8. 1.4.1. Quantification Method

So now once we have predicates we will be interested to convert them into propositions because then only we can apply the rules of inferences, rules of mathematical logic and do something meaningful with the resultant propositions. It turns out that there are two methods of converting your predicates into propositions. The method number one is you assign explicitly, manually the values to your underlying variables, but it is very less interesting.

In predicate logic, predicates need to be converted into propositions to utilize logical inference rules effectively. Although one method involves assigning values to variables, this approach is quite straightforward and not very engaging. Instead, the focus shifts to a more intriguing technique known as quantification, which allows for broader and more complex expressions.

Think of this process like baking: simply following a recipe (manually assigning values) might give you a cake, but using a technique like measuring out ingredients (quantification) allows for greater creativity in your baking, enabling you to adjust and adapt based on what you have at home.

What is a universal quantification? Well whenever we want to assert that a property is true for all the elements of my domain, then I use universal quantification. So very often you might have encountered this notation ( ꓯ ) for all x in your theorem statements.

Universal quantification is a way to express that a predicate holds true for every element in a specified domain. The notation '∀' is often used in formal statements to signify 'for all'. For example, if we want to say that all integers satisfy a certain property, we might write ‘∀x, P(x)’, indicating that property P is true for every integer x.

Imagine saying, 'All dogs are friendly.' Here, you are asserting a universal truth about the entire group of dogs. In logic, if any one dog is not friendly, the statement 'All dogs are friendly' would be false, similar to failing a universal quantification if just one value doesn't satisfy the condition.

Now, let us go to the next form of quantification which we call as existential quantification and this quantification asserts that a property is true for at least one element of my domain.

Existential quantification allows us to assert that there is at least one element in the domain for which a property holds true. This is represented by the notation ‘∃’, which conveys the notion of 'there exists'. For instance, in the statement '∃x, P(x)', we state that there exists some value of x such that property P is satisfied.

Consider the phrase, 'There exists a member in the team who speaks Spanish.' Here, the statement doesn't require everyone on the team to speak Spanish, just that at least one person does. This simplicity allows us to capture more nuanced situations in logical terms.

Now let us define what we call as bounded and free variables. So a variable is called bounded if there is a quantifier, which is applied on it.

A bounded variable in the context of quantification is one that is directly influenced by a quantifier, such as ‘∃’ or ‘∀’, which applies specifically to it. A free variable, on the other hand, is not constrained or defined by a quantification. For example, in the expression '∃x, P(x)' and 'Q(y)', y remains free while x is bounded.

Think of it like a group project at school. You can assign a specific task to a student (bounded variable) while others remain free to complete their own sections of the project without restrictions. Just like in logic, where understanding which variables are bounded or free helps clarify the scope and implications of statements.

Before proceeding further, I would like to stress here on the significance of the domain here. Whenever we are making quantified statements, it could be any form of quantification, it is very important that you clearly and explicitly say what is the domain of x.

The domain in which variables are defined is crucial for accurately assessing the truth of quantified statements. Changing the domain can drastically alter whether a statement is true or false. For instance, if we say 'for all x, P(x)' without specifying that x represents only positive numbers, adding zero could change the validity of the statement.

Imagine ordering a pizza for a group of friends. If you don't specify that only vegetarian toppings are acceptable, someone might add anchovies, making the order invalid for those who are vegetarian. Similarly, without a clear domain in logic, you might end up with contradictory or misleading statements.

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