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Infinite liars

🌈 Abstract

The article discusses various paradoxes and logical puzzles related to the concept of truth, including the Liar paradox, the Epimenides paradox, the Grelling-Nelson paradox, and Gödel's incompleteness theorems. It explores different strategies for resolving these paradoxes, such as the no-truth-value solution, the both-true-and-false solution, and the no-proposition solution. The article also delves into the concept of self-reference, the use-mention distinction, and the arithmetization of formal languages. It highlights the connections between these logical puzzles and the fundamental limitations of formal systems in capturing the full complexity of truth.

🙋 Q&A

[01] Liar Paradox and Proposed Solutions

1. What are the key features of the Liar paradox? The Liar paradox refers to the conundrum faced when trying to assign a coherent truth value to the sentence "This sentence is false." If the sentence is true, then it must be false, and if it is false, then it must be true, leading to a contradiction.

2. What are the different solution strategies proposed for the Liar paradox? The article discusses several proposed solutions:

  • The no-truth-value solution, which suggests that the Liar sentence does not have a truth value and is not meaningful.
  • The both-true-and-false solution, which embraces a paraconsistent logic where the Liar sentence can be both true and false.
  • The no-proposition solution, which argues that the Liar sentence does not express a meaningful proposition at all.

3. How do these proposed solutions fare against the "liar's revenge" versions of the paradox? The article suggests that the no-truth-value and no-proposition solutions are challenged by the "liar's revenge" sentences, which seem to undermine the effectiveness of these strategies in resolving the original Liar paradox.

[02] Self-Reference and the Grelling-Nelson Paradox

1. What is the Grelling-Nelson paradox, and how is it related to the concept of self-reference? The Grelling-Nelson paradox arises from the question of whether the word "heterological" (a word that does not describe itself) is heterological or not. This paradox, like the Liar paradox, involves self-reference and the difficulty in consistently assigning truth values to such self-referential statements.

2. How does the article discuss the relationship between self-reference and the paradoxes discussed? The article suggests that self-reference is not the root cause of the paradoxes, as evidenced by the fact that self-referential statements can be successfully handled in the context of Gödel's incompleteness theorems. The article argues that the paradoxes arise from the underlying theory of truth employed, rather than the self-referential nature of the statements themselves.

[03] Gödel's Incompleteness Theorems and the Arithmetization of Formal Languages

1. How does the article explain Gödel's first incompleteness theorem? The article explains that Gödel's first incompleteness theorem shows that every consistent formal system of arithmetic will admit true but unprovable sentences. This is demonstrated using the Gödel sentence, which asserts its own unprovability within the formal system.

2. What is the role of the arithmetization of formal languages in Gödel's results? The article discusses how Gödel's key insight of arithmetization, the ability to represent formal mathematical structures and assertions as numbers, allows for the construction of self-referential statements within the formal language of arithmetic, leading to the incompleteness theorems.

3. How does the article contrast the Gödel sentence with the Liar paradox? The article suggests that the success of the Gödel sentence analysis, leading to profound mathematical theorems, indicates that the problem with the Liar paradox lies not in the self-referential nature of the statements, but rather in the underlying theory of truth employed.

[04] Tarski's Theorem on the Nondefinability of Truth

1. What is Tarski's theorem on the nondefinability of truth, and how does the article prove it? Tarski's theorem states that there is no arithmetically expressible property that can define the true arithmetic sentences. The article provides two proofs of this theorem, one using the fixed-point lemma and another using the central idea of the Russell paradox.

2. How does the article connect Tarski's theorem to the Grelling-Nelson paradox? The article suggests that the proof of Tarski's theorem via the Russell paradox is essentially the same logic as the Grelling-Nelson paradox, demonstrating a fundamental unity among these various paradoxes.

[05] The Truth-Teller, Löb's Theorem, and Infinite Versions

1. How does the article analyze the "truth-teller" sentence and contrast it with the Liar sentence? The article argues that the analysis of the "truth-teller" sentence, which asserts its own truth, is not as straightforward as it may seem. It suggests that the fact that the "truth-teller" sentence does not immediately lead to a contradiction should not be taken as an indication that it can be either true or false.

2. What is Löb's theorem, and how does it relate to the analysis of the "truth-teller" and "Löb" sentences? Löb's theorem shows that the sentence asserting its own provability is, in fact, provable. The article uses this result to caution against being too quick in the analysis of the "truth-teller" sentence, suggesting that it may have a more nuanced resolution than initially apparent.

3. How does the article discuss the infinite versions of the "truth-teller" and "Löb" sentences? The article constructs and analyzes the infinite versions of these sentences, demonstrating that they can also be shown to be provable using a Löb-style analysis, similar to the single "Löb" sentence.

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