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What The Tortoise Said To Achilles
"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal ''Mind'', is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression. Summary of the dialogue The discussion begins by considering the following logical argument: * ''A'': "Things that are equal to the same are equal to each other" (a Euclidean relation) * ''B'': "The two sides of this triangle are things that are equal to the same" * Therefore, ''Z'': "The two sides of this triangle are equal to each other" The tortoise accepts premises ''A'' and ''B'' as true but not the hypothetical: * ''C'': "If ''A'' and ''B'' are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lewis Carroll
Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Some of Alice's nonsensical wonderland logic reflects his published work on mathematical logic. Carroll came from a family of high-church Anglicanism, Anglicans, and pursued his clerical training at Christ Church, Oxford, where he lived for most of his life as a scholar, teacher and (necessarily for his academic fellowship at the time) Anglican deacon. Alice Liddell – a daughter of Henry Liddell, the Dean of Christ Church, Oxford, Dean of Christ Church – is wide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logical Consequence
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (logic), valid logical argument is one in which the Consequent, conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is logical truth, necessary and Formalism (philosophy of mathematics), formal, by wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange loops, artificial intelligence, and discovery in mathematics and physics. His 1979 book ''Gödel, Escher, Bach, Gödel, Escher, Bach: An Eternal Golden Braid'' won the Pulitzer Prize for general nonfiction,"General Nonfiction" . ''Past winners and finalists by category''. The Pulitzer Prizes. Retrieved 17 March 2012. and a National Book Award (at that time called The American Book Award) for Science."National Book Awards – 1980" [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gödel, Escher, Bach
''Gödel, Escher, Bach: an Eternal Golden Braid'' (abbreviated as ''GEB'') is a 1979 nonfiction book by American cognitive scientist Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. It also discusses self-reference and formal rules, isomorphism, what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself. In response to confusion over the book's theme, Hofstadter emphasized that ''Gödel, Escher, Bach'' is not about the relationships of mathematics and art, mathematics, art, and music and mathematics, music, but rather ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rule Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as ''modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Regress Argument
Infinite regress is a philosophical concept to describe a series of entities. Each entity in the series depends on its predecessor, following a recursive principle. For example, the epistemic regress is a series of beliefs in which the justification of each belief depends on the justification of the belief that comes before it. An infinite regress argument is an argument against a theory based on the fact that this theory leads to an infinite regress. For such an argument to be successful, it must demonstrate not just that the theory in question entails an infinite regress but also that this regress is ''vicious''. There are different ways in which a regress can be vicious. The most serious form of viciousness involves a contradiction in the form of ''metaphysical impossibility''. Other forms occur when the infinite regress is responsible for the theory in question being implausible or for its failure to solve the problem it was formulated to solve. Traditionally, it was of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Münchhausen Trilemma
In epistemology, the Münchhausen trilemma is a thought experiment intended to demonstrate the theoretical impossibility of proving any truth, even in the fields of logic and mathematics, without appealing to accepted assumptions. If it is asked how any given proposition is known to be true, proof in support of that proposition may be provided. Yet that same question can be asked of that supporting proof and any subsequent supporting proof. The Münchhausen trilemma is that there are only three ways of completing a proof: * The circular argument, in which the proof of some proposition presupposes the truth of that very proposition * The regressive argument, in which each proof requires a further proof, ''ad infinitum'' * The dogmatic argument, which rests on accepted precepts which are merely asserted rather than defended The trilemma, then, is having to choose one of three equally unsatisfying options. Name The name ''Münchhausen-Trilemma'' was coined by the German philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homunculus Argument
The homunculus argument is an informal fallacy whereby a concept is explained in terms of the concept itself, recursively, without first defining or explaining the original concept. This fallacy arises most commonly in the theory of vision. One may explain human vision by noting that light from the outside world forms an image on the retinas in the eyes and something (or someone) in the brain looks at these images as if they are images on a movie screen (this theory of vision is sometimes termed the theory of the Cartesian theater: it is most associated, nowadays, with the psychologist David Marr). The question arises as to the nature of this internal viewer. The assumption here is that there is a "little man" or "homunculus" inside the brain "looking at" the movie. The reason why this is a fallacy may be understood by asking how the homunculus "sees" the internal movie. The answer is that there is another homunculus inside the first homunculus's "head" or "brain" looking at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Deduction Theorem
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A \to B, it is sufficient to assume A as a hypothesis and then proceed to derive B. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B is deducible from a set of assumptions \Delta \cup \ then the implication A \to B is deducible from \Delta ; in symbols, \Delta \cup \ \vdash B implies \Delta \vdash A \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Willard Van Orman Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He was the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978. Quine was a teacher of logic and set theory. He was famous for his position that first-order logic is the only kind worthy of the name, and developed his own system of mathematics and set theory, known as New Foundations. In the philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability argument, an argument for the Philosophy of mathematics#Empiricism, reality of mathematical entities.Colyvan, Mark"Indispensability Arguments in the Philosophy of Mathematics" The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), Edward N. Zalta (ed.). He was the main proponent of the view that philosophy is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conventionalism
Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicit or implicit) agreements in society, rather than on external reality. Unspoken rules play a key role in the philosophy's structure. Although this attitude is commonly held with respect to the rules of grammar, its application to the propositions of ethics, law, science, biology, mathematics, and logic is more controversial. Linguistics The debate on linguistic conventionalism goes back to Plato's '' Cratylus'' and the philosophy of Kumārila Bhaṭṭa. It has been the standard position of modern linguistics since Ferdinand de Saussure's '' l'arbitraire du signe'', but there have always been dissenting positions of phonosemantics, recently defended by Margaret Magnus and Vilayanur S. Ramachandran. Philosophy of mathematics The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of non-Euclidean geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
