Tautologies and Fallacies With Example

Basic Concept Of Tautologies and Fallacies

Tautologies or theorems are like axioms which are true for all values. In a truth table of tautology there will be only T in the last column. For example p OR~p= p, p AND p=p, and ~ ~p=p are all tautologies. As against these the Fallacies are the contradictions which will never be truth and obviously there will be only F in the output column of a truth table. For example p AND ~p is a contradiction, how can a statement and its negative both be true. These truth tables will further reveal this fact.

Since tautology is always true its negation is a fallacy and is always false.

• Example 1: Verify the following statement by constructing truth tables:

 

Solution:
(i)

Since there is T for all values of p, q in column 5, it is a tautology.

(ii)

Since there is F all values of p, q in column 6, it is a Fallacy

• Example 2: Using truth tables, show that (p AND q) =p, and p = (p OR Q) are both tautologies, where p,q are any two statements.

 Solution : 

Since all the entries in column 4 and 6 are T, the given proposition is both tautologies.

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Negation of Compound Statement

Today i Will write about Negation of Compound Statement. Here you will find the clear concept of Negation of Compound Statement and its Different Functions. When a compound statement is negated its connective changes from AND to OR and from OR to AND. For example,

We can say that the inversion of a function of several terms is obtained by inverting the individual terms and changing the connectives. This is the famous De Morgan’s theorem or law. It is very helpful in simplifying and rearranging the terms of a Boolean function which is sometimes better amenable to certain specialized operations. The truth of the De Morgan’s law can be verified from the following truth table:

The truth values of columns 4 and 7 are alike which proves the theorem.
The following are some relations based on the above law which can be verified by preparing truth tables:

Example: 1. Let p be the statement “the south-west monsoon is very good this year” and q be the statement “the rivers are rising”. Give the verbal translations for (a) and verify the statement (b).

Solution: 
(a)

1. The south-west monsoon is very good but the rivers are not rising.
2. It is not true that the south-west monsoon is not very good or the rivers are not rising. We can also state that the south-west monsoon is very good and the rivers are rising.

(b) The statement is false because if x2>1 then x>1 OR x<-1

So dear i think you understand this article very much. Best Of Luck. 

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Definition Of Disjunction And its Uses

A joint statement asserting that at least one of the constituent statements is true, so that more then one constituent statements can also be true, certainly all are not false is compounded by the use of ‘either…… or’ or simply OR. A compound statement of two statements will be true if either of them is true or both are true. It p stands for the optional Maths, and q for the optional Statistics, p OR q will signify;

P and q   Maths and Statistics
P and ~q Maths and not Statistics
Q and ~p Statistics and not Maths

This relationship will be brought out more clearly by the following truth table which may be compared with the previous one to grasp the difference.

As shown above p or q is false only when both p and q are false. This obviously is an inclusive type of disjunction which covers also the situation when both are true.
The plain circuit depicting the relationship is given here under:

The electrical impulse from initial point S_1 will not pass to the terminal point S_2 only when both the switches p and q are open (i.e. off)

The exclusive type of disjunction, which keeps out the situation when all the constituent statement are true, is indicated by p OR q or p+q. In this case the first row of the truth table 4 will have F in the output column. Its circuit will be of the following type :

The exclusive type of disjunction which conveys the sense of “p or q but not both“ is a special case of OR connective. Therefore, unless otherwise stated, p OR q will be taken in the inclusive sense which will always mean “p OR q or both” or “p and/or q”

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Conjunction : The Truth Value of a Compound Statement with Conjunction

A joint statement to the effect that each constituent of the statement is true, is compounded by the use of the connective AND. For example, if p stands for the statement “price is rising” and q for “the quantity of money is increasing”. Then the compound statement pq indicates that “the prices are rising and the quantity of money is increasing”. A compound statement of two statements will be true only when both the constituent are true and not when either of them is true or when both are not true. The truth value of a compound statement with conjunction will be as follows:

The first thing to note in the above is that with 2 statements there are 4 combination. In case there are three statements, the possible combination will be 8.

The second is that the basic truth values in the first two columns are 4, 3, 2, 1 in binary digits if the place 1 for T and F. In case there are three statements then the order will be 7, 6, 5, 4, 3, 2, 1, and 0. However, the order can be reversed there is no special sanctity attached to it.

The third is that in the output column, which is the third column in the above table the truth of the compound statement is indicate by T. wherever the alternative combinations are not in keeping with the relationship, F is written.

An electronic circuit in case of this operation will be in the same series so that the impulse from the initial point S1 will not pass to the terminal point S2 if either of the switches is open. See the circuit below….

I think you understand this topics about conjunction. Best of luck.

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Clear Concept of Compounding : What is Compound ?

Definition of COMPOUND

The method of combining statements is known as compounding; two or more constituent statements when combined into a joint statement is known as a compound statement. The compound connectives used for the purpose are AND, OR & NOT. Actually the Boolean Algebra recognizes only three operations by which a machine manages all other operations. Other connectives are also converted into these simple operators. The truth value of a compound statement will depend on the truth of the constituent statements. Compounding is done mainly through conjunction and disjunction.

• Conjunction:

• Disjunction:

I Will describe about conjunction and disjunction in my next post.

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Negation Is The Consideration Of The Statement

Basic Concept Of Negation

To assert a statement is to say that it is true and to deny it is to say that, it is false. Negation is the consideration of the statement which may be either an assertion or denial. If p is a statement then ~ p (rear as not p) will be its Negation. But if ~ p is a statement then p will be its Negation which we can also express ~(~ p) p. The following Truth table shows how when p is true, ~ p is false and vice-versa is also there.

An electronic circuit showing this type of relationship is as follows:

The above circuits shows that the electronical impulse passes from S1 to S2 in case of p and it will pass to S2 (S2 bar) and not to S2 in case of ~ p.

Double Negation is positive which can be verified from the following Truth Table.

 

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Truth table : Truth value of a compound statement constituting several statements

Definition Of Truth Table

Truth table is a table indicating the truth value of a compound statement constituting several statements. The statements are compounded by various connectives e.g. …… AND, OR & NEGATION etc. each one has a significance and therefore the truth of compound statement has to be considered taking these into considerations.

• A truth table has a number of columns and rows. The number of columns depends upon the number of constituent elements and how involved over their relationships. 

• The initial columns, two or three depending on the constituent statements. These give all possible combinations of the constituent statements.

• The remaining columns are the output columns giving the truth values of the compound statement as a function of certain relationship between constituent statements.

• The number of rows in a truth table is determined on the basis of the constituent statements. In case of 2 constituent statements, there are 4 rows and for 3 constituent statements, there are 8 rows.

The truth table is very useful in finding out the validity of an equivalence relation between function. For practical purpose, they help in designing and testing the electronic circuits to perform a given operation based on a certain relationship.

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Concept Of Logical Statements And Its Use

More About Logical statement

There are assertions in words or symbols which are either true or false but not both. The difference between an ordinary sentence and a logical statement is that it is not possible to say truth or otherwise of an ordinary sentence whereas true or false is an essential requisite of a logical statement. For example, the expressions: Oh God! ; How do you run fast? Are sentence all right but they are not logical statements.

The following are some logical statements:

• The number X is even
• x2-1 ≡ (x+1)(x-1) for all values of x
• The sum of the three angles of a triangle is equal to two right angles
• Economics is a dismal science.

I think this is a clear concept of Logical Statements which will help you to understand more about basic matter of Business Mathematics. So Best Of Luck.

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Concept Of Logical Statements and Truth Table : Chapter 1

Introduction of Mathematical Logic

Logical statements deal with the binary logic used extensively it modern mathematics. Based on Boolean algebra, the logical statements introduce mathematical logic to the students. There is great relevance of these statements in modern mathematics because the basic philosophy behind the digital computers is based on digits 1 And 0. The binary codes and logical circuits used in computer employ signals of true or false. The two stable states of an electrical switching circuit are ON and OFF. With these a large number of mathematical operations are performed through suitably designed circuits.

Structure:

•          Logical Statements
•          Truth Table
•          Negation
•          Compounding
•          Negation of Compound Statements
•          Tautologies and Fallacies
•          Propositions
•          Algebra of Propositions
•          Conditional Statements
•          Biconditional Statements
•          Arguments
•          Joint Denial

Objectives:  After studying this chapter You should be able to Understand…..

1.  Logical statements, Truth table, Negation, Compounding, Negation of compounding statements, Tautologies and Fallacies, Propositions
2.   Algebra Of Propositions
3.   Conditional and Biconditional Statements
4.   Arguments and Joint denial

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