Everyday speech rarely uses the exact words of the four forms. Before you can test reasoning, you rewrite each statement in standard form. The meaning must stay the same; only the wording changes.
| Everyday wording | Standard form |
|---|---|
| Every dog barks. / Dogs bark. | All dogs are barkers (animals that bark). |
| None of the guests left early. | No guests are people who left early. |
| A few campers sing. | Some campers are singers. |
| Not every runner is a swimmer. | Some runners are not swimmers. |
| Maya is a hiker. | All people identical to Maya are hikers. |
| All that glitters is not gold. | Some things that glitter are not gold things. |
Key idea
Three steps: (1) make the verb "are" plus a class ("bark" → "are barkers"); (2) choose the quantifier from the meaning: all, none, at least one, or at least one not; (3) write it as All/No/Some S are (not) P. "Many" and "most" become "some". A statement about one named thing becomes a universal about a one-member class.
Why it works
Standard form has only four quantifiers, so every everyday quantifier must map to the one that keeps what it guarantees. "Most" guarantees at least one, so it becomes "some" (we lose information, but we never add any). A singular statement such as "Maya is a hiker" says something about every member of a class with exactly one member, so it acts as a universal: "All people identical to Maya are hikers."
Worked examples
Worked example: A verb and a hidden quantifier
Put "Some skaters dance" in standard form.
- Turn the verb into "are" plus a class: those who dance are dancers.
- "Some" already means at least one.
Answer: Some skaters are dancers.
Worked example: Not every
Put "Not every hiker is a poet" in standard form.
- "Not every hiker is a poet" says that at least one hiker is not a poet.
- That is the particular negative form, O.
Answer: Some hikers are not poets.
Common mistake
Writing "All S are not P" for "No S are P". "All S are not P" is ambiguous: "All that glitters is not gold" means that not all that glitters is gold, an O statement. Never use "All … are not" in standard form: write "No S are P" for E and "Some S are not P" for O.