What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).

What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure

What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achillesand the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox thereforenaturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).

The Formal Logical Paradox and the Ontological Status of the Concept of “Pure (Strict) Individual”

The subject of the paper is the problem of the logi-cal status of the concept of “pure (strict) individu-al”. The author states that this problem has a huge impact on various spheres of human activity. There-fore, it is necessary to consider the existing para-doxes in this area in order to understand more clear-ly not only peculiarities of the term “individual” us-age, but also its essence as the philosophical phe-nomenon. At the end the author makes a conclusion that the concept of “pure (strict) individual” is para-doxical in its nature and its using in logical analysis is incorrect, therefore, the concept of “individual” cannot be uniquely determined. The author consider that such a conclusion can be applied in various fields of human activity, for example, it is incorrect logically to develop methodological approaches aimed at the development of individuality or individ-ual work with a person.

To the question on philosophical-methodological foundations of English legal positivism of the XIX century (legal teachings of J. Bentham and J. Austin)

The subject of this research is the  aggregate of philosophical ideas and methodological paradigms that underlie the concepts of the “first” legal (statist) positivism in England of the XIX century. The author traces the impact of certain philosophical trends and legal concepts of the XVIII – early XIX centuries upon the philosophical and methodological foundations of the positivist concepts of J. Bentham and J. Austin. The article describes the influence of social atomism, and exploratory rationality of Modern Age upon the “first” legal positivism of philosophical rationalism of the XVIII century. The impact of such philosophical and legal concepts as nominalism, the historical school of lawyers, and philosophical positivism of A. Comte upon the “first” legal positivism was reconstructed. The scientific novelty consists in reconstruction of the influence of an entire number of philosophical and legal ideas and concepts upon the development of “first” legal positivism. Correlation between the legal doctrine of J. Bentham, philosophical concepts of the XVIII century, and the legal teaching of T. Hobbes is underlined. The author draws the ideological parallels between the philosophical nominalism, logical paradox of D. Hume, and legal doctrines of J. Bentham and J. Austin. The author reveals the key “channels” of the impact of German Historical School upon legal positivism, describes the similarities and differences between the scientific positivism of A. Comte and the concepts of legal positivism of J. Bentham and J. Austin. The philosophical-methodological framework of the concepts of “first” legal positivism were subjected to a significant influence of the methodological paradigm of philosophical rationalism, social atomism, exploratory scientific rationality of Modern Age, and nominalism.

Intelligence and Self

The chapter considers self-knowledge or self-insight. The concept of self is an inevitable consequence of recursive social reasoning, but it is bound to cause logical paradox due to its self-reference. Broadly speaking, self is an example of metacognition, namely, a consequence of cognition applied to evaluate other cognitive processes, which includes the feeling of knowing and other abilities to select the optimal decision-making strategies. As the number and complexity of different learning strategies increase, this also produces undesirable side effects, including negative emotions, such as disappointment and regret, as well as potential failures of metacognition, which might manifest as mental illnesses.

Paradoxes of set and property

Emerging around 1900, the paradoxes of set and property have greatly influenced logic and generated a vast literature. A distinction due to Ramsey in 1926 separates them into two categories: the logical paradoxes and the semantic paradoxes. The logical paradoxes use notions such as set or cardinal number, while the semantic paradoxes employ semantic concepts such as truth or definability. Both often involve self-reference. The best known logical paradox is Russell’s paradox concerning the set S of all sets x such that x is not a member of x. Russell’s paradox asks: is S a member of itself? A moment’s reflection shows that S is a member of itself if and only if S is not a member of itself – a contradiction. Russell found this paradox by analysing the paradox of the largest cardinal. The set U of all sets has the largest cardinal number, since every set is a subset of U. But there is a cardinal number greater than that of any given set M, namely the cardinal of the power set, or set of all subsets, of M. Thus the cardinal of the power set of U is greater than that of U, a contradiction. (The paradox of the largest ordinal, discussed below, is similar in structure.) Among the semantic paradoxes, the best known is the liar paradox, found by the ancient Greeks. A man says that he is lying. Is what he says true or false? Again, either conclusion leads to its opposite. Although this paradox was debated in medieval Europe, its modern interest stems from Russell, who placed it in the context of a whole series of paradoxes, including his own.