8. 1.1. Motivation of Studying Predicate Logic

Even though propositional logic is very interesting, it cannot represent all kinds of mathematical statements that we are interested in. For example, consider the declarative statement 'x is greater than 3'. This is not a proposition because until the value of x is assigned, we cannot determine whether the statement is true or false.

Propositional logic deals with propositions that are either true or false. However, when a statement includes a variable, like 'x', it cannot be definitively categorized as true or false until 'x' is given a specific value. This limitation shows we need a more complex system to handle such statements.

Think of it like asking, 'Is it raining?' If you don't have a specific date or a time frame to refer to, you can't answer the question with a simple 'yes' or 'no'. The situation is similar with mathematical statements involving variables.

We want to characterize the property that whatever the value of x, it is either greater than 3 or not. Greater than 3 is the property we wish to express for the subject x. To represent these types of statements in predicate logic, we introduce a function called a predicate function.

In predicate logic, we can express more complex properties by using predicates, which are functions that involve variables. By introducing predicates, we can refer back to the variable 'x' in a way that allows it to take any value and still make a determination about whether that value satisfies the property we want to express.

Imagine you have a box of toys. You might want to say, 'Any toy in this box is a car.' However, without identifying which toy it is, you can't confirm whether that claim is true for a specific toy. Once you pick a toy (like assigning a value to x), you can say indeed whether it is a car or not.

The predicate function typically uses a capital letter to differentiate from propositional variables, which are symbolized by lowercase letters. When we assign a specific value to x in the predicate, it transforms into a proposition.

Predicates help us formulate statements that can later be evaluated as propositions when specific values are assigned. For example, a predicate P(x) allows us to create statements like P(4), which can then determine the truth status—such as '4 is greater than 3'—and turn that into a proposition.

Think of predicates like a recipe. The recipe states, 'If you put x cups of sugar, then you'll have a sweet dish.' Here, x represents the amount of sugar used. Only once you measure and pour the sugar can you say, 'I used 4 cups of sugar and now I have a sweet dish.'

We can define multi-valued predicate functions for statements involving multiple subjects. For example, 'x = y + 1 + 3' becomes P(x, y). Until we assign values to x and y, we cannot determine the truth of the proposition.

Multi-valued predicates allow for more complex statements involving more than one variable. This formulation enables us to express relationships between multiple quantities, representing conditions that may vary with different inputs or values assigned.

Consider a math class where you want to verify students' scores. The statement 'Student A's score equals Student B's score plus 5' cannot be evaluated without knowing the scores of the students. Once you have those numbers, you can confirm or refute the statement.

We are particularly interested in two methods of converting predicates into propositions: explicit assignment and the quantification method. The quantification method is preferred, as it allows us to express broader truths about all or some elements in a domain.

By using quantifiers, we can express statements that pertain to entire sets or domains without the tedious process of assigning specific values individually. This makes our logical expressions much more powerful and expresses general truths effectively.

Think of quantification like saying 'All students in the school must wear uniforms' versus checking each student one by one. The quantification saves time and effort while covering all students collectively.

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