Statements and truth values

“Close the door” can be obeyed or ignored, but it cannot be true or false. “The door is closed” can, and that one sentence of difference is where logic starts.

Sentences that can be true or false

Logic studies arguments: a list of sentences called premises, followed by a sentence called the conclusion that is supposed to follow from them. Not every sentence can sit in one of those places. A question such as “Is it raining?” asks for information. A command such as “Bring an umbrella” tells someone what to do. A cry such as “Ouch!” expresses pain. None of these says that something is the case, so none of them can be correct or mistaken in the way a claim can.

Compare “It is raining in Busan right now.” That sentence makes a claim about the world, and the world decides whether the claim is right. The open textbook forall x defines a sentence, in the logical sense, as a sentence that can be true or false. In maths classes and many textbooks the same thing is called a statement or a proposition. Being true or being false is called the truth value of the statement, so every statement has exactly one of two truth values.

A statement does not need to be one whose truth we know. “The number of grains of sand on Haeundae Beach is even” is a statement: it is either true or false, even though nobody will ever count. “There is life on a planet of another star” is also a statement. Logic cares whether a sentence has a truth value, and leaves the job of finding out which value it has to science, history or plain checking.

Opinions, vague words and open sentences

Here two traditions part ways. forall x warns readers not to confuse the idea of a sentence that can be true or false with the difference between fact and opinion: “Kimchi stew is the best food” still counts as a sentence that can serve as a premise. School maths in Korea is stricter. Its textbooks call something a proposition only when anyone can clearly decide whether it is true, so value judgements such as “best food” or “a beautiful song” are left out. In this course we follow the logic textbook, but on a school test you should follow the stricter rule.

Vague words cause a related problem. “Minho is tall” has no fixed truth value until we agree what counts as tall. A sentence with a variable, such as “x + 2 = 5”, is true when x is 3 and false otherwise, so on its own it is not yet a statement. Korean textbooks call such a sentence a condition. Pronouns behave the same way: “She was born in 2009” becomes a statement only once we know who she is.

Negation: turning a statement around

For any statement p, its negation, written ¬p and read “not p”, is true exactly when p is false. The negation of “The shop is open” is “The shop is not open.” That much is easy. The trap is with words like all and some. The negation of “All the buses are late” is not “All the buses are not late.” It is “Not all the buses are late”, which means at least one bus is on time. To check a negation, ask whether exactly one of the pair must be true in every situation. If both could be false together, you have written an opposite, not a negation.

Statements like “The shop is open” are called simple or atomic. Words such as and, or, not and if ... then join simple statements into compound ones, and the truth value of a compound statement depends only on the truth values of its parts. The next units work through those connecting words one at a time.