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Subsection 1.2.1 Truth Values

We are going to study two-valued logic.  This means that every logical statement is either:

  • True, or

  • False

Activity 1.2.1.

Here are some examples of true statements:

  1. 2 + 2 = 4.

  2. The Earth revolves around the sun.

  3. Paris is the capital of France.

  4. It never snows in Austin in August.

  5. The moon is in synchronous rotation with the Earth.

  6. If it’s raining, the sidewalks will be wet.

Every second grader agrees that 1 is true. 2 is pretty much universally agreed nowadays to be true, but Galileo [1564 – 1642] spent the last years of his life under house arrest after being tried by the Inquisition for arguing for it. 3 is true. 4 has solid empirical evidence behind it. 5 is also now known to be true. 6 corresponds to an if/then observation that we’ve all made about the world.

Activity 1.2.2.

Here are some examples of false statements:

  1. 2 + 2 = 5.

  2. The sun revolves around the Earth.

  3. London is the capital of France.

  4. It never snows in Fairbanks in January.

  5. The moon is made of green cheese.

There’s not much controversy about any of these nowadays.

Notice that, to be a statement, a sentence must have a truth value. We don’t require that we happen to know what that truth value is.

Activity 1.2.3.

Here are some examples of statements, each of which is true or false, but I, at least, don’t know which:

  1. The social security number of the President of the United States is 224-78-5742.

  2. Every even integer greater than 2 is the sum of two prime numbers.

  3. \(\displaystyle P \not = NP \)

  4. There was life on Earth 4 billion years ago.

Someone knows whether 1 is true. Just not me and, I’m pretty sure, not you. 2 is called Goldbach’s conjecture. It’s a famous claim in mathematics. No one knows for sure whether or not it is true. No one has yet been able to prove that it’s true or to find a counterexample that shows that it isn’t. 3 is a million dollar problem in computer science theory, in which P and NP are well-defined classes of problems. (No kidding – if you can prove either that it is true or that it is false, you’ll get $1,000,000.) To learn more about it, google \(P=NP\text{.}\) 4 might be true. It’s widely acknowledged that there was life on Earth 3.5 billion years ago and some studies suggest that it started more than 4 billion years ago.

Problems 1.2.4.

For each of the following claims, indicate what we know about its truth value. (Go ahead, Google if you like.)

(a)

There were people in North America 16,000 years ago:

  1. It is true and known to be so.

  2. It is false and known to be so.

  3. It is either true or false but no one (at least that the rest of us is aware of) knows which.

Answer.
c is the correct answer.

(b)

Before World War II, a “computer” was a person

  1. It is true and known to be so.

  2. It is false and known to be so.

  3. It is either true or false but no one (at least that the rest of us is aware of) knows which.

Answer.
a is the correct answer.

(c)

The Greeks were the only ancient people to study logic.

  1. It is true and known to be so.

  2. It is false and known to be so.

  3. It is either true or false but no one (at least that the rest of us is aware of) knows which.

Answer.
b is the correct answer.
Solution.
Explanation: We know, for example, that logic was studied in the ancient civilizations of China, India and Persia.

(d)

In the late 1930s, Alan Turing proved that the halting problem is undecidable.

  1. It is true and known to be so.

  2. It is false and known to be so.

  3. It is either true or false but no one (at least that the rest of us is aware of) knows which.

Answer.
a is the correct answer.
Solution.
Explanation: He did so in his 1937 paper, “On computable numbers with an application to the Entscheidungsproblem”.