Is 'London is the capital of UK' mathematically acceptable statement?

Question

A statement is called 'acceptable' if it's either true or false. In particular, an acceptable statement can't be paradoxical, ambiguous or subjective. (Here is a similar question asking about acceptable statements. I believe that the top answer there clarifies the definition.)

For example,

• $\text{'}2 + 2 = 4\text{'}$ is acceptable. It is always true.
• $\text{'James is a kind person.'}$ is not acceptable since 'kind' is a subjective word.
• $\text{'Today is Sunday.'}$ is not acceptable since 'today' is ambiguous.

But, I can't figure out this question:

Is $\text{'London is the capital of the UK.'}$ an acceptable statement?

(Consider that the words have their usual definition and the "range" of dates is specified so that entities 'London', 'UK' make sense)

It seems to me to be 'no' because the capital is something that can change.

For example, consider a scenario where the government has decided to change the capital of the UK to Birmingham on August $10, 2022$. Now, the statement is ambiguous since the time when the statement is spoken in ambiguous, and hence the statement can be either true or false depending on the time spoken. I can invent many other scenarios where things can change:

• $\text{Ellie listens only to Harry Styles's music.}$
• $\text{The Prime Minister of the UK is Boris Johnson.}$
• $\text{The total acceptance rate of Harvard University is 5%.}$

These statement can / will change with time. So, are these statements acceptable?

Thanks

Answer 1

an acceptable statement can't be paradoxical, ambiguous or subjective.

• $\text{'Today is Sunday.'}$ is not acceptable since 'today' is ambiguous.

Is $\text{'London is the capital of the UK.'}$ an acceptable statement?

• $\text{Ellie listens only to Harry Styles's music.}$
• $\text{The Prime Minister of the UK is Boris Johnson.}$
• $\text{The total acceptance rate of Harvard University is 5%.}$

The first example is not ambiguous due to its phrasing: it has neither semantic nor syntactic ambiguity, and ‘today’ is certainly not an ambiguous word. The uncertainty about the sentence's truth is due merely to a lack of context.

Similarly, the other four sentences^҂ are not ambiguously phrased. Each, given a context, has a definite truth value.

^{҂Some authors don't consider a sentence a statement unless a context has been specified. But this is distinction is observed only nichely.}

In the absence of an explicit definition of ‘ambiguous’ for the current treatment, I consider all the five sentences above unambiguous and ‘acceptable’.

A statement is called 'acceptable' if it's necessarily either true or false. In particular, an acceptable statement can't be paradoxical, ambiguous or subjective.

Consider the sentence $$\forall x\;\big(|x|=3 \implies x=\pm3\big).$$ Notice that it is true in the context of real analysis but false in the context of complex analysis; that is, it is not “necessarily” true or false, so it is apparently not an ‘acceptable’ sentence, even though Formal Logic disagrees. Not so fast: its truth value is definite once we assign a context/interpretation; in this sense, this mathematical sentence is indeed an ‘acceptable’ statement.

In the absence of an explicit definition of “necessarily”, and since “necessarily true” here clearly means neither logically true nor universally true, for example, the most conservative reading and a clearer rephrasing of the above definition is

• A sentence is acceptable iff it is either definitely true or definitely false.

  Thus, a paradoxical, ambiguous or subjective sentence is not acceptable.

So—without even needing to consider paradoxicality, ambiguity or subjectivity—a satisfiable but invalid sentence (consequently: a contingent sentence, i.e., a sentence that is neither a tautology nor a contradiction) cannot be ‘acceptable’ unless an interpretation/context has been assigned.