This is great for understanding why simpler logics (Boolean) are computable but too expressive ones (FO) are not. As soon as the logic is able to make statements about itself, things get 'weird'. A not formal(!) but imho still intuitive example:
"This sentence has five words."
This is a true statement. And we know that the conjunction of two true statements should also be true.
"This sentence has five words and this sentence has five words."
With eleven words each part of this statement is obviously false and here we can't rely on the "and", which we are used to from the simpler logics, anymore.
For example, if I say "it's sunny out now" or "today is a warm day" or "Berkeley is east of here" I also run into problems related to the prospect that references may not have the same referent in different contexts. (For example, two speakers can respectively affirm and deny each of these propositions even though they may both agree in all of their beliefs!)
The example that you give can also be interpreted as a deictic problem because the referent of "this sentence" can change from sentence to sentence, even though we normally don't expect the referents of noun phrases to change this way.
"This sentence has five words and this sentence has five words." In English "this" need not always be self referential, so I'll retort with "This sentence is imprecise".
The two "this"s could refer to two different things. However, the second this would normally become that to indicate comparison but grammatically the original is still sound [with notes]:
This sentence [indicates a sentence with five words in it] has five words and this sentence [indicates another sentence, also with five words in it] has five words.
"This sentence has five words" AND "this sentence has five words" seems to be logically consistent. Doesn't trying to construct the logical proposition in natural english violate some sort of rule?
Your reasoning only works for valid statements. That is, statements that have a truth value. As you’ve demonstrated P does not have one. For no admissible statement is P equal to not P.
The P, in your original post, is not a valid formula since it does not have a truth value. You can construct all the proofs and whatnot you want but if P is not a logical statement (one that is either true or false but not both) then the arguments are not valid.
Your Q was generic and could have been any statement. In particular one that has been provably shown to be false. If your arguments are correct then you have the ability to prove any false statement is true.
Do you really think you have found a way to demonstrate that the standard two-valued logic used in mathematics leads to a contradiction?
I see. The difference in our views comes from me being a mathematician. Curry’s Paradox is not a paradox in mathematical logic due to my objections. In mathematical logic we don’t allow statements that don’t have a well defined truth value. The premise in the start of Curry’s Paradox is a statement without a well defined truth value.
Curry’s Paradox shows the limits if naive set theory. Thus mathematically we have to be OK with the idea that not all naive set constructions produce valid sets.
In informal setttings we can argue that
If this sentence is true then Santa Claus exists
can’t be false. But then one usually assumes it must then be true. I would argue this isn’t the case. There are sentences that have no truth value. For instance,
"This sentence has five words."
This is a true statement. And we know that the conjunction of two true statements should also be true.
"This sentence has five words and this sentence has five words."
With eleven words each part of this statement is obviously false and here we can't rely on the "and", which we are used to from the simpler logics, anymore.