4.1.3. Checking Validity of Argument Forms

So, what do we mean by valid arguments in propositional logic? Suppose we are given a bunch of statements like this: the statements are; if you know the password then you can login to the network, and it is also given that you know the password; therefore, I am concluding that you can log on to the network. This is an argument which is given to you, and we have to verify whether this argument is logically correct or not.

Valid arguments consist of premises leading logically to a conclusion. In the provided example, if the premise states that knowing the password allows login (if p then q), and it is confirmed that the password is known (p), it follows that login is possible (q) based on logical reasoning. To validate the argument, we assess whether the conclusion is supported by the premises.

Think of a situation where you say, 'If it rains today, we will stay home.' Now, if you check the weather and see that it's indeed raining, you can conclude that you will stay home. The premises of knowing it's raining lead to the conclusion of staying home.

The bunch of statements before the conclusion is called the premises. You might be given one premise or multiple premises, and based on those premises, I am trying to derive a conclusion. So whatever is there before 'therefore' is called a premise, and whatever is there after 'therefore' is called the conclusion.

In logic, the premises are the statements or assumptions that provide the foundation for an argument. The conclusion is what follows from these premises. In the examples, statements that establish conditions for conclusions—like knowing the password—form premises, which lead to a logical outcome or conclusion.

Imagine a detective solving a case. The clues he gathers (premises) lead him to a conclusion about who the criminal is. Each piece of evidence contributes to linking the premises to the conclusion.

Well, if I view these two arguments in the English language sense, then they are different because we are talking about different things. But if I try to extract out these two arguments, then there is a similarity: both these arguments have a common structure, they have a common template.

Despite differences in content, arguments can share a fundamental logical structure. For instance, they may follow a format: if a condition (p) is true, then another condition (q) must also be true. Identifying this structure is essential in logic as it allows us to categorize and analyze arguments more systematically.

Consider different sports; everyone plays with a ball, yet the rules may differ. Basketball and soccer both use a ball following a 'if you position yourself right (p), then you score (q)' rule.

What I want to verify is whether this argument form is valid or not, whether it is correct or not. By valid, I mean whether it is correct or not. If I can say that the conjunction of premises implies the conclusion is a tautology, then the argument form is valid.

An argument form is valid if, whenever the premises are true, the conclusion automatically follows. The test for validity often involves determining if the statement formed by the premises leading to the conclusion consistently holds true (this is a tautology). This means that the truth of the premises guarantees the truth of the conclusion no matter the scenario.

Imagine a simple train schedule. If the train is scheduled to leave at 9 AM (premise), and you’re at the station by that time (premise), the conclusion follows—you will catch the train. If both premises are true, the conclusion must also be true.

Now how do I check whether a given argument form is valid? To check whether an argument form is valid or not, you check whether this implication is a tautology or not. The conjunction of premises implies the conclusion is a tautology.

One method to check the validity of an argument form is by constructing a truth table, which lays out all possible truth values for the involved propositions. The argument is valid if, in all scenarios where the premises are true, the conclusion is also true. This systematic approach helps visualize and confirm logical relationships.

Consider a vending machine: it operates on strict conditions. Each possible input (coin, selection) results in an output (snack). A truth table is like mapping all possible inputs to their respective outputs, helping you understand if it consistently works as expected.

So in general, I use rules of inference to prove the validity of large arguments. These rules of inference are simple argument forms whose validity can be easily established, and they serve as building blocks for larger arguments.

Rules of inference help simplify the verification of complex arguments by allowing us to apply well-established logical principles. These rules act as shortcuts, enabling easier proof of an argument's validity without needing to analyze every single premise directly.

Think of rules of inference like shortcuts in a recipe. Instead of measuring every ingredient for each dish, you use basic quantities established from previous cooking experiences. These shortcuts help you produce new dishes successfully.

There are some well-known fallacies which are incorrect arguments that might look valid at a high level. For instance, affirming the conclusion is one common fallacy where you might conclude p based on p → q and q, which is invalid.

Fallacies are logical errors that may appear correct but do not hold under scrutiny. In the case of affirming the conclusion, just because q is true does not mean p must also be true. Logical reasoning must ensure that the relationship is sound, and this specific combination can mislead people to incorrect conclusions.

Imagine someone claiming, 'If I eat all my vegetables, I will grow taller. I grew taller; therefore, I ate all my vegetables.' While it might seem plausible, they may simply have grown taller for other reasons, illustrating the fallacy.

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