1.3. Types of Mathematical Logic

It turns out that there are several applications of logic. So for example, we use it extensively in Boolean algebra, which forms the basis of computer architecture. We also use it for program verification and validation.

Mathematical logic is used in many fields, particularly in computer science and engineering. One of the primary applications is in Boolean algebra, which underlies the design of electronic circuits in computers. Additionally, mathematical logic is crucial for verifying and validating computer programs, ensuring that they perform the intended tasks correctly.

Think of programming like baking. Just like you must measure ingredients accurately to create a successful recipe, programmers must validate their code to ensure that it produces the correct outcomes under various conditions. If a recipe (program) doesn’t work, you need to identify what went wrong and fix it.

So let us begin our discussion on propositional logic. So we first define what is a proposition. So informally, a proposition is a declarative statement which is either true or false.

A proposition is a basic mathematically meaningful statement that can be definitively classified as true or false, but not both. For example, saying 'New Delhi is the capital of India' is a proposition because it can be checked and labeled accordingly. Understanding propositions is fundamental to engaging with logical operations and arguments.

If I say, 'The sky is blue', you can verify this statement by looking outside. It is either true or false. In contrast, asking 'What is your favorite color?' does not yield a definitive true or false answer, hence it's not a proposition.

A compound proposition is a larger proposition, which is obtained by combining many propositions using what we call as logical operators.

Compound propositions are formed by combining simple propositions with logical operators such as AND, OR, and NOT. These operators allow us to create more complex logical statements from basic ones. For example, if 'p' is 'It is raining' and 'q' is 'The ground is wet', then 'p AND q' means 'It is raining and the ground is wet'. Understanding how these operators function is key to effective logical reasoning.

Imagine you’re planning a picnic. You might say, 'We will go if it does not rain AND the picnic area is open'. This combines two propositions: one about the weather and another about the location, making it a compound proposition.

It turns out that since we are dealing with mathematical logic where each variable can take only two possible values, then each of this possible question marks can be either true or false.

In mathematical logic, particularly in the binary system we often use, each logical statement can be evaluated as either true or false. This simplicity leads to a variety of possible logical operators. For example, we can define up to 16 different logical operations with two propositions, thanks to the different combinations of true and false values.

Think of a light switch: it can either be 'on' (true) or 'off' (false). You can describe various conditions based on this binary state. If you combine the states of multiple switches in different arrangements, you can create a complex set of rules for your lighting system.

Let us define another logical operator, which is the conditional statement. This is also called as if-then statement and we use this notation p → q.

A conditional statement, often expressed in the form 'if p then q', establishes a relationship between two propositions where the truth of one proposition (p) guarantees the truth of another (q). The statement is logically false only when p is true and q is false. For example, 'If it rains, then the ground will be wet' is a conditional statement.

Imagine a school rule: 'If a student breaks the rules (p), then they will receive detention (q)'. This establishes that breaking the rules leads to a consequence, but if the student isn’t breaking the rules, it doesn’t matter whether they get detention; the condition is not set.

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