8. 1.7.1. De Morgan's Laws involving Quantified Statements

It turns out that we can define multi-valued predicate functions right? So the previous example was for statements where we had only one subject namely the subject x but now you might be dealing with statements where you have multiple subjects.

In predicate logic, we can define functions that depend on multiple variables, which means we can represent statements involving more than one subject. For instance, if we say 'x equals y + 3', we need both 'x' and 'y' to determine if the statement is true or false. By introducing predicate functions for each variable, we can check the truth of statements once we assign values to these variables.

Think about a recipe that requires both flour and sugar. Until you know how much of each ingredient you have, you can't determine if you'll have enough to make a cake. Here, the quantities of flour and sugar are analogous to the variable values in a predicate function.

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.

Universal quantification allows us to make statements like 'All integers satisfy property P'. We denote this with '∀x P(x)', meaning that for every value of x in a certain domain, the property P holds true. This is logical equivalence to saying that if the property is false for even a single element, then the universal statement is also false.

Imagine a teacher stating that 'All students in the class passed the exam'. If even one student failed, the entire statement is proven false. This exemplifies how universal quantification works in predicate logic.

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 states that there exists at least one value in the domain that satisfies a certain property. Written as '∃x P(x)', it means that for some x, the property P holds true. This form does not require that the property be true for all values, only for at least one.

Consider a box of assorted chocolates. If you say, 'There exists a chocolate that is filled with caramel', you only need one caramel chocolate in the box for your statement to be true, even if others are not.

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

In logic, a bounded variable is one that is within the scope of a quantifier. For example, in the expression '∃x P(x)', the variable x is bounded because it is governed by the existential quantifier. On the other hand, if you have a variable not related to any quantifier, it is called a free variable, which can cause confusion in expressions.

Think of a classroom where the teacher oversees certain students (bound variables) during a specific activity, while others are free to interact outside of that context. The free students are not bound by the activity's rules.

So now we can define logical equivalences even for the predicate world. We can have quantified statements and see whether the two expressions are logically equivalent or not.

Logical equivalence in predicate logic means two statements have the same truth value across all possible domains. For instance, you can prove that '¬∀x P(x)' is equivalent to '∃x ¬P(x)', showing that denying the universal truth implies there exists an element for which the property is false. Similarly, '¬∃x P(x)' is equivalent to '∀x ¬P(x)'.

Imagine two friends arguing about a movie. One says, 'Not every critic liked it' (equivalent to 'Some critics disliked it'), while the other claims, 'No critic liked it' (equivalent to 'All critics disliked it'). Understanding their statements relies on recognizing their logical relationships.

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