Are all non self-referential statements true or false?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

Answer 6

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).

Answer 7

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.

Answer 8

In mathematics, the truth or untruth of a proposition is fairly easy to determine and tends to be pretty unambiguous, but it gets a lot harder when we apply this to propositions expressed in natural language.

Consider the following argument:

Proposition 1: A stale piece of bread is better than nothing.

Proposition 2: Nothing is better than a big big juicy steak.

Conslusion : A stale piece of bread is better than a big juicy steak.

-- Land of the Blind

Most of us would probably agree that the conclusion is pretty nonsensical, yet struggle explaining why that's the case.

Logically, the conclusion should follow from proposition 1 and proposition 2. The reason it doesn't, is because the word "nothing" represents something different depending on whether it replaces the X or the Y in "X is better than Y".

What we see here, is not a limitation of propositional logic as such, but rather a limitation of natural language when using it in mathematical logic.

Take your proposition "Alice likes berries". Does it imply that Alice always liked berries? Does she like all berries or only some berries? As bytebuster explains pretty well in one of the other answers, "Alice likes berries" seems like a very simple proposition but is lacking so much nuance that it's a rather useless proposition in many practical contexts.

So are all non self-referential statements true or false? When applying them to propositions in a unambiguous mathematical language, I'm inclined to say yes. When applying them to propositions in natural language, I'm inclined to say no, only because of the limitations of natural language.