If we add more premises (true or untrue) in support of the conclusion, a valid argument will stay valid. If an argument is correct, it is an instance of a correct argument form. However, just adding more premises does not allow the argument form to have valid premises and a wrong conclusion. For example, if we add the premise "Socrates is mortal" to the argument "Homer is mortal", we get the incorrect argument "Homer is Socrates". We can see from this example that even though these arguments have the same number of premises, only the first one is valid.

A formal definition of logic is a method of reasoning that uses strict rules for inference. In other words, logic is the study of how conclusions can be drawn from propositions using strict rules. Logicians have developed many different forms of argument, but they all follow certain basic principles. One of **these principles** is called "the law of non-contradiction". It states that a proposition cannot be both true and false at the same time-either it is true or it is false. This means that no matter what evidence we use to prove any statement, we can be sure that there will be some way in which that statement is either true or it is false. For example, let's say that we want to prove that Homer wrote The Iliad. We could do this by showing that many events mentioned in The Iliad also happened in **ancient Greece**.

- Why adding more premises to a valid argument will not affect its validity?
- What is the valid argument form?
- Does a true conclusion guarantee validity?
- What makes an argument valid in logic?
- What is the relationship between sound and valid arguments?
- Is it possible to have a valid argument that is not sound?
- What is the difference between a valid and an invalid argument?

An argument form is valid if, regardless of whatever specific statements are replaced for **the statement variables** in its premises, the conclusion is also true when all of **the resultant premises** are true. For example, consider the argument: "All cats are mortal; Socrates is a cat; therefore, Socrates is mortal." Although it might not be as easy to see, this argument is actually valid because whether or not Socrates is a human remains true even when all other propositions in the argument are assumed to be true.

Here are some common argument forms:

1. All cats are mortal.

2. Socrates is a cat.

3. Therefore, Socrates is mortal.

4. All dogs are mortal.

5. Plato is a dog.

6. Therefore, Plato is mortal.

7. All horses are mortal.

8. Empedocles is a horse.

9. Therefore, Empedocles is mortal.

10. All birds are mortal.

11. Ornithes is a bird.

12. Therefore, Ornithes is mortal.

13. All fish are mortal.

14. Plutarch is a fish.

An argument is valid if it has **all true premises** and a true conclusion. That is, the process of arguing consists in establishing connections between ideas, and these connections can be demonstrated or proved. Thus, argumentation is a method for determining truth value of statements.

An argument is valid if and only if it is required that if all of the premises are true, then the conclusion must also be true; it is impossible for all of the premises to be true but the conclusion to be false.

Thus, an argument is valid if and only if it is correct that if all of **its parts** are true, then the whole argument is true too. This means that if you accept **these three conditions**, your task as a thinker will be done: if you want to show that something is true, you should only need to find a way to prove one of its parts. The rest of the parts can be assumed to be true and with them, the entire argument will follow.

These conditions are known as "logical requirements" for argument validity. There are other conditions that some arguments may or may not satisfy; for example, an argument might have several disjuncts, each one supporting **a possible conclusion**, so this would seem to make the argument invalid because there's no single conclusion that can be inferred from all the parts of the argument.

An argument form is valid if and only if all of the premises are true, as well as the conclusion. If an argument's argument form is correct, it is valid. An argument is considered sound if and only if it is valid and all of its premises are true. A sound argument leads to a valid conclusion; a sound argument form leads to **a valid argument**.

If an argument has one or more erroneous premises or is not valid, it is not sound. Only if all of the premises are true can a valid argument have a correct conclusion. As a result, a valid argument can have a false conclusion if at least one of the premises is incorrect.

For example, let's say I argue that all animals are mortal because humans are animals and humans are mortal. This argument is valid because if something is an animal it cannot be immortal, and if something is mortal it cannot be immortal. However, this argument contains a mistake: it assumes that because humans are animals they must also be mortal. Although this argument is valid, it does not follow that all animals are mortal. Some animals may be immortal such as angels or gods.

Here is another example. Let's say I argue that all dogs love their owners because my dog Baxter loves me. This argument is not valid because even though both dogs and their owners are members of the species "animal", that does not mean that every member of the species "dog" loves its owner and every owner loves his or her dog. Many dogs may not love their owners at all or maybe they do but that doesn't mean that all dogs share this trait. In fact, according to research some dogs exhibit dominant behaviors toward other dogs they perceive as threats and show submission only toward people they view as superior.

An invalid argument is one that is not valid. There are many ways to show that an argument is or is not valid.

See how we used our knowledge of validity to prove our theorem? In fact, this process can be applied to any logical argument to determine its validity. As you can see, logic plays an important role in mathematics as well as science. Mathieu, will you explain why invalid arguments can lead to true conclusions?

In order for an argument to be valid, each premise must contain at least one truth value (i.e., true or false). If one of the premises contains more than one truth value, then the argument is invalid. For example, suppose we were to argue that Obama is lazy by stating that he wears **dirty clothes**, hassles staff members, and spends most of **his time** watching TV. All of these statements are true, so our argument is valid. However, if we changed **the first statement** to say that Obama wears clothes that match his background color, then the argument would become invalid because now it contains two truths: that he wears dirty clothes and that his clothes match his background color.