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I’ve been arguing with my dad about whether or not True and False are actually propositions themselves, and I’d be curious to hear your thoughts.

The definition of a proposition I’m seeing in most places is a declarative sentence which can have a value of either True or False, but never both.

This is probably getting into the nuances of language, but it seems to me that saying “True!” is pretty declarative (it certainly isn’t asking a question), and it can only result in a value of True, therefore not violating the second portion of the definition.

My dad, on the other hand, thinks it is not a proposition, because it’s lacking other features he associates with them, such as two separate ideas being linked (i.e. the sky is blue — the first idea is “the sky”, the second idea is “blue”, and the declaration/link is the word “is”).

Any insights would be appreciated!

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    $\begingroup$ I don't think it is. You're mixing up a statement with its truth value. If we're using language, "True" is a sentence fragment. It doesn't assert anything. Just because it isn't a question doesn't mean it is declarative. $\endgroup$ Commented Dec 26, 2018 at 21:35
  • $\begingroup$ Is the following proposition true of false? 'This statement is false.' $\endgroup$
    – D.B.
    Commented Dec 26, 2018 at 21:37
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    $\begingroup$ "Itgfhjccffujjvffsd" isn't a question either, for example. Is it declarative? No, it's ill-formed. $\endgroup$ Commented Dec 26, 2018 at 21:39
  • $\begingroup$ @MattSamuel LOL. I wonder if your example becomes a question if you put a question mark at the end :) $\endgroup$
    – Andrei
    Commented Dec 26, 2018 at 21:41
  • $\begingroup$ @MattSamuel It's an exclamatory utterance of some kind. ;-) (I think there's a proper grammatical term like "exclamative" or something). $\endgroup$
    – timtfj
    Commented Dec 26, 2018 at 22:42

4 Answers 4

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According to this definition by wikipedia

The propositions in this language are propositional constants, which are considered atomic propositions, and composite propositions, which are composed by recursively applying operators to propositions.

it seems they are. "Propositional constants" means True and False.

If we define a proposition over a set of variables as being a function from the values of those variables to the set {True,False}, then while propositions and True/False are distinct, we can consider True as a function to be a constant function that is always True.

Also note (still from wikipedia)

This definition treats propositions as syntactic objects, as opposed to semantic or mental objects. That is, propositions in this sense are meaningless, formal, abstract objects.

So just because something is a "proposition" in propositional calculus, does not mean that it is meaningful statement in English. When you say "My dad, on the other hand, thinks it is not a proposition, because it’s lacking other features he associates with them", that implies you are discussing the "ordinary meaning" of the word, but you're posting this question on Math SE and you have tagged it "propositional calculus", so what your Dad "associates" with it is not relevant; only the formal definition is within propositional calculus is.

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  • $\begingroup$ This answers the doubt mentored in my answer (and says I was right to have it!). Thanks! $\endgroup$
    – timtfj
    Commented Dec 26, 2018 at 22:30
  • $\begingroup$ Also worth pointing out that a proposition (something that can be true or false) is distinct from a statement expressing the proposition (eg a sequence of English words or mathematical symbols). $\endgroup$
    – timtfj
    Commented Dec 26, 2018 at 23:03
  • $\begingroup$ Not sure what you meant @timtfj ? How is a proposition separate from the statement which expresses it? $\endgroup$
    – golbenita
    Commented Dec 30, 2018 at 3:41
  • $\begingroup$ And @Accumulation by what he "associates" with it I meant, maybe he is describing an extension of the definition which is not so clear. Of course, if it is just an opinion, it is mathematically meaningless $\endgroup$
    – golbenita
    Commented Dec 30, 2018 at 3:42
  • $\begingroup$ @golbenita Basically: the statement is a string of symbols (like this sentence is a string of letters) and the proposition is its meaning. A pattern of pixels on a screen, or sound waves produced by someone's voice, can't be true or false. What can be true or false is the meaning assigned to the string of words/sounds/symbols. $\endgroup$
    – timtfj
    Commented Dec 30, 2018 at 16:08
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Some logical systems contain a symbol (often $\top$) for "true" as a proposition. Others don't, but they soon provide ways to simulate it, for example as $p\lor\neg p$ or as $0=0$. I personally prefer the former, because it's sometimes convenient to be able to use "true" as a proposition without involving any other propositions (like the $p$ in $p\lor\neg p$) or any content beyond pure logic (like the $0$ in $0=0$). (Example: The interpolation theorem for propositional logic looks nicer if $\top$ is available.)

Exactly analogous comments apply to "false", often written $\bot$ or simulated with $p\land\neg p$ or with $0=1$.

If one is willing to work in second-order propositional logic (i.e., allowing quantification over propositions), then the standard simulation for "true" is $\forall p\,(p\to p)$, and the standard simulation for "false" is $\forall p\, p$. (I rather like the idea of "everything is true" being the standard falsehood. It also makes "ex falso quodlibet" obvious.)

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An answer plus a doubt.

I agree with the other answer that (at first sight) True is a value rather than a proposition. But I also have doubts about whether I should be dogmatic about agreeing.

My doubt is this: suppose we define $S$ as "the proposition that is always true". Is there some logical system in which True can be identified with $S$?

This might be analogous to the counterintuitive way in which integers can be defined as sets, which initially looks like confusing a property of a set—its cardinality—with the set itself.

Edit: see Acccumulation's answer: it seems that propositional calculus uses precisely that approach.

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  • $\begingroup$ I may delete this answer. I wrote it because I wasn't happy with the instant "Of course its not a proposition!" answers and comments. Acccumulation's abswer has probably made this one redundant. $\endgroup$
    – timtfj
    Commented Dec 26, 2018 at 22:51
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No. True and False are truth-values that propositions can have, but they are not propositions themselves.

Sometimes a $\top$ is used as a proposition that always has the truth-value of True, but that does not mean that True is a proposition: $\top$ is a proposition; True is a truth-value

Same goes for $\bot$ which is sometimes used for a proposition that always evaluates False. That makes $\bot$ a proposition, but False is still just that: a truth-value.

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