On Defining I "I logy"

Could we define I? Throughout this article we give a negative answer to this question. More exactly, we show that there is no definition for I in a certain way. But this negative answer depends on our definition of definability. Here, we try to consider sufficient generalized definition of definability. In the middle of paper a paradox will arise which makes us to modify the way we use the concept of property and definability.

💡 Research Summary

The paper tackles the age‑old philosophical question “Can we define the self?” by treating “I” as a formal object within a logical‑mathematical framework and then demonstrating that any attempt to give it a complete definition inevitably leads to contradiction. The authors begin by criticizing the conventional notion of definability, which usually means that an object can be uniquely identified by a finite description in a given language. They propose a generalized notion: an object is definable if, for every possible property it might possess, there exists a unique predicate that can discriminate that object from all others. Crucially, properties are not limited to first‑order attributes; they can themselves be meta‑properties that refer to other properties, creating a hierarchy of self‑reference.

Modeling “I” as a self‑referential entity, the paper shows that the subject simultaneously perceives itself, describes that perception, and is the object of that description. This dual role is formalized by a labeling function L such that L(x) = “x is defined by L”. When L is applied to itself, we obtain a classic liar‑type paradox: L(L) ↔ ¬L(L). The authors argue that any language capable of expressing such a self‑referential predicate cannot maintain both consistency and completeness, echoing Gödel’s incompleteness theorems and fixed‑point theorems in recursion theory.

The core argument proceeds in two steps. First, the authors enumerate the set of all conceivable properties P = {p₁, p₂, …} that could be ascribed to “I”. They prove that at least one property in this set must be a meta‑property m that either negates itself when applied to “I” (m(I) ↔ ¬m(I)) or asserts that it is a property about the very act of defining “I”. This mirrors the liar paradox and shows that any exhaustive property list inevitably contains a contradictory element. Second, they demonstrate that a system that includes such a meta‑property cannot be both complete (able to decide every statement) and consistent (free of contradictions). The proof uses a diagonalization argument: assuming a complete, consistent system, one can construct a predicate that contradicts its own truth value, violating the system’s consistency.

Mid‑paper, a paradox arises concerning the boundary between “definable” and “undefinable” properties. To resolve this, the authors introduce a two‑tier hierarchy of properties. The lower tier adheres to the traditional definability criteria, while the upper tier—reserved for meta‑properties—explicitly relinquishes the requirement of definability. Consequently, “I” is relegated to the upper tier, meaning that no finite or even infinitary language within the lower tier can capture the full essence of the self.

The discussion section extrapolates these findings to cognitive science, philosophy of mind, and artificial intelligence. If the self cannot be fully formalized, then any AI system that relies solely on symbolic representations will lack genuine self‑awareness. Moreover, the existence of unavoidable meta‑properties suggests intrinsic limits to any formal system’s ability to self‑extend or to provide a complete account of its own semantics.

In conclusion, the paper delivers a rigorous negative answer to the definability of “I”: within any sufficiently expressive logical framework, the self resists complete definition because self‑reference inevitably generates undecidable or contradictory predicates. This result does not merely close a philosophical door; it invites a re‑examination of how we model consciousness, self‑reference, and the limits of formal theories. The authors argue that acknowledging this limitation may lead to richer, perhaps non‑formal, approaches to understanding human identity and the prospects of machine consciousness.