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This work develops a new conceptual framework that questions assumptions long taken for granted in logic, such as self‑reference and same‑level evaluation. Its starting point is the principle of causality, understood as the reason why certain logical objects relate to one another in a unidirectional and asymmetric manner. This perspective inevitably leads to a dependent, hierarchical organisation that structures our thinking and shapes the way we understand the world.
The approach aims to offer a more general and integrative view than some of the historically most prominent solutions—among them, certain postulates of set theory, Russell’s type hierarchies, or Tarski’s distinction between language and metalanguage—highlighting that such formal restrictions are unnecessary if one attends to the very nature of ideas.
The text also shows how language, far from being an apparently innocuous instrument, has functioned as a double‑edged sword: its expressive power has been accompanied by ambiguities that have fostered level confusion and the illusion of self‑reference. Of particular relevance is logical homonymy, a linguistic phenomenon whereby the same symbol can denote entities or concepts of different orders.
From this standpoint, the work first undertakes a critical survey of those historical paradoxes whose origin lies in the ambiguity inherent in natural language, such as the Liar Paradox or the Grelling–Nelson paradox. It then examines some of the pillars of modern mathematical logic, especially Gödel’s first incompleteness theorem and Alan Turing’s halting problem, traditionally interpreted as paradigmatic instances of self‑reference. In Gödel’s case, it is argued that the encoded language merely simulates a self‑reference that, strictly speaking, never occurs; whereas, in the halting problem, the contradiction‑generating machine is required to operate in a way incompatible with the very assumptions of the argument.
This critical stance is not merely a philosophical or interpretive position, but a technical refutation that dismantles both theorems at the very core that sustains them: the supposed capacity for logical self‑reference and for same‑level evaluation. Moreover, it is shown that similar issues arise in other fundamental results, which reinforces the need to revisit certain traditional postulates and to move towards a logic that is more rigorous about the separation of levels.
It should be noted that the reader will not encounter complex mathematical formalisms in the text, for the aim is not to counter an excess of formalism with yet more formalism, but to show how language must always be interpreted against a prior, well‑defined logical structure.
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Title THE ILLUSION OF A SELF-REFERENCE THAT NEVER WAS: From the Liar Paradox to Gödel and Turing
This work develops a new conceptual framework that questions assumptions long taken for granted in logic, such as self‑reference and same‑level evaluation. Its starting point is the principle of causality, understood as the reason why certain logical objects relate to one another in a unidirectional and asymmetric manner. This perspective inevitably leads to a dependent, hierarchical organisation that structures our thinking and shapes the way we understand the world.
The approach aims to offer a more general and integrative view than some of the historically most prominent solutions—among them, certain postulates of set theory, Russell’s type hierarchies, or Tarski’s distinction between language and metalanguage—highlighting that such formal restrictions are unnecessary if one attends to the very nature of ideas.
The text also shows how language, far from being an apparently innocuous instrument, has functioned as a double‑edged sword: its expressive power has been accompanied by ambiguities that have fostered level confusion and the illusion of self‑reference. Of particular relevance is logical homonymy, a linguistic phenomenon whereby the same symbol can denote entities or concepts of different orders.
From this standpoint, the work first undertakes a critical survey of those historical paradoxes whose origin lies in the ambiguity inherent in natural language, such as the Liar Paradox or the Grelling–Nelson paradox. It then examines some of the pillars of modern mathematical logic, especially Gödel’s first incompleteness theorem and Alan Turing’s halting problem, traditionally interpreted as paradigmatic instances of self‑reference. In Gödel’s case, it is argued that the encoded language merely simulates a self‑reference that, strictly speaking, never occurs; whereas, in the halting problem, the contradiction‑generating machine is required to operate in a way incompatible with the very assumptions of the argument.
This critical stance is not merely a philosophical or interpretive position, but a technical refutation that dismantles both theorems at the very core that sustains them: the supposed capacity for logical self‑reference and for same‑level evaluation. Moreover, it is shown that similar issues arise in other fundamental results, which reinforces the need to revisit certain traditional postulates and to move towards a logic that is more rigorous about the separation of levels.
It should be noted that the reader will not encounter complex mathematical formalisms in the text, for the aim is not to counter an excess of formalism with yet more formalism, but to show how language must always be interpreted against a prior, well‑defined logical structure.
Work type Article
Tags filosofía, lógica, matemáticas
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Registry info in Safe Creative
Identifier 2605105595109
Entry date May 10, 2026, 1:19 AM UTC
License Creative Commons Attribution 4.0
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Author. Holder Ángel Huerta Álvarez. Date May 10, 2026.
Information available at https://www.safecreative.org/work/2605105595109-the-illusion-of-a-self-reference-that-never-was-from-the-liar-paradox-to-godel-and-turing
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