'Not the planet', 'Not the moon', 'Not the sun', oh. I see. I believe. It's something you have to go through yourself. It cannot be taught, but only taught as that which you have to go through yourself.

Follow me to the next article, more in store....

• Polysemy: Lyons, J. (1977). Semantics, Volume 2. Cambridge University Press, pp. 550-569. [Classic linguistic treatment of polysemy and semantic range]

• Polysynthesis: Fortescue, M. (1994). "Morphology, Polysynthetic." In R.E. Asher (Ed.), The Encyclopedia of Language and Linguistics. Pergamon Press. [Standard reference on polysynthetic structures] ↩

• Non-commutativity and dependency structures: This draws on work in formal ontology and conceptual dependency theory. See: Guarino, N., & Welty, C. (2002). "Evaluating ontological decisions with OntoClean." Communications of the ACM, 45(2), 61-65. [On formal methods for determining conceptual hierarchy through dependency relations] ↩

• Inductive reasoning and concept formation: Mill, J.S. (1843). A System of Logic, Ratiocinative and Inductive. Book III: On Induction. [Classic text on inductive methods for arriving at general concepts from particular instances] ↩

• Surah Al-Baqarah 2:67-71 - The passage about the cow, where Moses instructs the Israelites through a process of specification and abstraction. ↩

• Surah Al-An'am 6:75-79 - Abraham's logical progression from observing stars, moon, and sun to recognizing the one God. Demonstrates reasoning from particular instances to universal principle. ↩