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I am starting my self-study in Discrete Math, and I want to understand what propositions are. I thought of a few statements, and want to know if they are propositions or not (I do not care if they are true or not yet):

I am Batman

I think this IS a proposition because here I am unambiguously stating that I am Batman.

You are Batman

I think this is NOT a proposition because here we don't know who "you" are.

The dog is Batman

I think this is NOT a proposition because here we don't know who "the dog" is.

Rocky the Boxer Dog is Batman

I think this IS a proposition because here we are clearly being told who the dog is.

ryang
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Eramc_Ninja
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    They are all propositions, if the contexts in which the statements are made is taken into account. – coffeemath Apr 03 '23 at 02:34
  • @coffeemath so the context determines if these sentences are propositions or not? That's what I get from your answer (which thank you for answering by the way). My textbook basically is saying that propositions are declarative statements that are unambiguously true or false, but not both. I made up these Batman sentences. I assume they won't show up on a undergraduate math exam but I just want to know. Is it safe to say until we get more context, they are not propositions? – Eramc_Ninja Apr 03 '23 at 02:38
  • Eramc: A less ambiguous source of statements that are not propositions is to stick to math. Example $x >5$ in the universe of all integers. This is not a proposition since its truth value depends on $x.$ However if we add a quantifier like $\exists x (x>5)$ it becomes a proposition, in this case true. – coffeemath Apr 03 '23 at 02:44
  • @coffeemath I see. It does seem easier using math. Once you state "there exists an x such that (x>5)", it does seem to turn "x>5 in the universe of all integers" into a proposition. Which is why (as far as I am understanding) if we put context into those sentences I stated in the original post, they can be propositions. And I am assuming in no Discrete Math course will I run into sentences like that ones I posted. Though I wonder if some situations like my Batman sentences can be found when doing computer programming? – Eramc_Ninja Apr 03 '23 at 02:53
  • @Eramc_Ninja it does seem to turn "x>5 in the universe of all integers" into a proposition. $\quad$ No, coffeemath's proposed proposition-to-be is just "x>5"; "x>5 in the universe of all integers" is just a noun phrase like "the cats".$\quad$ Anyhow, all your examples are propositions; their truth values vary with the context. – ryang Apr 03 '23 at 03:05
  • @ryang may I ask why my Batman examples are propositions? – Eramc_Ninja Apr 03 '23 at 03:24
  • Your examples are propositions. Another example of a proposition: "Batman is Chainsaw Man." Whether or not it's true depends on whoever's imaginative (and possibly crazy) context. Obviously, most people would say that proposition is false because Batman and Chainsaw Man are from two completely separate stories, but it could be true in someone's fanfiction. Usually, in math, a proposition is a true statement but not as significant as other important statements. – Accelerator Apr 03 '23 at 05:26
  • @Accelerator My issue is since "You" and "The dog" are not well defined, they couldn't be propositions. I mean "He is a swimmer" isn't a proposition, right? But "Michael Phelps is a swimmer" is because we defined Michael Phelps specifically to be the swimmer. But now everyone is telling me "You are Batman" and "The dog is Batman" are propositions? "You" and "The Dog" aren't well defined. Just like x + 5 = 17 can' – Eramc_Ninja Apr 06 '23 at 19:27
  • can't be propositions because we don't know what "x" is. – Eramc_Ninja Apr 06 '23 at 19:27
  • @Eramc_Ninja I don't think it's that deep. You are looking too much into something that won't even matter in the long run. Plus, my textbook defines propositions as something slightly different anyways. By your reasoning, you might as well say "Michael Phelps is a swimmer" isn't a proposition because we never defined what a swimmer is, if Michael is a name, or some crazy stretch. You have to be working with a certain context to even be talking about propositions in the first place. Otherwise, common sense goes out the window. I really don't think this is something to stress that much over. – Accelerator Apr 06 '23 at 21:01
  • @Accelerator Ok. I understand. Yes....if I say "He is a swimmer", there likely was a context before I said that (which would make it a proposition). Ok. I'll think of it like that then. I just wanted to make sure to understand propositions as best as possible without having holes. – Eramc_Ninja Apr 06 '23 at 21:04
  • "Ok. I'll think of it like that then." This is exactly correct. Batman, Michael Phelps and the dog are all constants only within each context, just like the constant of integration C is constant for each antiderivative (here an antiderivative is a 'context'). But of course C varies as we investigate the family of antiderivatives, just as Michael Phelps is an accountant in another context and is non-existent in yet another context. The key phrse in my answer below is "can be (true or false)". – ryang Apr 07 '23 at 04:36
  • The idea in formalising sentences is to analyse their truth values as the context varies, rather than to fixate on their truth values for specific contexts. (In other words, we are interetsted in logical truth, not mere truth. Clearly, no atomic sentence can be logically true.) So, "Ommmpa is a unicorn" is indeed a proposition, and this statement makes perfect sense in the context of the story that I'm spinning. – ryang Apr 07 '23 at 04:45
  • @coffeemath The statements are getting made here in the context of math.stackexchange. I don't know who "the dog" is here in this context, and don't think anyone can know who "the dog" is here, so "the dog is batman" is not a proposition. "You are Batman" I dare say refers to anyone reading this, and "you" could mean the second person plural "You all are Batman". So, it's a proposition. But, we don't know if it's true or false, since conceivably there could be a program named "Batman" reading the original post. – Doug Spoonwood Apr 08 '23 at 05:09
  • And at a particular point in time the program named "Batman" could be the only reader of the post, so "you (all) are batman" could be true in this context, even if "Batman" couldn't know it as true. But, likely it's a false proposition in this context. It's not necessarily true or false in this context though, until we know who "you all" is exactly, and it's truth status could vary over time. – Doug Spoonwood Apr 08 '23 at 05:16
  • @DougSpoonwood The statements were simply invented by the OP. Just because OP's question was asked on stack exchange does not mean the context of the statements is math stack exchange. – coffeemath Apr 08 '23 at 11:37
  • Thank you for all your help Ryang, coffeemath and @DougSpoonwood. Yes I made these Batman sentences up. But from what I am gathering from the undergraduate Discrete Math books on this, it looks like what Doug said is the answer I am looking for. Since we don't have context yet, we can't call some of my sentences a proposition (in particular the "You" and "The Dog" ones) YET. But there likely is a context for me to have said those sentences, so we COULD say they're propositions. But if we are ONLY given these sentences ("You" and "The Dog" ones) and nothing else, they aren't propositions. – Eramc_Ninja Apr 09 '23 at 02:29
  • @ryang Just did it. What you added made sense :) – Eramc_Ninja Apr 09 '23 at 08:30
  • Can she be a Batman? – Bob Dobbs Apr 09 '23 at 13:53

1 Answers1

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may I ask why my Batman examples are propositions?

That your examples don’t have definite truth values is not because they aren’t propositions, but because you have not fixed the context.

For example, in our current reality and “you” and “the dog” as specified constants, all four statements are false. In a different universe, exactly two of them might be true.

So, the criteria “is either true or false” that you are working with is more accurately revised to “can be either true or false”.

P.S. The other posted answer says

"The first female US President was blonde" we have not had a female president yet so this statement doesn't make sense and therefore is not a proposition.

I disagree that the above example isn’t a proposition just because it is not meaningful or well-defined in the current context; consider the proposition ∀x x>0; by the above reasoning, it is not a proposition because in the universe of complex numbers it makes no sense.

Addendum

Thank you for all your help ryang, coffeemath, Accelerator and DougSpoonwood. Yes I made these Batman sentences up. But from what I am gathering from the undergraduate Discrete Math books on this, it looks like what Doug said is the answer I am looking for. Since we don't have context yet, we can't call some of my sentences a proposition (in particular the "You" and "The Dog" ones) YET. But there likely is a context for me to have said those sentences, so we COULD say they're propositions. But if we are ONLY given these sentences ("You" and "The Dog" ones) and nothing else, they aren't propositions.

You want to believe that

  • "My mother is Wonder Woman" and "Julia Roberts acted in The Price is Right", being clearly false to you, are propositions,

and that

  • "You are Batman", having no clear truth value to you, is not a proposition.

After reading the replies on this page, you now also believe that lacking context is the problem with the latter, while the first sentence has a clear context.

However, this is fruitless cherry-picking. Once I point out that there is another Julia Roberts for which "Julia Roberts acted in The Price is Right" is true, is "Julia Roberts acted in The Price is Right" suddenly no longer a proposition (due to your realisation of insufficiency of context and that "Julia Roberts acted in The Price is Right" has no definite truth value)?

"My mother is Wonder Woman" appears to have a clear context only because you are reading it self-centrically; however, in the story that I am spinning, it is actually a true, not false, statement.

Do you not consider "The square of every nonzero number is positive" a proposition until I inform you that the discourse universe is $\mathbb R$ (then True) or that the discourse universe is the set of purely imaginary numbers (then False)?

My point is that:

  1. the classification system that you want is not useful. Notice that I keep clunkily quoting that Julia Roberts string in its entirety, because I am having to refrain from calling it a 'sentence/proposition' since you think that whether it is a proposition or not depends on context. Isn't more useful to simply be able to just call it a proposition then analyse its truth value (or well-definedness) as the interpretation varies? Isn't this the goal of the formal-logic chapters of your Discrete Mathematics course?

  2. actually, when discussing the truth of non-tautological non-contradiction propositions, the context/interpretation, even if not explicit, is always at least tacitly in the background.

ryang
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  • "the context/interpretation, even if not explicit, is always at least tacitly in the background" THIS summed it up better. – Eramc_Ninja Apr 10 '23 at 06:44
  • @Eramc_Ninja Okay, but my initial comment (before posting this answer) and the main part of this answer, and coffeemath’s comment, all already say as much; your summary response forced it to become more explicit. -) – ryang Apr 10 '23 at 08:51