Quantifier Words: All, Some, None and More
What Are Quantifiers?
Quantifier words tell us how many things a statement is really about. For example:
All kids love to play.
No one arrived on time.
Only one person knows the answer.
All the kids except two went outside.
Someone is knocking at the door.
No cat likes taking a bath.
In math, there are special symbols that stand for these ideas:
∀ is read as “for all” or “for every”. What comes after this symbol is meant to be true for every object in the group.
∃ is read as “there exists”. What comes after this symbol is true for at least one object.
∃! is read as “there exists exactly one”. What comes after this symbol is true for one and only one object.
Mathematicians call these symbols quantifiers.
Understanding this kind of language is a key part of both mathematical and language literacy.
Why Does It Matter for Kids?
A large cross-linguistic study looked at how kids understand quantifier words like none, some, all, most. It included 768 five-year-olds and 536 adults across 31 languages. The results showed that kids, no matter what language they speak, tend to master these words in a particular order.
Around age 3–4, kids usually understand sentences like “All the dogs are running” or “All the cars are red” quite well.
Around age 4–5, they begin to reliably tell the difference between none and some, and understand that “No dogs are running” means that not a single dog is running.
Around age 5, kids start to grasp that some means “one or more, but not necessarily all.” For a long time, many kids treat “some” as “some but not all.” They think “some” excludes “all,” even though in logic “some” includes the possibility that it might actually be all of them.
The hardest part is noticing subtle differences in sentences like:
All the boys lifted the box (all the boys together lifted one box).
Each boy lifted a box (every boy lifted his own box).
To figure out which word applies to which part of the sentence, kids need to track several layers of meaning at once. Most begin to do this reliably only around 8–9 years old, when their ability to hold and compare nested statements in mind becomes stronger.
Another study looked at Czech math textbooks for grades 3–5. The researchers searched for problems that required understanding quantifiers. They found surprisingly few of them. The authors of the books seemed to assume that kids already understood what “all,” “each,” or “none” meant and did not need practice. Other studies show that this is not always true. This part of language and logical competence develops unevenly, and many kids need explicit support.
A Dutch study tested 65 kids aged 5–8 on problems containing the word “all.” Twenty-five of them made systematic errors. These kids then played a short 15-minute game with a tiger puppet who “spoke Dutch badly.” The tiger kept making mistakes, and the kids were asked to correct him. For example: “Look, it is important that each girl has a car. The toys that are left over do not count.” More than half of the kids improved significantly on standard “all/each” tasks right after this brief training, and for most of them the effect was still visible five weeks later.
All of this research points in the same direction: quantifier words like “all,” “some,” “none,” and “each” look simple, but they actually sit at the crossroads of language, logic, and math. Many kids need clear, playful practice to really understand them.
How Do We Teach?
Kids learn quantifier words best when they use them while doing real things. For example:
“Take all the cars out of the box.”
“We did not clear all the plates from the table. Look, your favorite plate is still there. We need to put it in the sink too.”
Over time, tasks can get a bit more complex. You might ask a kid to pick out only the blue T-shirts or to put all the socks into matching pairs. You can also start using the “there exists” idea in everyday language:
“Do you have any blue blocks in your drawer?”
“Is there a cookie left?”
“If you find a pencil, bring it here.”
“There’s someone at the door.”
Once kids understand the words same and different, you can talk about whether there are any matching pairs in a set, or whether all the items are different. You might ask questions like:
“Did all the kids go in the same direction?”
“Are there two identical cups, or are they all different?”
This is also where the odd one out game fits in. The kid not only has to choose the object that does not belong, but also explain why. For example:
“All these blocks are blue, and this one is red.”
“These cups have the same shape, and this one looks different.”
You can look for sets with two or more identical objects, or sets where all the items are different.
Gradually, you can move on to tasks that involve several sets at once:
“Which toy appears in all the gift bags?”
“Which toy does not appear in any of them?”
“Which toy appears in exactly one bag?”
It is helpful to talk about equal sets as well. Kids can solve problems such as:
“We made equal sets, but one toy is missing from one of them. Which toy is missing?”
“What do we need to add to each set so that all the sets become the same?”
The most challenging tasks compare one type of object with another. For example:
“Is it true that every red ribbon is longer than every green one?”
Here kids have to think carefully about what every and any really mean.
First Steps
Talking About What Is in a Set
The best way to learn these words is to follow instructions while doing real actions. For example, you tell a kid to take all the socks off the washing line. If one sock is still hanging there, it means they did not take all the socks, but all except one.
Operations
What's the order?
In many games and tasks, kids have to select all the objects in a set that have a certain property, such as all the birds in red hats or all the striped circles.
Besides all, another important word is some. In logical terms it matches the quantifier “there exists.”
If a picture shows three squares and some of them have eyes, that means one, two, or all three squares might have eyes.
If a problem says that there are at least two red apples, that means there cannot be zero or one red apple. There must be two, three, four, or more.
All and some
Find the described elements
All and some
Find the described elements
Sets
All and some
Quantifiers
All and some
Deep Understanding
Statements About Pairs of Objects
Sometimes a statement about all the objects in a set can be checked by looking at each item separately. But not always.
If you want to know whether all the fish are swimming in the same direction, you cannot just inspect each fish on its own. You have to compare each fish with others and check them in pairs.
True or false
All and some
The same is true when you ask, “Are there two identical cars in this set?” You cannot answer this by looking at the cars one by one. You need to compare them with each other.
In some tasks, kids have to find the odd one out. This means that all but one of the objects share some property, and one object does not. For example, several cups may have the same shape, and one cup has a different shape.
Other problems ask whether all the objects in the set are different from one another.
Quantifiers
All and some
Quantifiers
Different and same
Quantifiers
Different and same
All and some
Statements about pictures
Confident Mastery
Statements About Several Sets
Some statements talk about more than one set at once. To decide whether such a statement is true, kids have to compare sets with each other, not just individual items. For example: “Which gift is in just one bag?”
Sets and subsets
Equal sets
We can also ask questions like which gift is in every gift bag and which gift is not in any of the bags.
Sometimes kids need to add objects to different sets so that the sets become the same. To do this, they first have to look at all the sets together and figure out which items appear in at least one of them. Then they build a “complete” set and see what is missing in each of the smaller sets.
A particularly interesting kind of problem asks kids to compare each element of one type with every element of another type. For example: “Every green ribbon is longer than every red ribbon. True or false?”Here kids are really working with the idea of two quantifiers, “for every X and for every Y,” even if they do not see the symbols yet.
Sets and subsets
Equal sets
Sets and subsets
Equal sets
Sets
All and some
Sets and subsets
Equal sets
Big Ideas
Understanding words like all, some, none, and exactly one helps kids read math precisely. Later, these quantifier words become part of the language of sets, algebra, analysis, and geometry. They also matter in statistics and science, where the difference between “in some cases” and “in all cases” changes conclusions.
Math often stacks quantifiers. In calculus, the idea of a limit is written with three of them, so students learn to read layered phrases like “for every… there exists… for all…”. Game-winning strategies could be described with quantifiers as well: there exists my move, for every reply from my opponent there exists my next move, for every reply… and so on, until I win the game! There are so many alternating steps that we use an ellipsis to show the pattern continues.
Quantifiers also matter in law. They set the scope of rights and obligations. If a law says that all kids have the right to an education, all really means all. It includes kids with significant health needs and kids who do not speak the national language. Any exceptions have to be written explicitly in the law. Families are entitled to expect schools and education authorities to create the conditions that make learning possible for every kid covered by that “all.”
