A commonly studied paradox is the liar's paradox. The liar's paradox is to determine whether "this statement is false".
The usual resolution is to state this the sentence is not actually a statement at all. For example, the majority of mathematicians and philosophers take this approach (although they do not all do so). The reasoning mathematicians and philosophers give for it not being a statement is different, however.
- A philosopher will usually approach the problem from the point of view of self reference. The reason "this statement is false" is not a statement is because it contains "this statement", a reference to itself. Most paradoxes related to the liar's paradox can be eliminated this way.
- A mathematician, on the other hand, does not generally take issue with the self reference. The Diagonal lemma states that a statement is perfectly permitted to reference itself just as it would any another statement. Instead, they take issue with semantic aspect of the liar's paradox. The reason "this statement is false" is not a statement is because "false" is a reference to semantics. Tarski's undefinability theorem establishes that there is no mathematical statement stating another is true, unless the former is using a more powerful language. This is in contrast to the philosopher who has no problem stating the statements are true or false.
My question is whether or not there are semantic paradoxes in philosophy that do not involve self reference. That is, a sentence which is paradoxical because it talks about things like truth, and not involving any self reference that can be blamed.
There are of course some close examples.
Quine's Paradox: "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
This statement clearly refers to itself, but not by using any self-referential language. It shows that a there is no special "key words" needed for self-reference, just the ability to manipulate them as strings of symbols. This is quite similar to how the Diagonal Lemma works. (Also, the semantic element in this case is "yields falsehood".)
Berry's Paradox: "The smallest positive integer not definable in under sixty letters" can not equal itself.
In this example, the phrase does reference itself, but only through the set of all phrases. This shows that a self-referential paradox does not need to reference itself specifically even if it does not use keywords. It can just reference some set that contains it. (Also, the semantic element in this case is "definable".)
Y_1 := Y_i is false for all i>1
Y_2 := Y_i is false for all i>2
Y_3 := Y_i is false for all i>3
Y_n := Y_i is false for all i>n
Is Y_1 true?
This shows that a self-referential paradox does not even have to have self-referential statements. Rather, you can have a set of statements such that the set is self-referentially defined, but none of the individual statements are. So simply eliminating self-referential statements is not enough to eliminate self-reference. One would also need to abandon self-referential sets, and likely many other instances of self-reference. (Note too that banning self-referential language will not work, since you could simply combine Quine's Paradox with Yablo's paradox. Or you could just say "All statements that are x characters long, for some x > 200, are false if they imply that all statements longer than x are false." Also, the semantic element is "false" again.)
That being said, all of these examples have at least some aspect of self-reference. Can we have a semantic paradox with no self-reference at all?