9.2.2. Translating English Statements using Predicates

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 start by identifying the English statement we want to translate into predicate logic. The statement talks about 'every student in course CS201 has studied calculus'. We define the context or 'domain', which is all students enrolled at a college. In this case, we also specify what CS201 is, making it relevant to our example.

Imagine a teacher wanting to keep track of all the students in a specific class. The teacher knows that some students are in a math class (CS201) and needs to ensure they all have a prerequisite (calculus). The teacher's job is to check if this condition is met.

I want to assert or relate properties of a student x with respect to whether he has studied calculus or not and whether he has enrolled for CS201 or not. So let me first introduce a predicate here S(x)... and it will be true if student x in your domain has studied calculus else, it will be false.

Here, we introduce predicate variables, which help us express the logical relationships and properties of students. We define S(x) to represent whether student x has enrolled in course CS201, and C(x) to check if student x has studied calculus. This mapping from statements in English to predicate logic allows us to work with the statement more precisely.

Think of predicates like labels that you can attach to items. For students, one label (S) indicates if they are in the course, and another label (C) indicates if they have taken calculus. This organization helps the teacher quickly assess where students stand.

So 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... if student x has enrolled for CS201, then student x has studied calculus.

We create a universal statement by connecting our predicates: ∀x (S(x) → C(x)). This means 'for every student x, if x is enrolled in CS201, then x also has studied calculus'. This expresses a general rule about the entire domain, encapsulating the logic of our earlier English statement into a formal structure.

Consider a rule in a library: 'Every member who attends the reading class has read at least one book'. This statement applies to all members who join the class, just like our statement applies to all students in CS201.

Now an interesting question here is whether the statement that I want to represent is represented by the first expression or is it represented by the second expression?

In this portion, we differentiate between two logical expressions to clarify which correctly represents the original English statement. We analyze potential misunderstandings if students mistakenly choose the second expression instead of the first. This exploration illustrates the nuances of logical representation and is critical to understanding predicate logic.

Consider two signs: one reads 'if you buy a product, then you get a discount' versus 'if you have a discount, you must have bought a product'. The first sign accurately reflects the promotion, while the second confuses the cause and effect, similar to how misinterpreting logical expressions can lead to incorrect conclusions.

So let me demonstrate that why the second expression is an incorrect expression and it is the first expression which represents the statement every student has studied calculus in CS201.

This section involves examining examples with specified domains (Ram, Shyam, Balram) to validate the truth of our expressions. By assigning truth values to the predicates based on real-world conditions, we can ascertain whether the statements hold true based on the defined logic.

Imagine trying to determine whether all students passed calculus based on their enrollment status. You gather data on each student and find that while some passed, others did not; thus, the enrollment may not guarantee passing, similar to how we test our expressions against actual conditions.

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 assertion will be represented by this existentially quantified statement namely there exists some x in my domain such that the property S(x) and C(x) are simultaneously true for the same x.

Here, we shift focus to existential statements, which assert the existence of at least one student who fulfills both predicates (enrollment and studying calculus). We use existential quantification (∃x) to encapsulate this idea, marking a key distinction from universal statements.

Think of a treasure hunt scenario where someone claims, 'There is at least one hidden treasure in the park'. This statement shows that one treasure exists, without specifying how many there are, just like our existential statement focuses on at least one student meeting the criteria.

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