# What are the steps and thought process behind translating Exclusive Propositions into Standard Categorical Form?

Source: p 256, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

Many propositions that involve the words “only,” “none but,” “none except,” and “no . . . except” are exclusive propositions. Efforts to translate them into categorical propositions often lead to confusing the subject term with the predicate term. To avoid such confusion keep in mind that language following “only,” “none but,” “none except,” and “no . . . except” goes in the predicate term of the categorical proposition. For example, the statement

1. “Only executives can use the silver elevator”    is translated
[✓] 2. “All people who can use the silver elevator are executives.” [✓]

If it were translated
[✘] 3. “All executives are people who can use the silver elevator,”
the translation would be incorrect. [✘]

I accept that 2 is the correct, and 3 is the incorrect, translation of 1; but I wish to dig deeper: What are the steps and thought processes behind translating 3 (the generalisation of 1) into 4 (the generalisation of 2)? Can a Venn Diagram depict why only 4 is correct, and 5 is incorrect?

1. Only A are B.       ✓ 4. All B are A. ✓      ✘ 5. All A are B. ✘

Also note that many English statements containing “only” are ambiguous because “only” can be interpreted as modifying alternate words in the statement. [...]

• Only A are B == No non-A is B == Every B is not a non-A == All B are A – user9166 Jan 2 '16 at 19:02

I'll simplify a little bit the example :

Only Greeks are Boethians.

This is translated into logical form as :

All Boethians are Greek,

i.e. according to the schema : "All B are A".

This can be visualized as a Venn diagram where the set of Bs is (fully) included into the set of As; being so, there is no B that is not "an A", i.e. "Only A are B".

"All A are B", instead, will be visualized with the set of As (fully) included into the set of Bs; this is compatible with the fact that we can have some B "outside" the set of As, i.e. some Beothian, call it Bob, that is not A, i.e. it is not a Greek.

In this case it is not true that "Only A are B", because Bob is a Beothians which is not an A, i.e. not a Greek.