# What's the difference between "All A are B" and "A is B"?

I'm having trouble catching the difference between these two forms. I tried to draw some Venn diagrams but it's not helping.

"All A are B" means A is contained in B.

Does "A is B" mean A is also contained in B? Or that A is equal to B (the set is the same)?

Thank you!

• "A is B" is ambiguous between "A=B" and "A is included in B". The source of the problem is whit the ambiguity of "is" itself in natural Language. Sep 27, 2019 at 11:38
• Regarding Aristotelian Syllogism, see The Structures of Assertions Sep 27, 2019 at 11:41
• One of the problems with mapping language onto semantics is that you need to keep track of the types of your variables. A is B means A = B if A and B are both sets (The set of integers is the set of rational numbers with unit denominators), but B(A) if B is a property (The ball is red) and it may mean All A are B if A and B are categories (A man is an animal). This is one of the reasons a lot of logicians adopt font or case distinctions to clarify their expressions or they declare everything with a type. There is no answer to your question that applies to all options for the type of A and B
– user9166
Sep 27, 2019 at 17:43

There are four standard forms of categorical propositions from which syllogisms are constructed. Here is Wikipedia's description:

The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called A, E, I, and O). If, abstractly, the subject category is named S and the predicate category is named P, the four standard forms are:

All S are P. (A form)
No S are P. (E form)
Some S are P. (I form)
Some S are not P. (O form)

Let's consider the question:

"All A are B" means A is contained in B.
Does "A is B" mean A is also contained in B? Or that A is equal to B (the set is the same)?

"All A are B" can be easily paraphrased to fit the A form given above just by substituting S for A and P for B.

However, "A is B" does not specify a quantifier. It could be paraphrased as "All S are P", "No S are P" or "Some S are P". It is ambiguous as it stands and one would need more information to decide which categorical proposition best represents it.

Wikipedia contributors. (2019, September 2). Categorical proposition. In Wikipedia, The Free Encyclopedia. Retrieved 12:08, September 27, 2019, from https://en.wikipedia.org/w/index.php?title=Categorical_proposition&oldid=913715701

• Thank you ! It really helped me ! Sep 27, 2019 at 12:33

First, try good grammar. All A are B is just bad grammar. You should say: All A's are B. That is, "all" implies a plural here, whereas in "A is B", "is" implies a singular.

An example of A is B may be: Joe is Irish. That is, one thing, Joe, has the one quality, property or attribute of being Irish.

An example of All A's are B would be: All Irish Catholics are Irish.

Please note the difference with for example: The population of Irish Catholics living in Britain is quite large. That is, the population is quite large, not necessarily the Irish Catholics themselves. So, it is an instance of A is B, not All A's are B.

The form A is B is also used to say, for example "Superman is Kent Clark". Here, this is a relation between two things, not between a thing and a quality, but we can also do good logic with that: "If Superman is Kent Clark, then, if Kent Clark is hungry, then Superman is hungry, too". The word "is" here is used in the two different ways I just explained but, in spoken languages, we are seldom confused as to which is which because the context usually gives us the necessary clues.

• Please. Clark Kent. Not Kent Clark. Sep 28, 2019 at 19:28
• @gnasher729 Oops, sorry! Not me, guv, my brain. I guess it may have had Kenneth Clark in mind, also a superhero, of British politics, so the confusion is understandable. And, the logic of my example does not rest on historical accuracy, let alone on comic strips' lore. Sep 29, 2019 at 7:24