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).

Revisiting ‘Third-Person’ Narrative Unreliability

Taking its cue from the critical treatment given to unreliable narration by Wayne C. Booth and his early followers, and in contrast to the claims often made in the field of authentication theory, this paper seeks to join the debate on “third-person” narrative unreliability by outlining an inclusive approach to this phenomenon in which the “person” parameter need not be a determining factor. To theorize and illustrate this approach, a methodological context is first developed by juxtaposing Genette’s revisionist stance on voice and perception with Booth’s 1961 dismissal of the vocal issue and his controversial assimilation of tellers and observers. Then Ryan’s dissenting views are addressed by identifying common ground between her idea of the impersonal narrator and the principles of inclusivity which precisely rest on the impersonating potential of that figure. Finally the inclusive conception of unreliability is shown at work in three Jamesian tales – “The Aspern Papers” (1888), “The Liar” (1888), and “The Beast in the Jungle” (1903) – whose different vocal options do not seem to immunize their narrators against charges of untrustworthiness.

Why Can the Brain (and Not a Computer) Make Sense of the Liar Paradox?

Ordinary computing machines prohibit self-reference because it leads to logical inconsistencies and undecidability. In contrast, the human mind can understand self-referential statements without necessitating physically impossible brain states. Why can the brain make sense of self-reference? Here, we address this question by defining the Strange Loop Model, which features causal feedback between two brain modules, and circumvents the paradoxes of self-reference and negation by unfolding the inconsistency in time. We also argue that the metastable dynamics of the brain inhibit and terminate unhalting inferences. Finally, we show that the representation of logical inconsistencies in the Strange Loop Model leads to causal incongruence between brain subsystems in Integrated Information Theory.

Why Can the Brain (And Not a Computer) Make Sense of the Liar Paradox?

Ordinary computing machines prohibit self-reference because it leads to logical inconsistencies and undecidability. In contrast, the human mind can understand self-referential statements without necessitating physically impossible brain states. Why can the brain make sense of self-reference? Here, we address this question by defining the Strange Loop Model, which features causal feedback between two brain modules, and circumvents the paradoxes of self-reference and negation by unfolding the inconsistency in time. We also argue that the metastable dynamics of the brain inhibit and terminate unhalting inferences. Finally, we show that the representation of logical inconsistencies in the Strange Loop Model leads to causal incongruence between brain subsystems in Integrated Information Theory.

THE LIAR’S PARADOX –A WILD INTERATION. PART II. DIFTHONG «ARISTOTEL-ANOKHIN» (CONTINUATION)

There are more and more precedents with offended infants of 30-40 years old — they are not emotionally abstinent, because they are in an artificial coma of infantilism, in which ‘desire’ has replaced ‘sacrifice’, and are clearly hypocritical, which is why the Holiday of Disobedience, hanging around the planet with a blinking garland of conflicts and wars, creates a turbulent zone in which the bifurcation points are taken out — beyond the orbit of common understanding, turning Consciousness into the quietest Sphinx, producing hypotheses. The saying, willingly or unwittingly, can become a “winged missile” — and destroy the whole world, good or bad, but the theory created by the presentiment of scientific research can help keep it in health and in the flesh of a divine plan, but on one condition: while maintaining peace and the will of Consciousness — the indispensable parity of the Mind, which multiplies both entities and doubt as paradox, whose mental albatrosses format our understanding. Thus, a hypothesis based on a paradox forms the Image of the Concept, and thereby builds a fundamental frame of the worldview, without belittling the elephants, and without forgetting the whale. In our world, a liar as Caesar’s wife turns out to be beyond suspicion, and, therefore, discussion, and his figure is so transparent and nano-technological that it has long been soldered into the ‘scale of errors’ of all perception — and this is the toothless sperm whale that substituted its back for the pillars of thinking, which is why not only looms as a wise turtle, but is also perceived by a cheerful Buddha. From time to time, the whale opens its mouth — and we all find ourselves in its throat, and the liar himself is outside the Law, outside the conflict, but in the Law: in the legal field of the Absolute, who knows only the doctrine of exclusiveness and the purple of shamelessness is accustomed.

Précis of Uncut

Uncut is a book about two kinds of paradoxes: paradoxes involving truth and its relatives, like the liar paradox, and paradoxes involving vagueness. There are lots of ways to look at these paradoxes, and lots of puzzles generated by them, and Uncut ignores most of this variety to focus on a single issue. That issue: do our words mean what they seem to mean, and if so, how can this be? I claim that our words do mean what they seem to, and yet our language is not undermined by paradox. By developing a distinctive theory of meaning, I show how this can be.