|Paradoxes: are contradictions within formally correct statements or sets of statements that lead to an existence assumption, which initially seemed plausible, to be withdrawn. Paradoxes are not errors, but challenges that may lead to a re-formulation of the prerequisites and assumptions, or to a change in the language, the subject domain, and the logical system. See also Russellian paradox, contradictions, range, consistency._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
False Conclusions/Thiel: only of interest if they are intentionally induced as "fallacies", or if they smuggle supposedly legitimate conclusions into an argument in the form of "sophisms", or as in Kant's case so-called "paralogisms" which have their reason "in the nature of human reason" and are therefore "inevitably though not indissoluble". Example: arithmetic fallacy: 5 = 7 (I 321 +).
Example: Syllogism with a quaternia terminorum (hidden occurrence of four instead of three allowed terms in a final schema)
Flying elephants are fantasy imaginations.
Imaginations are part of our reality.
So, flying elephants are part of our reality.
Paradoxes are something contrary to ordinary opinion (doxa). Other form: fact wrapped in a puzzle solution.
For example, that a strap placed tightly around the equator would suddenly protrude by 1/2π, i.e. by about 16cm, after being extended by only one meter.
In everyday use, paradoxes are often only corny things, like the hypochondriac who only imagines himself to have delusions (question of definition) or "Murphy's law" that everything lasts longer, even if one has already considered it.
Since the English scientific literature "paradoxically" compromises both paradoxes (not real antinomies) and antinomies, a distinction has not yet prevailed.
Example "crocodile conclusion" (already known in ancient times): a crocodile has robbed a child, the mother begs to give it back. The crocodile places the task of guessing what it will do next. The mother (logically preformed) says: you won't give it back to me. Hence stalemate.
Because the mother now argues that the crocodile must give the child back, because if the statement is true, she gets it back on the basis of the agreement, but if it is false, then it is just wrong that she does not get the child back, so because it is true that she gets it.
The crocodile, on the other hand, argues that there is no need to give the child back, because if the mother's statement is false, she will not get it back because of the agreement, but if it is true, it means that she will not get the child back.
Only a careful analysis reveals that the agreement made does not yet provide a rule for action.
If "z" stands for giving back, "a" for the mother's answer (which is still indefinite and can therefore only be represented schematically by a), the agreement does not yet provide a rule system that can be followed, but rather the rule schema.
"a" ε true >> z
"a" ε false >> ~z
If the range of variability of a is not restricted, then one can also make choices of a that are incompatible with Tarski's condition of adequacy for truth definitions.
This states that for a predicate of truth "W" and any statement p, from which it can be meaningfully stated, always
"p" ε W <> p
has to apply. In the crocodile conclusion, the mother selects ~z for a, thereby turning the rule scheme into the rule system.
(R1) "~z" ε true >> z
(R2) "~z" ε false >> ~z
The crocodile now concludes to R2 and Tarski (with ~z for p) to ~z. The mother, on the one hand, deduces after R1 and on the other hand metalogically from the falsity of "~z" and from there (after Tarski) further to z.
Since the argumentation makes use of a predicate of truth and a predicate of falsehood as well as the connection between both, the crocodile conclusion is usually counted among the "semantic" antinomies.
One can see in it a precursor of Russell's antinomy.
Thiel I 328
One should not hastily deduce from this that the antinomies and paradoxes have no meaning for mathematics. Both Poincaré's criterion (predictiveness) and type theory force a restriction of the so-called comprehension axiom, which determines the conditions permissible as defining conditions for sets of forms of statement._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995