Arguments in Discrete Mathematics

Arguments are an important part of logical reasoning and philosophy. It also plays a vital role in mathematical proofs. In this article, we will throw some light on arguments in logical reasoning. Logical proofs can be proven by mathematical logic. The proof is a valid argument that determines the truth values of mathematical statements. The argument is a set of statements or propositions which contains premises and conclusion. The end or last statement is called a conclusion and the rest statements are called premises.

The premises of the argument are those statements or propositions which are used to provide the support or evidence while the conclusions of an argument is that statement or proposition that simply defines that premises are intended to provide support.

An argument is denoted by the following expression as follows.

 P1, P2, ..., Pn $ Q
 Where P1, P2...... Pn is the premises and Q is the conclusion.

Example of Arguments :

Example-1 :

• Every student of Information Technology studies Data Structures.
• Data structures necessarily contain the study of arguments.
• Therefore, every student of Information Technology studies arguments.

Example-2 :

• Every parent is a mature person.
• Children should listen to mature people.
• Therefore, every child should listen to their parents.

Example-3 :

• Every mother is a woman.
• All women are caring.
• Therefore, every mother is caring.

Types of Arguments :

1. Deductive Argument –

We can say that an argument where the truth of the premises always results in the truth of the conclusion. The true value of premises never gives a false value of conclusion, where no such condition occurs, is called deductive arguments.

Example –

All men are busy
Ram is a man
______________
Ram is busy

2. Inductive Argument –

An argument where the premises point to a few instances of some pattern and the end expresses that this pattern will hold as a rule generally. An inductive argument won’t be deductively valid, in light of the fact that regardless of whether a pattern is discovered ordinarily, that doesn’t promise it will consistently be found. Consequently, an inductive argument gives weaker, less reliable support for the conclusion than a deductive argument does.  

Example –

We have seen 800 ducks, and every one of them has been white  
________________________________________________  
All ducks are white  

Validity and Soundness of argument :

An argument is said to be valid only if it’s not possible for the premises to have true value and the conclusion to have false value. If the above statement does not hold then it is called invalid. Arguments that are invalid are also called a fallacy. If the truth of the premises logically confirms the truth of the conclusion then the argument is valid.

Note –

A deductive argument is said to be sound if and only if it is both factually correct and valid. Otherwise, deductive arguments are unsound.

Uses and Application :

• Arguments are used in computer programming.
• Arguments are used in critical thinking.
• Arguments are used to test logical ability.
• Arguments offer proof for a claim or conclusion.
• Argument mapping is useful in philosophy, management reporting, military, and intelligence analysis, and public debates.

Conclusion :

An argument is a set of statements, including premises and the conclusion. The conclusion is derived from premises. There are two types of argument; valid argument and invalid arguments and sound and unsound. Apart from these, arguments can be deductive and inductive. There are many uses of arguments in logical reasoning and mathematical proofs.

Methods of checking whether an argument is valid or not:

Method 1: (using critical rows)

             1. Make truth table and make all columns of the different premises p₁,p₂,……….pn and also make a column of Q.

2. Mark the rows in which all of p₁,p₂,……………..p_n (i.e all premises) are true. such rows are called critical rows.

3. Then, in the critical rows, check the value of conclusion ‘Q’, if Q is true in all the critical rows, then the conclusion is valid

Note: 1. There can be more than one critical row.

2. If there is anyone critical row in which Q is false, then the argument is invalid.

3. For the argument to be valid, ‘Q’ should be confirmed in all the critical rows in the truth table.

Method 2: ( Tautology Method )

• Just make a truth table in which make the columns of p₁,p₂,p₃………..p_n, Q , p₁∧p₂∧p₃………p_n and finally p₁∧p₂∧p₃∧……p_n->Q.
• if p₁∧p₂∧………….p_n -> Q is a Tautology, then the argument is valid.