Editing Curry's paradox (section)

==In natural language==

Claims of the form "if ''A'', then ''B''" are called [[indicative conditional|conditional]] claims.  Curry's paradox uses a particular kind of self-referential conditional sentence, as demonstrated in this example:

{{block indent|If this sentence is true, then Germany borders China.}}

Even though [[Germany]] does not border [[China]], the example sentence certainly is a natural-language sentence, and so the truth of that sentence can be analyzed. The paradox follows from this analysis. The analysis consists of two steps. First, common natural-language proof techniques can be used to prove that the example sentence is true ''[steps 1–4 below]''. Second, the truth of the sentence can be used to prove that Germany borders China ''[steps 5–6]'':

# The sentence reads "If this sentence is true, then Germany borders China" &nbsp; ''[repeat definition to get step numbering compatible to [[#Sentential logic|the formal proof]]]''
# If the sentence is true, then it is true. &nbsp;  ''[obvious, i.e., a [[tautology (logic)|tautology]]]''
# If the sentence is true, then: if the sentence is true, then Germany borders China. &nbsp; ''[replace "it is true" by the sentence's definition]''
# If the sentence is true, then Germany borders China. &nbsp; ''[contract repeated condition]''
# But 4. is what the sentence says, so it is indeed true.
# The sentence is true ''[by 5.]'', and ''[by 4.]'': if it is true, then Germany borders China.<BR>So, Germany borders China.  &nbsp; ''<nowiki>[</nowiki>[[modus ponens]]<nowiki>]</nowiki>''

Because Germany does not border China, this suggests that there has been an error in one of the proof steps. The claim "Germany borders China" could be replaced by any other claim, and the sentence would still be provable. Thus every sentence appears to be provable.  Because the proof uses only well-accepted methods of deduction, and because none of these methods appears to be incorrect, this situation is paradoxical.<ref>A parallel example is explained in the Stanford Encyclopedia of Philosophy. See {{Cite SEP |url-id=curry-paradox|title=Curry's Paradox|first=Lionel|last=Shapiro|last2=Beall|first2=Jc|date=2018}}</ref>

=== Informal proof===

The standard method for proving [[conditional sentence]]s (sentences of the form "if ''A'', then ''B''") is called "[[conditional proof]]".   In this method, in order to prove "if ''A'', then ''B''", first ''A'' is assumed and then with that assumption ''B'' is shown to be true.

To produce Curry's paradox, as described in the two steps above, apply this method to the sentence "if this sentence is true, then Germany borders China".  Here ''A'', "this sentence is true", refers to the overall sentence, while ''B'' is "Germany borders China".  So, assuming ''A'' is the same as assuming "If ''A'', then ''B''". Therefore, in assuming ''A'', we have assumed both ''A'' and "If ''A'', then ''B''".  Therefore, ''B'' is true, by [[modus ponens]], and we have proven "If this sentence is true, then 'Germany borders China' is true." in the usual way, by assuming the hypothesis and deriving the conclusion.

Now, because we have proved "If this sentence is true, then 'Germany borders China' is true", then we can again apply modus ponens, because we know that the claim "this sentence is true" is correct. In this way, we can deduce that Germany borders China.