9.5. Further Examples

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. Let me retain the same two predicates S(x) and C(x) from the previous example. So again, we have to understand whether this statement is universally quantified or is it existential quantified whether it involves any kind of 'if then' or not and so on.

In this example, we are considering the statement 'some student in CS201 has studied calculus.' To convert this to predicate logic, we need to decide how to quantify this statement. The statement speaks about at least one student, which indicates it should be represented as an existential quantifier. Therefore, we express it as 'there exists some x in my domain such that S(x) and C(x) are both true.' The use of existential quantification signifies that we're only interested in at least one instance (or one student), who meets both conditions: being enrolled in CS201 (S(x)) and having studied calculus (C(x)).

Think of a classroom with a mix of students. If you say, 'At least one student in the room is wearing glasses,' you are making a statement that does not require every student to be wearing glasses, just that one student meets the criteria. Similarly, the statement about 'some student in CS201' does not require all students in that course to study calculus, just at least one does.

Now an interesting question here is why cannot we represent this assumption by this second expression there exists x such that S(x) → C(x) might look that this second expression also can represent the same assertion but that is not the case because if you closely see here this second expression, this expression becomes true even for an x who is not enrolled for CS201.

The issue with using the expression 'there exists x such that S(x) → C(x)' lies in the nature of the implication. In predicate logic, an implication (P → Q) is considered true if P is false, regardless of Q's truth value. Therefore, if we find an x such that the student is not enrolled (S(x) = false), the statement will be true irrespective of whether that student studied calculus or not (C(x)). This does not accurately capture our original statement, which specifies that we want a student who is both enrolled and has studied calculus.

Imagine someone saying, 'At least one of my friends is nice.' If that person adds, 'or if he is not nice, then he is still a good friend,' it suddenly allows for the possibility that the friend could be mean, which contradicts the initial idea. Thus, the original statement fails to hold true under this changed logic.

So to make my point more clear, our goal is to identify whether it is the first expression or whether it is the second expression which represents my assertion that some student in CS201 has studied calculus or not.

To validate which expression is correct, we utilize a scenario with three students: Ram, Shyam, and Balram. If we assume that none of these students are enrolled in CS201, we must evaluate the truth values of both expressions under this assumption. The expression 'there exists x such that S(x) and C(x)' results in false because there isn't any student who meets both criteria. Conversely, the expression 'there exists x such that S(x) → C(x)' produces true regardless of students' enrollment statuses because of the nature of implications in logic. Thus, by demonstrating that only the first expression accurately reflects the original assertion, we reinforce the notion that accuracy in logical representation is critical.

Consider going to a party where you declare, 'At least one person at this party brings a gift.' If it's true that nobody brought gifts, but later you say, 'Someone didn’t bring a gift, but it’s still a good party,' it undermines the original expectation of the statement about gifts. Being precise in language reflects our intentions clearly.

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