Part b

Today, we will explore how to assess the validity of logical arguments using predicate functions. Can anyone explain what a predicate is?

A predicate is a statement that can be true or false depending on the values of its variables.

Exactly! So in our example, when we say 'some math majors left,' we can express that using the predicate function M(x). Who can tell me what it means?

It means there is at least one student who is a math major.

Great! Also, we will introduce W(x) to denote whether a student left for the weekend. We use these predicates to form logical statements.

What about the existential and universal quantifiers?

Good question! The existential quantifier expresses that 'some', while the universal quantifier covers 'all'. Let's move on to an example. If we say: 'Some math majors left and all seniors left for the weekend,' how would we express this?

We could write it as ∃x (M(x) ∧ W(x)) and ∀x (S(x) → W(x)).

Exactly! This will help us analyze the validity of the argument.

Let’s analyze a sample argument. If I say 'Some seniors are math majors,' what do you think about its validity given our earlier premises?

We need to check if any student satisfies being both a senior and math major.

Exactly! Now, let's consider counterexamples. If we say all seniors left and there were senior students who weren't math majors, how would that affect our conclusion?

Then the argument could be invalid since our conclusion doesn't necessarily follow from the premises.

Correct! Having just one counterexample disproves the validity of the argument.

Could you demonstrate with a specific example?

Sure! In an example college with students x1, x2, and x3 where x1 is a math major but not a senior, we can see how the premises can still hold while the conclusion fails.

Now let’s create predicates for a different scenario: a stamp collector who has stamps from African countries. How do we express that using the predicates I(x) and F(x,y)?

We want to say that she has exactly one stamp for each country.

Right! To achieve that, we are looking for a combination of existential and universal quantifiers. Who remembers how to ensure exactly one condition?

We must include two parts: one stating there’s at least one and a second preventing others.

Exactly! So if we say, for every African country y, there exists a stamp x such that I(x) and it's issued by that country, but no other stamp x’ issued by y exists in her collection, that encapsulates our requirement.

But isn’t it more complicated to write?

It can be, but once you practice more, it becomes clearer. Always remember the condition for uniqueness!