4.2. Summary

What 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 in propositional logic are those that logically follow from their premises to their conclusions. In this example, the argument states that knowing the password (premise) leads to the ability to log on to the network (conclusion). The structure is crucial; if the premises are true, the conclusion must also be true for the argument to be considered valid.

Imagine you're planning to bake cookies. Your premises could be: 'If I have flour, then I can bake cookies' and 'I have flour'. The conclusion would be, 'Therefore, I can bake cookies.' This argument is valid because if both premises are true, the conclusion logically follows.

Therefore, the bunch of statements before the conclusion are called premises. If you might be given one premise or two premises or multiple premises, based on those premises I am trying to derive a conclusion.

Premises are the foundational statements upon which an argument is built. They provide the necessary information to draw a conclusion. The conclusion is the result or inference based on those premises. For example, if we have two premises about knowing a password and its implications, they collectively support the conclusion about accessing the network.

Think of premises as ingredients in a recipe. If you have the right ingredients (premises), you can create the dish (conclusion). For instance, 'If I have eggs and flour, then I can make a cake.' Here, the premises are the eggs and flour, and the conclusion is the cake.

If I try to extract out these two arguments, then there is a similarity that both these arguments have a common structure, they have a common template.

Many valid arguments can be represented by a common structure in propositional logic, exemplified by 'If p, then q; p; therefore q'. This template allows us to generalize the validity of arguments without focusing solely on their specific content. By evaluating this structure, we can determine the overall validity of various logical arguments.

Consider a school rule: 'If a student studies hard (p), then they will pass the exam (q)'. If you know that a student studied hard (p), you can conclude they will pass (q). This structure can be applied to many situations, reinforcing how this format provides clarity in logical reasoning.

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. What do I mean by argument form is valid my definition here is I will say my argument form is valid if I can say that the conjunction of premises implies conclusion is a tautology.

A valid argument form is one where, if the premises are true, the conclusion must also be true. This validity can be checked by determining if the relationship between the premises and conclusion always holds true in every possible scenario, establishing it as a tautology. For example, knowing that 'If p then q' holds, we need to ensure that in every case that p is true, q must also be true.

Think of setting a table for dinner. Premise 1: If it's dinner time (p), then people will be hungry (q). If dinner time is reached, it always follows that people will start eating (the conclusion). This holds true as long as the established premises are correct.

How do I check whether a given argument form is valid? My definition says to check whether this implication is a tautology or not. I can say that I can use rules of inferences, which are very simple argument forms whose validity can be easily established using the truth table method.

Rules of inference are logical rules that dictate how conclusions can be derived from premises. By applying these rules, we can simplify complex arguments to determine their validity without constructing lengthy truth tables. These rules provide a systematic approach to evaluating arguments based on previously established logical forms.

Consider rules of inference as shortcuts in a math class. Just as you learn formulas to quickly solve equations, these rules help you efficiently reach conclusions in logic based on known premises, bypassing the need for time-consuming evaluations every time.

Now there are some well known fallacies which are incorrect arguments but on a very high level it might look a valid argument but they are very subtle incorrect arguments.

Fallacies are logical errors in reasoning that can often seem plausible. For example, 'Affirming the consequence' and 'Denying the hypothesis' can mislead us into thinking an argument is valid when it is not. Understanding these fallacies is crucial for critical thinking as they help us identify flawed reasoning.

Think of a popular myth: 'If you eat healthy food, you will never get sick.' If someone gets sick despite eating healthy, the fallacy illustrates a flaw in the reasoning. Just because eating healthy reduces the likelihood of illness doesn't guarantee invulnerability—similar to how fallacies can mislead in logic.

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