Categorical Logic · Translation

Translating into
Standard Form

Standard form categorical propositions have a precise structure: quantifier, subject term, copula, predicate term. Many ordinary English sentences express categorical claims without following this structure. Translation is the skill of finding the underlying logical form.

What counts as standard form?
Every standard form categorical proposition has four required elements in the right order.
1 — Quantifier
All · No · Some
2 — Subject Term
A noun phrase naming a class
3 — Copula
are · are not
4 — Predicate Term
A noun phrase naming a class
1
The Four Standard Forms
A
All S are P
universal affirmative
E
No S are P
universal negative
I
Some S are P
particular affirmative
O
Some S are not P
particular negative
2
How to Translate
1
Identify what claim is being made. Is it universal (about all members of a class) or particular (about some)? Is it affirmative or negative? This tells you which form — A, E, I, or O — you're aiming for.
2
Identify the subject and predicate classes. Both must be noun phrases that name groups of things — not adjectives, verbs, or proper names. You may need to rephrase (e.g. "mortal" → "mortal beings").
3
Choose the right quantifier and copula. The copula must be a form of "to be" — "are" or "are not." Other verbs must be absorbed into the predicate term (e.g. "Dogs bark" → "All dogs are barking animals").
4
Write it out and check. Does it follow the pattern exactly? Quantifier — subject term — copula — predicate term? Both terms name classes of things? If yes, you're done.
3
Four Tricky Sentence Types
Type 1
Statements with implicit quantifiers  ·  "Dogs bark"
Sentences like "Dogs bark" or "Politicians are liars" make a universal claim without stating a quantifier. The word "all" is implied. Additionally, the verb must be converted into a predicate noun phrase using "are" as the copula — absorb the verb into the predicate term.
Original
"Dogs bark."
Standard form A
All dogs are animals that bark.
Original
"Politicians lie."
Standard form A
All politicians are people who lie.
Type 2
"The" statements  ·  "The whale is a mammal"
Sentences beginning with "the" followed by a general kind — "The whale is a mammal," "The raven is a black bird" — make a universal claim about all members of that kind. Treat "the [kind]" as shorthand for "all [kinds]." Do not confuse with definite references to a specific individual.
Original
"The whale is a mammal."
Standard form A
All whales are mammals.
Original
"The raven is a black bird."
Standard form A
All ravens are black birds.
Type 3 · Tricky!
"Only" statements  ·  "Only A are B"
"Only A are B" does not mean "All A are B." It means "All B are A" — the subject and predicate switch. "Only" restricts what can be B: only things that are A qualify. So B-ness requires A-ness, which means B goes in the subject position of an A proposition.
Memory tip: "Only" introduces the predicate term of the A proposition. The class after "only" ends up as the predicate, not the subject.
Original
"Only citizens can vote."
Standard form A
All voters are citizens.
Original
"Only the brave deserve the fair."
Standard form A
All those who deserve the fair are brave people.
Type 4 · Tricky!
Exclusive statements  ·  "None but A are B"
"None but A are B" works exactly like "Only A are B" — it means "All B are A." "None but citizens can vote" = "Only citizens can vote" = "All voters are citizens." The subject and predicate switch, just as with "only."
Other exclusive phrases that work the same way: "No one except," "None except," "Only." All of them introduce the predicate term of the resulting A proposition.
Original
"None but the lonely know my sorrow."
Standard form A
All those who know my sorrow are lonely people.
Original
"No one except members may enter."
Standard form A
All those who may enter are members.
4
Worked Examples
Worked example 1
"Happy people rarely suffer."
1
Claim: universal — about all happy people. Negative — they don't suffer. → E proposition
2
Subject class: happy people. Predicate class: "suffer" is a verb → convert to "people who suffer."
3
"Rarely" implies "almost never" — treat as universal negative for logical purposes.
Standard form E
No happy people are people who suffer.
Worked example 2
"Some of the students passed."
1
Claim: particular ("some of"). Affirmative. → I proposition
2
Subject class: students. Predicate class: "passed" is a verb → convert to "students who passed."
3
Assemble: Some + students + are + students who passed.
Standard form I
Some students are students who passed.
Worked example 3
"Only licensed drivers may operate a vehicle."
1
"Only" → the class after "only" becomes the predicate. Predicate: licensed drivers.
2
Subject: those who may operate a vehicle — the class being restricted.
3
Universal affirmative with subject and predicate in the right positions. → A proposition
Standard form A
All those who may operate a vehicle are licensed drivers.
Worked example 4
"There are dishonest politicians."
1
"There are" signals existence — a particular affirmative claim. → I proposition
2
Subject class: politicians. Predicate class: "dishonest" is an adjective → convert to "dishonest people."
3
Assemble: Some + politicians + are + dishonest people.
Standard form I
Some politicians are dishonest people.
?
Challenge Problems
Translate each sentence into a standard form categorical proposition. Identify which form (A, E, I, or O) your translation takes.
1
"Mammals nurse their young."
2
"The tiger is an endangered species."
3
"Not all that glitters is gold."
Hint: "not all" translates to a particular negative.
4
"Only native speakers are fluent."
5
"None but the guilty need fear the law."
6
"There are no honest used-car salespeople."
7
"A few students failed the exam."