Introduction to Mathematical Logic

Mathematical logic deals with the logic in mathematics. Mathematical logic operators and laws define various statements in their mathematical form. In this article, we will explore mathematical logic along with the mathematical logic operators and types of mathematical logic. We will also solve some examples related to mathematical logic.

The study of mathematical logic in mathematics is called mathematical logic. The basic mathematical logic used are the conjunction (∧), disjunction (∨), and negation (¬). Some other mathematical logics are implication and double implication.

The basic mathematical logic operators are:

• Conjunction
• Disjunction
• Negation
• Implication
• Double Implication

Conjunction

In mathematical logic conjunction of two statements results in true when both the statements are true otherwise false. Conjunction is also known as AND operator and is represented by ∧.

Disjunction

In mathematical logic disjunction of two statements results in false if both the statements are false otherwise true. Disjunction is also known as OR operator and is represented by ∨.

Negation

In mathematical logic negation of two statements results in the not of the given statement i.e., if the statement is true it results in false and if the statement is false it results in true. Negation is also known as NOT operator and is represented by ~ or ¬.

Implication

In mathematical logic implication of two statements results in false if the first statement is true and second statement is false otherwise true. Implication is also known as conditional operator and is represented by → or ⇒. Implication X→Y is read as If X and then Y.

Double Implication

In mathematical logic double implication of two statements results in true when either both statements are true or both statements are false. Double implication is also known as biconditional operator and is represented by ↔ or ⇔. Double implication X↔ Y is read as Y iff X or Y if and only if X.

Some of the basic mathematical formulas are listed below:

The different types of mathematical logic include:

• Set Theory
• Model Theory
• Proof Theory
• Recursion Theory

Set Theory: Set theory is a part of mathematical logic that deals with the sets which means collection of elements. The set theory is the theory consisting of sets, sets formulas and many more.

Model Theory: Model theory is a part of mathematical logic that deals with the models of different theories of mathematics. The model theory provides different models describing the complex theories making it easy to understand.

Proof Theory: Proof theory is a part of mathematical logic that deals with the proofs. The mathematical proofs provide easy analysis of different mathematical methods.

Recursion Theory: Recursion theory is a part of mathematical logic used to construct computable functions, Turing machines and recursively enumerable sets.

The truth table in mathematical logic is a table which takes inputs and provides output when a logic is applied to it. The truth table for different mathematical logic operators are given below.

Negation

The truth table for negation is given below.

Conjunction

The truth table for conjunction is given below.

Disjunction

The truth table for disjunction is given below.

Implication

The truth table for implication is given below.

Double Implication

The truth table for double implication is given below.

Example 1: Consider the statement x < 5 → x – 2 < 5 is true or false?

Solution:

If x < 5 is true then, x – 2 < 5 is also true.

T → T is true

So, the given statement x < 5 → x – 2 < 5 is true.

Example 2: For given two statements compute the truth table for conjunction.

P: a is divisible by 4

Q: a is divisible by 10

Solution:

Given the two statements P and Q.

P: a is divisible by 4

Q: a is divisible by 10

Example 3: Find the negation of the given statement P: It is a rainy day.

Solution:

Given statement,

P: It is a rainy day.

Negation of P = ¬ P: It is not a rainy day.

What is Meant by Mathematical Logic?

Mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power.

Who is the Father of Mathematical Logic?

An English mathematician George Boole (1815–1864) is consider the father of Mathematical Logic.

What is the Basic Mathematical Logic?

Basic mathematical logic includes AND, OR and NOT.

What is Mathematical Logic Used For?

Mathematical logic is used for logical reasoning which helps to solve mathematical problems.

What are the Applications of Mathematical Logic?

Various applications of mathematical logic are providing analysis of mathematical techniques, proving different mathematical theorems and helps to verify the computer programs.