Did you know that there are many types of logic? In this section, we will introduce classical logic.
Let us assume truth values can take on two values: true or false. Then, we can talk about three classical laws of thought over propositions and truth values:
- Law of Identity:
- “A proposition’s truth value is equal to the same propositions’ truth value.”
- Also: “Whatever is, is.”
- Law of Non-Contradiction:
- “Propositions can never be both true and false.”
- Also: “Things can’t be both true and false.”
- Law of Excluded Middle:
- “Propositions are either true or false.”
- Also: “If something is not true, it is false. If something is not false, it is true.”
After thinking about the three laws about, you’ll probably think that they’re obvious. But let’s break it down and dive into these ideas:
Dive: Law of Identity
Suppose “it is raining.”
However, it may stop raining later or you may want to talk about a time in the past where “it is not raining.”
This brings us to our first idea: when we talk about logic, we are usually talking about things in an instant of time or things that do not change with time to prevent our premises from changing (unless we say so).
There does exist systems of logic where time is considered as part of logic (temporal logic). However, because we are trying to learn logic simply, let us assume that our statements and propositions will not change once we accept them as a premise.
Dive: Law of Non-Contradiction
Think about it: can things ever be both true and false? Certain schools of philosophy say reality itself as an illusion, and yet, still reality itself. You could say that they hold that “reality is not real and reality is also real” as true. According to their kind of logic, that sort of statement is valid. So thinking about “the truth” of a statement can be complicated, because it depends on what rules of logic you are using.
Because we want to deal with a simple logic — classical logic — to make things simple, we will assume that things cannot be both true or false.
Dive: Law of Excluded Middle
Certain things could be thought about as being neither true nor false. Take the statement: “I am smart.” This would depend on who is saying this statement, why somebody is saying it, who that person is comparing themselves to and how they are comparing themselves to other people. Often, in real life, it is too simple to reduce such a complex statement to a just a “true/false” statement.
However, since we are talking about classical logic, let us try to simplify things such that things are either true or false, and nothing else. Although this may seem too simple, you’ll quickly find that things can still get quite complex even though we are simplifying things.