The paradox of the liar can be expressed by the statement “this sentence is false”. Refer to a hypothetical solution to the paradox of the liar with the variable x. Then, the follow-up revenge statement can be phrased as follows: “this sentence is x or false”. Thus, the generalization of liar’s revenge statement could be stated as: “this sentence is false. Also, this sentence is ‘this sentence is false’ or false”.
The “revenge sentence” corresponds to the following truth tables:
- If the sentence were true, then it would be “this sentence is false” or false. If the sentence were “this sentence is false”, it would not be true, and if the sentence were false, it would not be true.
- If the sentence were false, then it would be true, because it would be both described as false and stated to be false.
- If the sentence were “this sentence is false”, it would be both “true” and “this sentence is false”. The sentence “this sentence is false” is not true
The natural language general liar’s revenge sentence can be translated into Boolean algebra. Boolean algebra is a classical formal symbolic language. A formal symbolic language represents logical equations with symbols. The classical reading of logic represents logic via a computational framework, with constructions like “true or false is true” and “true and false is false”.
A paradox can be formulated within a classical formal symbolic language. The workings of a formal symbolic language has implications for logic writ large. Therefore, classical logic is paradoxical. Paradoxical logic has a third primitive truth value – paradoxical – both true and false. Paradoxical objects could be, for example, physical – perhaps a particle.
The framework of logic is important for a number of reasons – logic functions at a high level of abstraction, logic is a powerful system, logic is very basic to thought. Logic matters. That a paradox has import to logic is profound. The paradox is an ontological category that seems similar to both true and false. It reframes the field of logic, and raises the possibility of fundamentally new proofs within a three-value logical system. Disruption of the edifice of logic is also important, because logic would describe itself as logical. The paradox would have been considered impossible to the classical logician. What impossible technologies will the paradox render possible?
Generalizing the revenge sentence to the antinomy of the liar itself comprehensively results in a paradox. The truth tables have a single, contradictory, solution. It is possible to define the generalization of the antinomy of the liar as a “paradox”. The general liar’s revenge sentence accounts for all possible alternatives to the paradox. The paradox is literal, factual, physical, and certain; the paradox is real. The paradox has been proven to be a paradox within a classical logic framework, definitively.
The paradox is difficult to imagine and conceptualize; the paradox is also difficult to understand. The paradox resists facile attempts to characterize it. Extant languages describe lived experience, an example would be “a blue bluejay landed on the tree branch”. The paradox would resemble a classic subject verb and object inasmuch as it would be what we typically encounter as reality, and, simultaneously, would not be reducible to that normative expression.
Paradoxical physics connotes possible human interactions with paradoxes. Paradoxes will be seen, tasted, described in language, represented in paintings. A three-value logic proposes three categoiries true, false, or paradoxical – these categories manifest at the level of logic itself. This is important – the paradox functions at the level of truth itself. A particular physics system, for example, could be described as true, false, or paradoxical. A particular mathematical proof could be evaluated as paradoxical. Furthermore, logic would describe itself in terms of the true, the false, and the paradoxical. Because the paradox functions at the level of logic, logical interpretation necessitates a paradoxical truth-value. Any application of logic, for example, systems of physics, needs to be interpreted within a logical framework of true, false, and paradoxical, along with the corresponding logic gates.
The paradox is both true and false. For something to be true and false simultaneously is impossible. This state requires an explanation that admits the impossible. A hard-line approach to interpreting the paradox as both true and false materially dismisses reductive explanations that try to recuperate describing the paradox in terms of the possible and the normative.
The explanation for how the paradox can be both true and false is that the paradox is magical. That the paradox is both true and false is impossible, yet needs to be explained somehow. Taking the impossibility of the paradox seriously mandates an impossible system. Being both true and false, physically, actually, must function via magic.
The magical is a category as basic as true and false. Magic is structured by a Boolean algebra logic circuit – the impossible is built from the possible. Moreover, magic is accessible in this way. Magic can be performed in a similar manner to, for example, eating a bowl of cereal. Logic magic has three truth-values – the true, the false, and the magical – and the associated logic gates. Logic magic can be viewed as an extension of Boolean algebra because magical logic can be scripted in Boolean algebra. We could see magical phenomenon similar to how we see the sky.
Logic magic is analogous to normative logic with the truth-values of both true and false. Logic magic would describe itself in a similar way to how Boolean algebra might describe itself as “true”. Logic magic would be scripted in a magical formal symbolic language, like a magical Boolean algebra. Formal magical symbolic logic, like classical logic, is a powerful language.
Working Notes:
Plain Explanation, Regarding the Translation of the Formalism to Formal Symbolic Logic, Slightly Clearer Phrasing, Modern Phrasing, Formal Symbolic Language Translation, Natural Language Phrasing, Correcting a Major Error, Updated Formalism, In Vacuo, Identity Property Continued, Formalism
December 15, 2025