9.6. Arguments in English Statements

I want to represent a statement that every student in course number CS201 has studied calculus. If you are wondering what is this CS201 well at my institute IIIT, Bangalore the course number for discrete maths course is CS201 and say my domain is the set of all students in a college.

In this chunk, we are introduced to the concept of predicates and how to formulate logical statements using them. The example given illustrates that we need a clear understanding of the statement we want to express. Here, we're considering a specific course (CS201) at an institute and the students enrolled in that course. The goal is to represent the statement - 'Every student in CS201 has studied calculus' - using predicate logic.

Imagine a teacher who wants to verify if all students in a particular class passed a specific exam. They might say, 'Every student in my class has passed the exam.' Just as the teacher's statement can be made precise using logic, the course example uses predicates to define the relationship between students and their success in calculus.

So, let me first introduce a predicate here S(x) while you can use any predicate variable but I am using S(x) for my convenience. And, remember in the predicate world we use variables in capital letters for denoting predicate functions.

This chunk discusses how to represent the logical statements using predicates. Here, we define two predicates, S(x) for 'student x is enrolled in CS201' and C(x) for 'student x has studied calculus.' By using these predicates, we can create a logical formula that expresses the original English statement. We introduce the universal quantification, indicating the statement applies to every student in the defined domain.

Think of the predicates as labels. If I have a box of fruits and label 'A' for apples and 'O' for oranges, I can easily state characteristics about these fruits. Similarly, predicates allow us to label students according to whether they are enrolled or have studied a subject.

When I am saying that every student in my domain who is enrolled for CS201 has studied calculus the interpretation of that is that I am making a universal statement.

In this section, the concept of universal quantification is explained. A universally quantified statement means that the assertion applies to all members of the specified domain. In our case, if 'all students x' that are enrolled in CS201 must have studied calculus, it underscores the necessity that every individual in that group meets the criteria.

Consider the rule in a club where 'All members must attend monthly meetings.' As a rule, it applies to every member—there aren't any exceptions.

So I am writing down here two expressions. One expression is for all x, S(x) → C(x) this represents that for every x in the domain here domain is the set of all students in my college.

This chunk presents the logical expressions derived from the English statement. The first expression, 'for all x, S(x) → C(x),' signifies that if a student is enrolled in CS201 (S(x)), it implies that they have studied calculus (C(x)). This establishes a direct relationship between enrollment and studying calculus, where the implication holds true across all students in consideration.

Think of a rule in a game: 'If you score more than 50 points, you win.' This means for every player, having a score above 50 leads to winning. In this example, we relate two conditions with an implication.

Indeed S(Ram) is true and C(Ram) is true, so true implies true is true. Now S(Shyam) is true, C(Shyam) is true, so true implies true is also true.

In this section, the validity of the first logical expression is established through specific examples with named students. Using concrete instances like Ram and Shyam, the truth values of predicates S and C are evaluated. The analysis demonstrates that if the conditions of enrollment are satisfied, the conclusion about having studied calculus also holds true.

Imagine students attending a workshop: if every student (like Ram and Shyam) attends, you can conclude they all learn something. But if some don’t attend (like Balram), you can’t conclude they learned just because others did.

Let us see another example, so my domain is still the students of my college and I want to represent the statement that some student in class CS201 has studied calculus.

This section contrasts universal quantification with existential quantification. Here, the statement shifts to 'some student has studied calculus,' which does not require all students to meet the criteria. Instead, it suffices for at least one student in the domain to have studied calculus. This shifts our logical approach to existential quantification, allowing for a different predicate representation.

Think of it this way: in a classroom, if the teacher says, 'At least one student answered correctly,' it is enough for one individual to get the right answer for the statement to be true, as opposed to every student needing to answer correctly.

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