# Is there a natural example of a non-self-referential semantic paradox in philosophy?

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?

• `A philosopher will usually approach the problem from the point of view of self reference.` is an extremely bold statement. Philosopher differ greatly on both what qualifies as a philosophical question and what qualifies as the best way to answer it. – virmaior Jan 13 '19 at 7:27
• @virmaior I said "usually". I guess I should say "most philosophers" instead. I am know about alternate approaches, but disallowing self-reference seems to be more common than the rest (although I could not really prove it without a poll). – PyRulez Jan 13 '19 at 7:28
• @virmaior The Euthyphro dilemma would count as a paradox, but it is not semantic. It is not about statements at all. – PyRulez Jan 13 '19 at 8:02
• My guess is that self-reference paradoxes are called "semantic" because Tarski developed his formal semantics to block Liar-like sentences, then variations on the theme were joined in by family resemblance. So "at least some aspect of self-reference" is just the vague condition for belonging to the "semantic" family. There are plenty of paradoxes that have nothing to do with self-reference though, sorites, epistemic paradoxes, Zeno's, etc. What is your "semantic"? – Conifold Jan 13 '19 at 9:17
• IMO, the issue is that waht we usualy call Semantic Paradoxes is considered a subset of the Paradoxes of Self-Reference. – Mauro ALLEGRANZA Jan 13 '19 at 9:23

Paradoxes involving vague predicates, such as the sorites paradox as Conifold mentioned in a comment, may be examples of semantic paradoxes that are not self-referential.

Dominic Hyde and Diana Raffman describe such paradoxes as semantic:

Most theorists of vagueness conceive of vagueness as a semantic phenomenon, as somehow rooted in the meanings of words like ‘tall’ and ‘old’.

We usually know enough to make a claim that someone is tall or old, but do not have a definition that allows us to make such a claim unambiguously. Hence the paradoxical nature of determining when a person ceases to be tall. These paradoxes do not involve self-reference.

Hyde, Dominic and Raffman, Diana, "Sorites Paradox", The Stanford Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2018/entries/sorites-paradox/.

• But note: "The Sorites paradox is a paradox that on the surface does not involve self-reference at all. However, Priest (2010b, 2013) argues that it still fits the inclosure schema and can hence be seen as a paradox of self-reference, or at least a paradox that should have the same kind of solution as the paradoxes of self-reference. This has led Colyvan (2009), Priest (2010) and Weber (2010b) to all advance a dialetheic approach to solving the Sorites paradox." SEP on Self-Reference – Eliran Apr 11 '19 at 1:37
• I wonder if Sorites really is a paradox. It seems be simply a definitional ambiguity. – PeterJ Sep 9 '19 at 13:01
• @PeterJ It is listed as a paradox in the link I cited. One can't assign such predicates a truth value without first assuming a definition. That assumption may be too arbitrary to get more than one person to commit to it. I think that is what underlies this paradox. That would mean there are many declarative sentences that do not have truth values. I don't mind that, but vague predicates do function as a challenge for those who want to assign truth-values consistently. – Frank Hubeny Sep 9 '19 at 15:58
• @PeterJ: I argue here that Sorites is less of a paradox in its own right, and more of a framework for constructing paradoxes in other settings. This is particularly the case when you have bivalent predicates that don't admit a "halfway true" state, such as with paradoxes of identity (compare and contrast the Ship of Theseus). – Kevin Sep 10 '19 at 0:00
• @Kevin - I share your view. . – PeterJ Sep 10 '19 at 10:34

I'm not sure if Moore's paradox should count. It does "talk about things like truth," but probably not in the way you intend. It can be constructed from any simple declarative statement. Here is the presentation which SEP quotes from Moore originally:

I went to the pictures last Tuesday, but I don’t believe that I did.

The "I did" refers to the first clause of the sentence, but this can be trivially eliminated:

I went to the pictures last Tuesday, but I don't believe that I went to the pictures last Tuesday.

Both conjuncts could plausibly be true: It could indeed be the case that I went to the pictures last Tuesday, and it could simultaneously be the case that I don't believe that I went to the pictures last Tuesday (because I forgot, or suffered a bout of amnesia, or whatever). Yet it nevertheless seems intuitively "wrong" to assert this statement. So, the question arises, if this statement could plausibly be true, why can't I assert it?

On the other hand, SEP points out that the statement seems entirely reasonable when recast into third person ("He went to the pictures..., but he does not believe that he did.") or even past tense ("...but I did not believe that I did."), which suggests it has rather more to do with epistemology than semantics. Most philosophers would categorize it as such, certainly. However, the typical approach to solving it tends to focus on semantics, for example by using a more elaborate form of logic to "properly encode" the relationship between beliefs and underlying propositions. So you might consider it as both a semantic and an epistemic problem.