# How do I interpret “All A are only B”? (Categorical logic)

I'm having trouble interpreting "All A are only B".

At first, it seems like it's a redundant form of "All A are B". The conditional relation between A and B seems the same (if A then B). But when trying to put it into a Venn diagram or into standard-form syllogism, I have trouble.

Example:

A mother states a rule that "All kitchens are for eating". Her child dances in the kitchen. And so the rule strictly speaking hasn't been violated. But if the mother states "All kitchens are only for eating" and the child dances in the kitchen, are they breaking the rule? How would this look like in standard-form, or how could I diagram it?

• This is logistically troublesome. Example: can you breathe in a kitchen? sit? stand? Those things are not eating. Formally, you're saying "for all possible conditions on a set S (there are 2^S of these), all members of subset A have property B and no other property". I don't think you can construct a non-trivial example where this could be true. – barrycarter Mar 24 '18 at 14:52

Yes, if the mother says that the kitchen is for eating only, then the child has broken the 'law' when dancing in the kitchen.

In general, 'All A are only B' means that all A's can be B's but they cannot be anything that is not a B. As such, notice that this claim is quantifying over predicates; not something you can express using your typical first-order logic statements.

Still, if you have a Venn diagram with A, B, and C, then you can indicate that there cannot be any A that is a C but not a B, i.e. the area inside A and inside C but outside B must be empty.

I could not reach the statement “All A are only B” by manipulating the statements “If A then B” or “Not-A or B”. The answer to this question seems to be a definition: “kitchen” is a room where the only correct activity is eating.

-If this room is a kitchen, then it is only for eating.

-This room is a kitchen.

-Therefore this room is only for eating.