Introduction

These notes are for absolute beginners. They cover the following topics. Propositional logic cannot adequately express the meaning of all statements in Mathematics and in natural language. That is why we have Predicate Logic. Predicate Logic is used to express the meaning of a wide range of statements in Mathematics and Computer Science in ways that permit us to reason and explore relationships between objects.

I introduce mathematical notation as and where appropriate, disregarding the chronology and history of their actual development. Every section accompanies simple and complex exercises to help cement the concepts presented in that section.

Concept of a variable

Variables are placeholders which are replaced by concrete values. Another way of saying this is, Values are substituted for variables. Variables are the first level of abstraction which lead to the development of algebra.

$$2 + 3$$

I can generalize the above expression by replace $`2`$ with $`x`$ and create an abstraction.

$$x + 3$$

The proper way to do this is to use a variable binder which is usually the Greek letter lambda. I will use variable binder which is generally called lambda.

$$\lambda x.x + 3$$

Now this expression has much wider applicability which is what mathematical abstractions are intended to achieve.

Concept of Predicate

Consider the common mathematical expression $`x > 3`$.

$$x \text{ is greater than } 3$$

$`x`$ is the variable and subject of the statement. "is greater than 3" is the predicate and refers to the property that subject of the statement can have.

Suppose P = "is greater than 3", then $`P(x)`$ is the propositional function P at x.

When a value is assigned to the variable $`x`$, $`P(x)`$ becomes a proposition gaining a truth value.

Quantifiers

Quantification expresses the extent to which a predicate is true over a range of elements. In English language, the words "all", "some", "many", "none", "few" are frequently used in quantification.

Why do we need Quantifiers?

Suppose we know these two facts:

1. Fact 1 "Every computer connected to the University Network is functioning properly.

2. Fact 2: "MATH3 is functioning properly" where MATH3 is one of the computers connected to the university network.

No rules of propositional logic allow us to conclude the truth of statement 2 from 1.