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It is well known that there exists self-referential statements which are neither true or false, such as "I am lying".

Is it possible to have statements neither true or false which are not self-referential?

Consider the statement, "Alice likes berries". If Alice likes some berries (say grapes), and dislikes other berries (say tomatoes) is the statement "Alice likes berries" neither true or false? My reasoning is that if the statement is true, then one could argue because Alice likes berries, and tomatoes are berries, then Alice likes tomatoes, but this is false. Similarly if the statement is false, then one could argue since Alice doesn't like berries, and grapes are berries, then Alice doesn't like grapes, but this is also a false conclusion.

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Your example is vague because you're mixing a language that uses quantifiers with one that doesn't. Colloquially, everyone knows that "Alice likes berries" is true whether she likes most berries or all berries. Formally, "Alice likes berries" is vague and thus can't be evaluated easily. Does she like some berries? All berries? –  danielm Oct 6 '12 at 9:06
    
There are no perfect answers in ordinary sense. Information provided needs be less blurred so that answers can be sharp. –  user2411 Oct 6 '12 at 12:33

7 Answers 7

up vote 9 down vote accepted

Saul Kripke, in his paper "An Outline of a Theory of Truth", talks a little about sentences that don't receive determinate truth values in virtue of their logical structure, such as the self-referential examples you're talking about. Kripke calls these "Ungrounded" sentences. The general intuition behind the idea of ungroundedness is that the valuation of some sentences might be totally unsettled by the way the world is taken to be, and that this shouldn't stop us being able to account for a theory of those sentences that do receive a determinate and positive truth value.

On the face of it, "Every kind of berry is something that Alice likes" doesn't appear to be ungrounded in Kripke's sense. If a "kind of berry" is something that Alice can have an attitude towards, and Alice has a positive attitude towards every such kind of berry, then the sentence works out true, and otherwise, it works out false. Now we can reasonably discuss the possibility that neither kinds of berries nor the attitude of liking are usefully described with the particular logical structure that underlies the standard modelling view I've supposed here, but then we need to ask more about what it is you mean when you're talking about berries and liking.

But! There may well be other sentences that work out as ungrounded that aren't purely self-referential. Consider the following example Kripke gives. Richard Nixon is well known for having told a lot of lies about his involvement in the Watergate affair. So suppose that someone, named Jones, who has otherwise nothing to say about the scandal, claims of Nixon that:

All of the claims made by Nixon about Watergate are false.

Let's also suppose that Nixon knows Jones personally, and, generally thinking that Jones will be supportive of him, asserts that:

Everything that Jones says about Watergate is true.

Both of these sentences have meanings such that we know where we would need to go and look, independently, in order to determine whether they came out true or false. However, since it turns out that they are in fact mutually dependent on one another for their truth value, it will turn out that they are both Ungrounded sentences when we theorise about them in a Kripkean theory of assigning truth values.

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Even the phrase "Alice likes berries" may be impossible to evaluate.

  • "Alice": colloquially, it means "a person from a certain community whose name is Alice". From the formal point of view, it may mean, "any person whose name is Alice". Look what happens if there's no person with such name. "Anyone whose name is Alice likes berries" + "the set of Alices is empty" ==> neither true nor false;
  • "Likes" - described in the excellent @danielm's comment;
  • "Berries", really? Tomato is a berry. Watermelon is a berry. Alice may like commonly known berries, but not tomato. Depending on a classification of "berry", the statement may become true or false;
  • Other statements, although not self-referred, may also be true or false depending on various understanding of only terminology.
  • Since the statements are expressed in natural languages, they may fall into various linguistic pits and also be true or false or indeterminate depending on a language.
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The phrase "for any x in the empty set P(x)" is actually true for any P. –  Marco Oct 6 '12 at 15:41
    
@Marco It's mathematically correct. In philosophy, linguistic pitfalls may apply. E.g., a naive substitution of Alice with Empty set of people produces "An empty set of people likes berries" ==> "Nobody likes berries" –  bytebuster Oct 6 '12 at 16:26
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"Alice" in "Alice likes berries" does definitely not mean "Anyone whose name is Alice". It means (refers to) a particular person with that name -- which one, the context and the intentions of the speaker will fix. –  Schiphol Oct 10 '12 at 16:10
    
@Schiphol "definitely" by who? –  bytebuster Oct 10 '12 at 18:44
    
By anyone using "Alice". If you mean what philosophers of language have defended that the relevant kind of uses of "Alice" refer to concrete Alices, all of them have; since Frege till just about any contemporary theorist. Burge, Kripke, Evans, Soames, from the top of my head, but really, there are no exceptions. Do you know of any theory of the meaning of proper names that asserts or implies what you claim in your answer? –  Schiphol Oct 10 '12 at 19:49

In provability logic, either a theorem is false, true or unprovable. Turing machines either halt accepting, halt rejecting or doesn't halt.

If I understand it correctly, the provability of a theorem/halting behaviour of a program is not connected to it's self-reference. Consider programs printing their own source code or "This sentence is self-referential".

Provability logic
Gödel's Incompleteness Theorem
Curry-Howard correspondence
Halting Problem

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What is the truth of this statement 'Tommorrow I am going to eat an apple'?

Its truth is indeterminate now. By the following day its truth will be determined.

This example was discussed by Aristotle.

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On the other hand, if you're an eternalist, then that statement really is either true or false. We just don't know which yet. –  Mechanical snail Feb 6 '13 at 10:39

Yes:

It possible to have statements neither true or false which are not self-referential.

For example:

Consider a piece of card. On one side it says, "The statement on the other side of this card is true". On the other side it says, "The statement on the other side of this card is false."

Neither of these sentences is self-referential, but both are indeterminate.

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Consider the Continuum Hypothesis: the assertion that the second smallest infinite cardinality is exactly that of the continuum (i.e. the cardinality of points on the real line, or equivalently of sets of integers). It's been demonstrated that there are models of ZFC set theory in which the Continuum Hypothesis is true; and that there are models of ZFC set theory in which the Continuum Hypothesis is false. If we suppose that ZFC is consistent, this means that the Continuum Hypothesis is neither provable, nor disprovable, in standard set theory; in which case I would say that in that context, it is neither true nor false.

Conventionally we would say that the Continuum Hypothesis is undecideable within ZFC; but on what other basis does one hope to "establish" truth or falsehood in a formal theory? It cannot have a truth value in this case, except by recourse to refinements of the theory, or other informal criteria (such as various empirical arguments; though these are unlikely to be of any use in the case of the Continuum Hypothesis).

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The following sentence (Russell's paradox) is not self-referential, but can be held to be neither true nor false:

{x : x ∉ x} ∈ {x : x ∉ x}

This is a syntactically valid sentence in the formal language of naïve set theory. (Of course, whether this constitutes a "statement" is debatable. A common resolution to the paradox is to restrict set comprehensions so that {x : x ∉ x} does not exist as a set, and you may consider a sentence describing a nonexistent entity to not constitute a real statement.)

Here, the self-reference is effectively hidden behind the set comprehension. The paradoxical self-referential entity is the set of sets that do not contain themselves. The sentence itself, rather than being self-referential, merely refers to a self-referential entity.

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