@ Tero,

Your question is very smart and important------it touches the bottom of our science (supposed things we are talking about are in science).

I think logics are relationships between or among concepts------concepts and logics are basic stones for a theory system, and logical paradoxes sourly disclose critical defects in the related theory system. It is for sure that for scientists, logical paradoxes hide “Treasures” and I am now busy with “infinite concept” related paradoxes.

Our cognitive abilities improve as we human evolve, we may find that some of the “old” concepts and logics are not that scientific any more; this is the right time for certain logical paradoxes come to being urging us to re-cognize some concepts and logics, improve even reconstruct the related theory system.

So, logical paradoxes are accompanying us all the time until no human and no human science. Theoretically, there is infinite number of paradoxes but it does means that logic is not working.

What do you think?

I'm not sure your question is well-posed.  The answer depends upon both a formal definition of paradox and a classification of paradoxes in to "types".  Quine gave a classification of paradoxes (at the level of rigor common in analytic philosophy, rather than mathematics), but is it unclear that it is exhaustive, applicable to whatever formal setting you had in mind, or either optimal or unique as a classification of paradoxes.

Back to Quine:  is Quine's paradox, that is,

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

of the same "type" as "This statement is false" ?

In terms of Quine's coarse classification they are, but it can be argued that Quine's is of a different type because it is not self-referential.

Dear David,

Do you think following presentation is a paradox?

There are two reasonable limit operations in present science theory system:

（1） During the whole process in dealing with infinite substances (infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonious Series is divergent, so we can produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity. Here we have one of the modern versions of ancient Zeno’s Paradox.

（2） During the process in dealing with infinite substances (infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from some time and Harmonious Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series. But if it is convergent, another paradox appears.

But when and why should or should not people treat infinitesimals appearing in infinite numeral cognitions that way?

Sincerely yours,

Geng

Could we have a short break for following dish?

When we study ”the meaning of zero" and the location of zero in “number spectrum” in our mathematics, an unbalanced defect can be easily discovered: “zero" appears on one side of the “number spectrum” as a kind of mathematical language telling people a situation of “ nothing, not-being,…”; but on the other side of the “number spectrum” we lack of another kind of mathematical language telling people an opposite situation to “zero”------“ something, being,…”.

We need a new number symbol with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the existence of “zero”.

Sincerely yours,

Geng