# Renouned Set Theory

Consider the following ten statements:

1. $A$ do not have $B$.
2. Some $C$ are $D$, but no $E$ are $D$.
3. $F$ that are $A$ can still be $G$ if they have $H$.
4. No $C$ that do not have $G$ can be both $H$ and $B$.
5. All $D$ are $B$, and all $B$ that are $A$ are also $G$.
6. If $E$ are $B$, they have $H$.
7. Some but not all $F$ are $H$.
8. $C$ that have $B$ that are $E$ are not $A$.
9. Some $F$ have $G$ even if they do not have $D$.
10. All $F$ that are $G$ and are not $C$, are $E$.

The eight variables $A$ through $H$ represent plural nouns. Your task is to determine a value for each noun so that all ten statements hold true.

Generally speaking, a statement is true if and only if:

• it requires no creative interpretation (e.g. the statement "All bishops are containers." is invalid even though bishops' bodies contain organs, blood, etc. and hence 'technically' qualify as containers)

• it is reasonably objective (e.g. "All elephants are giants." is a reasonably objective statement, while "All elephants are weirdos." is not)

• it is unambiguous and can be reasonably verified as correct (e.g. "Not all foods are bean bags." is valid despite being obvious, while "Not all powders are chemicals." might be true or untrue depending on how literally one takes the statement)

Note that if the above conditions are met, only the condition that is explicitly stated needs to hold true. For example, "Some rock stars are either umbrellas or artists." is true even though the statement might imply that some rock stars are umbrellas, which is false, or that not all rock stars are artists, which is also false.

Since the question is open-ended, the voters will ultimately determine if any given answer is stretching the truth. When in doubt: don't risk it!

The nouns must be made up of real English words. They can be abstract, and they can be qualified as little or as much as desired, but they must be plural nouns. Some valid examples include "dogs", "big dogs", "big dogs' ideas", "people that like big dogs", etc. Note that nouns capable of possessing/containing/having other nouns as well as nouns capable of being possessed/contained/had are strongly recommended since many of the statements employ the verb "have".

An answer's score is the total number of characters (including any punctuation and spaces) in the eight nouns. Lower scores are better.

The winning answer is the answer with the lowest score where all ten statements are true.

Good luck! And may the most re-nouned answer win!

• I love this question! :-D – Rand al'Thor Jun 20 '15 at 9:49
• Does "All Stack Exchangers are weirdos" count as a reasonably objective statement? – Rand al'Thor Jun 20 '15 at 11:48
• But... not all rock stars are artists... – Ian MacDonald Jun 20 '15 at 12:15
• I think one clarification is required - are we replacing the letters with words, or with concepts? For instance, can the same Letter be used to mean "a symbol used in a word" in one statement, and "text sent by mail" in another? If we're replacing the letters with words, then the word can be interpreted as appropriate for the sentence, whereas if we're replacing the letters with concepts, it has to mean the same thing every time it's used. – Glen O Jun 20 '15 at 18:03
• @GlenO: I like the idea of having words with multiple meanings from a creativity standpoint, so we'll say it's 100% legal. – COTO Jun 20 '15 at 19:42

Here's a solution that seems to work (inspired by Glen O's mention of letters and words in a comment):

$A=$ letters
$B=$ words
$C=$ sentences
$D=$ pronouns
$E=$ articles
$F=$ words
$G=$ members of the set {a,I}
$H=$ letters

Going through the ten statements one by one:

1. Letters do not have words. Of course not - letters having words doesn't make sense!
2. Some sentences are pronouns, but no articles are pronouns. Many pronouns can function as whole sentences ("Who did it?" "Her."), but articles and pronouns are disjoint classes of words.
3. Words that are letters can still be a or I if they have letters. In fact, single-letter words must be a or I, and the "if they have letters" part is superfluous (but doesn't invalidate the statement).
4. No sentences that do not have the words a or I can be both letters and words. A sentence that consists only of a single word and also only of a single letter must be either a or I (in fact, it must be I), so a sentence that doesn't contain these words can't be both a letter and a word.
5. All pronouns are words, and all words that are letters are also members of the set {a,I}. Pronouns are certainly words, and a and I are the only single-letter words.
6. If articles are words, they have letters. Articles are words, and they do have letters. That was easy!
7. Some but not all words are letters. Some words (a, I) are single letters, but others (stack, exchange) are not.
8. Sentences that have words that are articles are not letters. The only way a sentence can be a single letter is if it's a or I, but a is not a sentence. So if a sentence has articles, it can't be a single letter.
9. Some words have members of the set {a,I} even if they do not have pronouns. The word a.
10. All words that are members of the set {a,I} and are not sentences are articles. I can function as a sentence ("Who is it?" "I.") but a cannot, so so a word in {a,I} that's not a sentence must be a, which is an article.

• Very nicely done. – COTO Jun 21 '15 at 13:06

Making the following assumptions:

• The predicate "Some F have G even if they do not have D", could be understood as "Some F have G but don't have D" (it could be more laxly understood as "Some F have G regardless of having D or not")
• The word have can be used as in "I have a mother", not just traditional possession
• Nouns can be repeated
• All employees have bosses

Here's my solution:

$A$ = pots
$B$ = children
$C$ = people
$D$ = children
$E$ = adults
$F$ = men
$G$ = bosses
$H$ = employees

1. Pots do not have children.
2. Some people are children, but no adults are children.
3. Men that are pots can still be bosses if they have employees.
4. No people that do not have bosses can be both employees and children.
5. All children are children, and all children that are pots are also bosses.
6. If adults are children, they have employees.
7. Some but not all men are employees.
8. People that have children that are adults are not pots.
9. Some men have bosses even if they do not have children.
10. All men that are bosses and are not people, are adults.

I know there's a way to fill half of the words with wide concepts like "men" or "people", making it easier to find short-named subsets. Or nouns that you can have but are still broad, like "friends".

Edit: Considerably shortened $A$

• Welcome, and nice first post! A few of your statements are vacuous though, like 6 and 10 and the second half of 5, which isn't great... – Rand al'Thor Jun 20 '15 at 22:03
• @randal'thor you are right, the more I tried to fit nouns the less concerned I became with giving it meaning. I'll try to improve it and shorten it in the process. My previous A made more sense, but was too long. – Emilio Martinez Jun 20 '15 at 22:25