Learn Propositional Logic and Truth Tables: Pure Mathematics for Beginners

Propositional logic also known as statement logic is a very popular technique to represent the idea clearly and accurately. The semantics of logic specify the meaning of a proposition. This story is exploring what are the theories behind the proportional logics and how to solve them. This is a formal way of knowledge representation in Mathematics which is using also in Artificial Intelligence(AI). Logic is a language. Therefore It has lexicon, syntax, semantic mechanisms.

Table of Content

1. Statements with words
2. Statements with symbols
3. Truth Tables

1. Statements with words

A statement(proposition) is a statement where we can definitely say whether completely true or false. But not both together. I will show you how to solve this kind of statements, using semantics.

Truth Values

True - 1 or 'T'
False - 0 or 'F'

Examples for propositions; Also these are the atomic statements. They are expressing a single idea.

1.Joe Biden is the current president of US -> True
2.Canberra is the capital of Australia -> True
3.Everyone born in February is intelligent -> False
4.3+4 = 7 -> True

Another way you want to identify what we can’t consider as a proposition as well. Therefore these are not propositions.

1.What is your name? - This is a question
2.Good Morning!!! - This is a greeting
3.Please switch on the computer - This is a command

Compound statements consist of more than one statement which is logically connected together. There have many connectors to know.

And, Or, If...then, If and only if, Not

First four connectives are calling binary connectives and NOT is a unary connective which means it describes one statement. Let’s look at some examples of compound statements.

1.13 is a prime number and 5 > 6. -> False, because of one statement is false. That's the behaviour of the AND
2.Today is Friday or water is a liquid. -> True, because of one statement is true. That's the behaviour of the OR
3.If Jonathan has a pet, then fish have lungs. -> This depends on the first statement. Second is already false. If T, then F is always F. If F, then F is always T.That's the behaviour of If...then

4.Michael Jackson is alive today if and only if 2*5 = 10. -> The First one is false and the second one is true. The entire statement is false because F if and only if T is F
5. 64 is not a perfect square. -> False, because not T, is always false.

2. Statements with symbols

we will use symbols such as A, B, C, p, q, r to represent single statements. These symbols are known as propositional variables. A truth assignment of a list of propositional variables is a choice of T or F for each propositional variable in the list.

Think about those five connectives again. They also have symbols for each.

And (˄) - conjunction 
Or (˅) - disjunction 
If..then (→) - conditional or implication 
If and only if (⟷) - biconditional
Not (¬) - negation
≡ - Sysmbol of Logical equivalence
Truth table of the implication
If you are hard working(P) then You will study hard(Q)
P   Q   P→Q
T   T   T
T   F   F
F   T   T
F   F   T

This following exercise will describe how to determine the truth value of a statement.

Given That,
A - True
B - True
C - False
D - Unknown
Find the truth value of each of the following.
1. (A˄C)→(D˅C)
 = (T˄F)→(Unknown˅F)
 = F→(Unknown˅F)
 Therefore statement is True because of left side is False

2. D → ((A˅C) ≡ A)
 = D → ((T˅F) ≡ T)
 = D → (T)
 Therefore statement is True because of right side is True
 
3. (¬B→D) → ¬(¬C ˄ A)
 = (¬T→Unknown) → ¬(¬F ˄ T)
 = (T) → ¬(T)
 = (T) → (F)
 = F
 
4. (B˄D) → ¬((C˄A)→(B˅C))
 = (T˄Unknown) → ¬((F˄T)→(T˅F))
 = (T˄Unknown) → ¬(F→T)
 = (T˄Unknown) → ¬(T)
 = (T˄Unknown) → F
 = Unknown

3. Truth Tables

A truth table can be used to display the possible truth values of a compound statement. As an example, we will see the truth table of the main five connectives.

1. And (˄) - conjunction

        P   Q   P˄Q
        T   T   T
        T   F   F
        F   T   F
        F   F   F

2. Or (˅) - disjunction

        P   Q   P˅Q
        T   T   T
        T   F   T
        F   T   T
        F   F   F


3. If..then (→) - conditional or implication

        P   Q   P→Q
        T   T   T
        T   F   F
        F   T   T
        F   F   T
        
4. If and only if (⟷) - bi-conditional

        P   Q   P⟷Q
        T   T   T
        T   F   F
        F   T   F
        F   F   T

5. Not (¬) - negation

        P   ¬P
        T   F 
        F   T

Let’s see an example, how to solve a compound statement using these truth tables.

Example: Let p, q and r be proportional variables with p and q true, and r false. compute the truth value of; ¬p ˅ (¬q → r)

    p   q   r   ¬p  ¬q  (¬q → r)    ¬p ˅ (¬q → r)
    T   T   T   F   F      T            T
    T   T   F   F   F      T            T - Appropriate Row
    T   F   T   F   T      T            T
    T   F   F   F   T      F            F
    F   T   T   T   F      T            T
    F   T   F   T   F      T            T
    F   F   T   T   T      T            T
    F   F   F   T   T      F            T
Final answer is True

Conclusion

At the end of the tutorial, you have learned the basics of propositional logic and truth tables. In this way, you can address any complex argument. I hope this short guide will help you to begin a happy journey with Logic. Comment you’re great ideas or ask any question has with this story in the response section below.
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