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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

BrianO
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MangoPizza
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    It is not mathematical statement. We have not time in mathematical object(s) definitions. – zkutch Aug 04 '22 at 15:26
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    So fix a date, e.g. "London was the capital of the UK on 2022-08-04." – Qiaochu Yuan Aug 04 '22 at 15:31
  • (Of course there are further ambiguities. If we go either back or forwards in time far enough it may become quite ambiguous in what sense there exists a political entity that deserves to be called "the UK," in what sense it has a "capital," and in what sense there exists another entity that deserves to be called "London" that might or might not be that capital. This sort of thing is fundamentally inescapable. A famous example: when did the Roman empire end? It depends which political entities you consider to deserve the name "the Roman empire"!) – Qiaochu Yuan Aug 04 '22 at 15:43
  • @QiaochuYuan I see your point, but consider that the words have their usual definition (and the "range" of dates is updated so that entities 'London', 'UK' make sense). Without fixing a specific date / range, are these statements "acceptable"? – MangoPizza Aug 04 '22 at 15:49
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    Probably not. I mean, your definition of "acceptable" is itself quite ambiguous (what does "necessarily" mean? What do "paradoxical," "ambiguous," and "subjective" mean?) so it's hard to be too precise about this. – Qiaochu Yuan Aug 04 '22 at 15:52
  • @QiaochuYuan. Ok, I found another question that asks about acceptable statements: link. See if this helps to clarify the definition. – MangoPizza Aug 04 '22 at 15:53
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    You know.... I don't care what you answer as long as you can justify it. I'd even accept "Today is Sunday" as acceptable if it understood that "Today" means the date that sentence was conceived and utter. (Okay, that could be stretching it)... what about "Elephants eat grass" 60 million years ago there were no elephants and 5 billion years from now there (probably) won't be any either. – fleablood Aug 04 '22 at 16:02
  • Let f(t) denote the capital city of the UK at time $t$. Then f(2022-08-04) = "London" (or "Westminster", depending on your definition of "city"), f(2022-08-11) = "Birmingham" (in your hypothetical), and f(1706-07-01) is undefined (because there was no "UK", England and Scotland still being separate countries that just happened to share a monarch). Then you can make logical statements about $f(t)$ for a specific $t$. – Dan Aug 04 '22 at 16:03
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    I say it's all relative to assumptions. You can define a mathematical system in which there is no number $4$ and $2+2=0.$ But if the domain of discourse is "events between 1970-01-01 and 2022-08-04" then "London is the capital of the UK" seems acceptable. – David K Aug 04 '22 at 16:23
  • For this to be a mathematical sentence, you would need to define a logical system that has the objects "london", "uk" and the relation "being the capital of". If you can do that, for example by accepting laws of physics as axioms, and deriving the definition of those objects using only those laws of physics you have axiomatized, then you will have a mathematical statement. (not sure that logic will be coherent though) – tbrugere Aug 04 '22 at 17:15
  • Today is not always Sunday, of course. Also, if it is Sunday in one time zone, it could be Saturday in another time zone further west or Monday in another time zone further east. – Geoffrey Trang Aug 04 '22 at 17:17
  • All of these time periods would arguably still be ambiguous, given the relativity of simultaneity – ajd138 Aug 05 '22 at 00:00

1 Answers1

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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.

ryang
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