13.2. Question 1

Here, you are supposed to find out whether the following argument is valid or not. So you are given some premises and conclusion. So the first thing that we have to do is we have to convert everything in terms of predicate functions. So we introduce appropriate predicates here. So of course, the domain is explicitly not given here. But domain, the implicit domain here is the set of students.

The argument revolves around determining the validity of a statement based on given premises. First, we need to express these premises using predicates, which are functions that return true or false based on certain conditions or inputs. In this case, the domain we analyze consists of students, meaning all statements will pertain to students and their characteristics.

Think of it like investigating whether a group of students fulfills certain criteria for a project. To clarify who's who, you might label each student with tags like 'math major' or 'senior,' instead of stating each characteristic in plain language. This simplifies our analysis.

The first statement here, the premise here is some math majors left the campus for the weekend. So it is easy to see that this is an existential quantified statement, it is not making an assertion about all the math majors. But let us first decide what are the predicates that we need here. So the assertion is about math majors. So let M(x) be the predicate which is true if the student x is a math major. And we are saying something regarding whether he has left the campus for the weekend or not. So that is the second property for the subject x. So that is why I introduce a predicate the W(x) which is true, if the subject x or if the student x is left for the weekend.

Here we analyze the first premise: 'Some math majors left the campus for the weekend.' This is an existential statement, meaning it asserts the existence of at least one math major who left. We represent this using predicates, where M(x) indicates 'student x is a math major' and W(x) indicates 'student x has left for the weekend.' This sets the foundation for our logical analysis.

Imagine a classroom where the teacher asks if any students went on a trip. The teacher doesn't mean all students but just wants to know if at least one student went. Here, instead of asking students directly, we label them as 'student who went on a trip,' allowing us to keep track of the information.

The second statement here or the premise here is that all seniors left the campus for the weekend. So this is a universally quantified statement. And if you see clearly or closely here, the interpretation of this statement is that, if a student x is senior then he has left the campus. So there is an implicit implication here and that is why this premise can be represented as ꓯ x, S(x) → W(x).

The second premise states that 'all seniors left the campus for the weekend,' which is a universally quantified statement, meaning it applies to every student who is a senior. We represent this using predicates as well, where S(x) denotes 'student x is a senior.' Thus, we can express it in logical terms as 'for all x, if x is a senior, then x has left for the weekend.' This helps in evaluating the conclusions based on these premises effectively.

Imagine all seniors taking a school trip. The teacher can say, 'Every senior today is out of class.' This generalization makes it clear to the teacher which students aren't available based on their senior status.

The conclusion that I am making here is, some seniors that means existentially quantified statement, are math majors. That means at least one student is there for which the property that he is a math major as well as, he is a senior are true.

The conclusion derived from the premises is stated as 'some seniors are math majors.' This is again an existential statement, implying that there exists at least one student who is both a senior and a math major. It's important to note that the validity of this conclusion relies heavily on the truth of the previous premises.

Think of it like piecing together a puzzle. If I know two pieces fit together (the premises), it doesn't guarantee that they create a complete picture (the conclusion). Just knowing there are math majors and seniors doesn't mean there must be a math major who is also a senior without additional information.

Now we have to verify whether this is a valid argument and as per the definition it will be a valid argument if, based on the premises I can draw the conclusion for every possible domain. However, it turns out that this is not a valid argument and we can give a counterexample. You can give multiple counter examples here. Even if you show one counter example that is sufficient to show that this argument form is not valid.

To determine if our conclusion is valid, we need to check if there are situations where the premises could be true while the conclusion is false. A valid argument guarantees that if the premises are true, the conclusion must also be true in all cases. We illustrate this by creating a counterexample where the premises hold true, but the conclusion does not.

Consider having a few students classified into different groups. Just knowing that some students in one group (math majors) and all in another group (seniors) left does not prove that any member of the math majors group overlaps with the seniors. Imagine a classroom where some seniors are involved in arts, not math; therefore, the argument fails.

So the domain that I consider is the following imagine you have a college where you have 3 students x1, x2, x3. And say with respect to those 3 students the status of the 3 predicate functions are as follows. For x1, the property M is true, W is true, but S is false. Student x2, the property M is false, property W is true and the property S is true and so on.

In our constructed counterexample, we have three students: x1, x2, and x3, evaluated based on their properties as math majors (M), left for the weekend (W), and are seniors (S). By checking various combinations of these properties, we demonstrate that the premises can still be true while the conclusion remains false, thus concluding that the argument is invalid.

Think of this as testing a recipe with different ingredients. If we have one ingredient (math major) mixed correctly but don’t have the essential seasoning (seniors), we can Produce a dish (true premises) that lacks the final flavor we desire (false conclusion) because not all parts fit together.

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