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True or false

Logic and puzzles · level 1, lesson 4 of 6

A question to start

Is this sentence true?

All birds can fly.

Most people say "yes!" straight away. Robins fly. Ducks fly. Eagles fly. Pigeons fly.

But think about a penguin. A penguin is a bird. It has feathers, wings and a beak, and it lays eggs. But it cannot fly at all. It waddles and it swims.

So is "all birds can fly" true, or false? And how many penguins do you need to find before you can decide?

The idea

Sentences that tell you something

Some sentences tell you something about the world. "A cat has four legs." "Snow is cold." "The sun is hot."

A sentence like that is either true or false.

  • True means it matches the world. A cat really has four legs. True!
  • False means it does not match the world. "Snow is hot" is false, because snow is cold.

A false sentence is still a sentence. It is just a wrong one.

Sorting sentences

You can sort sentences, just like buttons, with the rule "Is it true?"

true false a cat has four legs snow is cold 2 and 2 make 4 a fish can ride a bike snow is hot 3 is more than 5

Each sentence goes into one pile. Look at "a fish can ride a bike". It is a silly sentence, and it makes us laugh. But it is still a sentence about the world, and it is false. Silly sentences are usually false. That is what makes them silly!

How do you know?

For many sentences you look or check.

  • "This cup is full." Look in the cup.
  • "2 and 2 make 4." Count it on your fingers: 2 fingers and 2 more fingers is 4 fingers. True.
  • "3 is more than 5." Count to 5: 1, 2, 3, 4, 5. You say 3 first, so 3 comes before 5. It is less. So the sentence is false.
0 1 2 3 4 5 6 7 8 9 10 3 comes first 5 comes later

For other sentences you need to know something, or ask someone who knows, or look it up.

  • "Mars has one moon." This is false. Mars has two small moons, named Phobos and Deimos. You cannot see this from your window, but scientists have looked with telescopes and with spacecraft.

The word "all" is a big word

Let us go back to the birds. "All birds can fly" is a big promise. It says: pick any bird in the whole world, and it can fly. No bird is left out.

To show the promise is broken, you only need one bird that cannot fly. The penguin is one. So:

"All birds can fly" is false.

robin flies ✓ duck flies ✓ eagle flies ✓ pigeon flies ✓ penguin cannot fly ✗

It does not matter how many flying birds you find. A hundred robins and a thousand ducks cannot fix it. One penguin is enough to make it false. A thing that breaks an "all" sentence like this is called a counterexample (say "COUNT-er-example"). The penguin is a counterexample to "all birds can fly". (There are more: ostriches, emus and kiwis cannot fly either.)

"Some" is a smaller word

What about this one?

"Some birds can fly."

"Some" means "at least one". To show it is true, you only need one bird that flies. A robin flies. So this sentence is true.

The two words work in opposite ways:

Sentence One example that fits One example that does not fit
"All birds can fly." does not prove it proves it false (the penguin)
"Some birds can fly." proves it true (the robin) does not matter

Not every sentence is true or false

Some sentences do not tell you anything about the world, so they are not true and not false.

  • Questions. "Is it raining?" You can answer it, but the question itself is not true or false.
  • Orders. "Sit down!" "Please pass the salt." These ask someone to do something. They are not true or false.
  • Opinions. "Ice cream is the best food." You may think so, but your friend may love pizza more. People can disagree and both be fair. In logic we stay away from opinions, because there is no way to check.
Is it raining? a question Sit down! an order Ice cream is the best food. an opinion not true and not false

Some sentences change

"It is raining." Is that true? It depends on when you say it. Today it might be true, and tomorrow false. "I am 6 years old" is true for one year and then false! When a sentence changes like this, you have to say when (or who, or where) before you can decide. You will meet sentences that are "sometimes true" in level 2.

Fixing a false sentence

You can often turn a false sentence into a true one by changing a word.

  • "A cat has six legs." False. Change it: "A cat has four legs." True.
  • "All birds can fly." False. Change it: "Most birds can fly." True. Or: "Some birds can fly." True.
  • "3 is more than 5." False. Change it: "3 is less than 5." True.

Worked examples

Example 1. True or false? Decide for each one, and say how you know.

  1. "A triangle has 3 corners." Count them: 1, 2, 3. True.
  2. "Ice is hot." Touch an ice cube: it is cold. False.
  3. "5 and 1 make 6." Count 5 fingers, then 1 more: 6. True.
  4. "A dog can talk." Dogs bark, but they cannot say words. False.

Example 2. Find a counterexample. "All fruit is sweet." True or false?

  1. This is an "all" sentence. One fruit that is not sweet will break it.
  2. Think of fruits: apple (sweet), strawberry (sweet), lemon: a lemon is very sour!
  3. So a lemon is a counterexample. "All fruit is sweet" is false.
  4. Fix it: "Many fruits are sweet." True.

Example 3. Some or all? Are these true?

  1. "All numbers are bigger than 1." Is there a number that is not? Yes: 1 is not bigger than 1 (it is the same), and 0 is smaller. So it is false. 1 is a counterexample.
  2. "Some numbers are bigger than 1." Is there one? Yes: 2. So it is true.

Example 4. Is it a true or false sentence at all?

  1. "Close the window." An order. Neither.
  2. "The window is closed." Look at the window. This is true or false: it is a real sentence about the world.
  3. "Is the window closed?" A question. Neither.
  4. "Windows are beautiful." An opinion. People can disagree. Neither (in logic we do not count it).

Try it

  1. True or false: "A spider has 8 legs."
  2. True or false: "4 is more than 7." Fix it if it is false.
  3. True or false: "All animals have fur." If it is false, give a counterexample.
  4. True or false: "Some animals have fur."
  5. Is "Go to bed!" true, false, or neither?
  6. True or false: "All shapes with 3 sides are triangles."
  7. True or false: "Mars is called the red planet." (Ask a grown-up or look at a picture of Mars.)
  8. Make up one silly false sentence and one true sentence. Ask someone at home to sort them.
Answers
  1. True. Spiders have 8 legs. (Insects, like ants and bees, have 6.)
  2. False. Count: you say 4 before 7, so 4 is less. Fixed: "4 is less than 7." Or "7 is more than 4."
  3. False. A fish has no fur. So does a snake, a bird, a frog. Any one of these is a counterexample.
  4. True. One example is enough: a cat has fur.
  5. Neither. It is an order. It tells someone to do something; it does not say how the world is.
  6. True. That is what the word "triangle" means: a flat shape with 3 straight sides. There is no shape with 3 sides that is not a triangle.
  7. True. Mars looks reddish orange, because its dust and rocks have rust in them, like an old nail.
  8. For example: "My shoe can sing" (false) and "I have two hands" (true, for most people). Any pair is fine.

Common mistake

Thinking lots of examples make an "all" sentence true. "I saw ten birds and all of them flew, so all birds can fly." But there are millions of birds, and you have not seen them all. One penguin is enough to break it. Many examples that fit do not prove an "all" sentence. One example that does not fit does break it.

Calling a question or an order false. "Sit down!" is not false. It is not true either. It is not that kind of sentence.

Where it leads