Russell paradox: a famous example of formal logic paradox is the Russell paradox( by B.A.W. Russell, 1901).
It would be a logical paradox if, as has been claimed, it cannot be resolved by purely logical means, e.g. it is claimed that using theory of types to resolve the paradox makes use of some extralogical assumptions. Whether or not the paradox depends on a mistaken definition depends on whether one adopts the principle that any definable collection is a set. One might argue that given that principle, the definition "the set of all sets that are not members of themselves" cannot be mistaken, so the fault must lie with the principle. However, Wittgenstein in his Tractatus (333.3), gives a short argument that can be interpreted as implying that the principle is either defective or, relative to other assumptions, not properly deployed.
https://www.gutenberg.org/files/5740/5740-pdf.pdf
An ancient Chinese idiom, self-contradictory, is an example for Russell paradox.
Philadelphia, PA
Dear Jaing & readers,
It strikes me that the prior question here is, "What is a genuine (logical) paradox?"
Traditionally, and in later accounts, the Russel paradox is treated as a paradox of set theory, and it can certainly be argued that set theory is mathematics, and not strictly within the domain of (first-order) logic. But whether logical or mathematical, one still needs to rely on some account of what a genuine paradox is supposed to be.
One way to look at that question is that a paradox is genuine, if it genuinely puzzles the researchers in the field where it arises. This is to discount the notion that we only have a genuine paradox, if it doesn't adequately respond to particular sorts of proposed solutions --whether they be though of as, say, modification of mathematical or logical principles or assumptions. Since proposals for analysis such as "the set of all sets not members of themselves" is shown by Russell to lead to contradiction, it follows that not every formulatable or proposed condition on set membership can actually name a mathematical set. That's an interesting result. Solutions have varied, but they all involve restrictions on designations of sets. This would seem to clearly be a modification of the mathematics of set theory. Does it also involve a modification of a definition? That seems less important.
H.G. Callaway
Yaozhi Jiang RE: "An ancient Chinese idiom, self-contradictory, is an example for Russell paradox."
By itself that's not helpful. Tell us what the idiom is and, if it is not self-evident, please explain how it is an example.
Russell's paradox is a problem of unreasonable reasoning, which is due to the unboundedness of thought processes.
A long, long time ago, a merchant in the State of Chu sold both spears and shields. He bragged about his shields one day. 'My shield is the strongest in the world and nothing can pierce it,' he said. Then he bragged about his spears, 'My spear is the sharpest and can pierce anything.' Someone then asked, 'What would happen if you use your spear to pierce your shield?' The merchant did not know what to say after hearing this.
Above is the story about ancient Chiese idiom: self-contradictory.
Well, Yaozhi Jiang , you could've explained this better if you had also mentioned that the Chinese ideogram for contradiction is 矛盾 , which contains the character for spear (矛) and the character for shield (盾). Okay, so the point is that defining something as all-piercing is incompatible with also defining something as unpierceable which leads to the self-contradictory conclusions that something is both pierceable and unpierceable, or that something is both all-piercing and not all-piercing. I guess one way of resolving the paradox would be to insist that piercing is not a reflexive relation and that appearances to the contrary, all the merchant's shields are also spears and vice versa. We already know that some spears and shields could have the same topology, so all we'd need would be to add a bit of bizarre perceptual illusion. 😜😍
Very good is my praise for your Chinese, Karl Pfeifer. The main failure point of Russell paradox is not satified the self-reflexive law, i.e. the law of identity established by Russell logic. I have made a proof to avoid the paradox, and expand, if the original Russell paradox is called as 1-order Russell paradox, to i-order Russell paradox. It is a pity what the file is restricted by format that can not been attached.
Yaozhi Jiang where is your proof? So what is your conclusion? is it a a real paradox or definition flaw?
I think we should first determine here what you mean by "real paradox". I take it that a paradox is an inconsistent proposition or set of propositions each of which appear true. The Liar's Paradox - "This sentence is false" - is an example of a single inconsistent proposition which seems true, and so a paradox. Russell's Paradox strikes me as an example of an inconsistent set of propositions each of which appear true. Specifically, the propositions:
(1) For any property of objects, there is a set whose members satisfy that property, i.e. unrestricted comprehension
(2) The membership relation can be reflexive
(3) The relata of the membership relation ranges over sets and elements of sets
Proposition (1) entails the existence of any set definable by a property, e.g. 'the property of being red' defines the set of all red things, i.e. has members who satisfy the property of being red. Assuming (2) and (3) as characterizing membership, and observing the property 'is not a member of itself' is well-defined, the existence of the set whose members are not members of themselves, must exist. But that is a contradiction.
As described, I think this is a "real paradox".
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