9.5. Example of Validity in Argument Forms

Now let us try to understand Nested Quantifiers, so there is very often we encounter statements where we need to have a nested form of quantification and this is similar to nested loops in programming languages. So let us see an example here, so say the predicate M(x, y) is defined in such a way that it is true if person y is the mother of person x.

Nested quantifiers involve using one quantifier inside another, similar to nested loops in programming. The example introduces a predicate M(x, y), where M(x, y) is true if y is the mother of x. This sets up the foundation for understanding how we can express relationships involving multiple entities.

Think of nested quantifiers like a family tree. Each person has a mother, but if you want to describe this in a programming context, it's like having a loop that goes through each person and then looks for their mother, much like asking a series of questions about each family member.

And, I want to represent a statement that every person in this world has a mother. So my claim is that this can be represented by this expression for all x there exist y such that M(x,y) is true and this is an example of nested quantification.

The expression 'for all x there exists y such that M(x,y) is true' captures the idea that for every person (x), there is a mother (y). This demonstrates how nested quantification allows for expressing universal truths using logical predicates.

Imagine standing in a room full of people. If I say, 'every person has a parent,' it means for each individual here, you can find an example of someone who is their parent. This is expressed mathematically as nested quantification.

And why this is the expression representing every person in this world has a mother; well, this is equivalent to saying that for all person x, there is some y, such that person y is the mother of person x which is indeed what is represented by this expression... because if you change the order of the quantification then the logical interpretation of the statement changes completely.

The order of quantification is crucial. Changing 'there exists y for all x, M(x,y)' means we're saying there is one specific mother for everyone, which is a different interpretation than saying each person can have their unique mother. This emphasizes how structure in logic significantly influences meaning.

Consider two distinct questions: 'Does everyone have a best friend?' versus 'Is there one person who is a best friend to everyone?' The first question allows for individual answers, while the second limits friends to just one person, totally changing the meaning.

Hence they are represented by two different nested quantifications. So that is why swapping of quantifications are not always possible, it is possible only when you have the quantifications of the same type occurring throughout the expression.

When quantifications are of the same type (e.g., all 'for all'), swapping them does not change the statement's validity. However, if one is universal and the other is existential, such as 'for all x' and 'there exists y', swapping alters the meaning and logic of the proposition significantly.

Think of two teams in a relay race. If swapped, the position of runners determines the outcome. If both teams have equal runners, changing the order might not change the race results, but if one team is faster at various parts, the outcome could shift drastically with a simple change in order.

So, now let us see some more examples here how we can start translating statements using the help of nested quantification. Suppose I want to represent a statement that if a person is female and is a parent then this person is someone's mother.

This chunk introduces how to translate specific statements into logical expressions using nested quantification. It highlights creating predicates that define properties and then using them to build a logical expression that reflects the intended statement.

Imagine trying to convey that 'if someone is an adult and is a driver, then they have a vehicle.' Here, we create specific definitions that correspond to real-world characteristics, resembling how you categorize types of people in social groups.

So let me introduce this predicate F(x) which is true if person x is female and I also need a predicate P(x) to represent that person x is parent and I had already introduced a predicate M in the previous slide which I am retaining here.

Defining predicates F(x), P(x), and M(x,y) allows us to describe complex relationships logically. Here, F identifies females, P identifies parents, and M connects mothers to their children. This is foundational for building logical proofs using predicates.

Creating definitions in logic is similar to defining terms in any situation. For instance, in a sports event, defining 'offside' or 'foul' helps clarify rules of the game for players and spectators alike.

This is a universally quantified statement because I am making a statement about all persons here... and this is an if statement of the form if-then your premise is for all person x in the domain...

The statement's structure is crucial; it’s universally quantified, which means it applies to everyone in the specified domain. Expressing this as an 'if-then' condition further categorizes it as a logical assertion that can be evaluated.

A simple rule, like 'if it rains, then I will carry an umbrella,' is universally applicable. It sets up a condition that can be assessed, just like logical statements in programming.

Of course now, if you do want to apply the De Morgan’s law of quantifications, you can take the negation, this negation that is here, and you can take it inside and then conjunctions get converted into disjunctions and so on...

Applying De Morgan's laws helps in transforming statements involving negations. Understanding this is vital for deriving equivalent logical statements, enabling clarity when asserting properties under negation.

Think of De Morgan's laws like providing alternatives in a restaurant: instead of 'no soup or salad,' you can say 'either you can have the soup or the salad.' This reframing can simplify understanding the options available.

Now let us do some rules of inferences for quantified statements, so which are very important the first rules of inference is universal instantiation...

Rules of inference, like universal instantiation and generalization, guide how we derive conclusions based on premises within logical frameworks. Understanding these helps in logical reasoning across various scenarios.

It’s like a recipe; if you follow the recipe (rules), you can create a dish accurately. Similarly, following logical rules leads to valid conclusions, ensuring you don’t stray from intended logic.

So here you are given two premises and conclusion here. So I am retaining the same predicates S(x) and C(x) that we have defined in some earlier slides...

Here we analyze an example using predicates S and C. We evaluate if the argument is valid by using established rules of logic such as universal instantiation and Modus Ponens.

Picture this like an exam: you have premises as questions, and your answer needs to logically follow them. If your conclusion lines up correctly with the premises, you’ve successfully constructed a valid argument.

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