Tautology can be defined as a compound statement made up of two or more simple statements which always answers truth, no matter what combinations of truth are used in the individual statements. While, fallacy is a compound statement that always gives false as the output no matter what combination of truth is used in its individual statements.
Generally, Tautology is considered to be the valid equation, while fallacy is the invalid argument. Although, this does have its exceptions, as Tautology is not always valid.
The word Tautology has Greek roots, as the word ‘tauto’ means the same while ‘logy’ means science. In mathematics, it is used to decide whether the answer obtained is true and accurate or not.
Read on as we delve deeper into the different operators that make up Tautology statements in this study material notes on Tautology.
Truth Tables
A statement in mathematics often only has two outcomes: that is, it can either be True or False. These values are known as Boolean values and are used in Boolean Algebra. A truth table encapsulates the different input true and false components of a statement and has a Boolean output accordingly.
It is denoted by the symbols T and F. It is also denoted by the symbols 1 (for the truth) and 0 (for false).
Now, a truth table containing only truth as the output in its last column can be said to have Tautology.
Example of Tautology
Let us take two simple statements a and b and combine them using the “OR” operator. The OR operator is used to combine two simple statements to form a compound statement. The combining of two statements using the OR logical operator is known as a Disjunction or Alternation.
The OR operator is denoted by the symbol “∨”
So if two statements a and b are disjunct, then they are symbolically represented as a ∨ b
So let’s say the statements are:
a = I will get paid
b = I will NOT get paid ( ~a – this symbolises the NOT operator, which is essentially the opposite of a statement )
so if we find the disjunct of the two statements, then it will be represented as a ∨ b
Since b = ~a, therefore we can also write the above formula as
a ∨ ~a
Now, according to the OR operator, the output shall be true if any one of the inputs is true. Let us create a truth table to see the tautology in this operation.
a | ~a | a ∨ ~a |
T | F | T |
F | T | T |
As we can see, the last column of the truth table consists of only Ts, hence it’s a Tautology.
Other Operators Of Tautology
A Tautology statement is a complex statement made up of more than one simple statement and many logical operators like AND, OR, IF…THEN, NOT, etc.
Example: ((a ∧ b)∨ f)⇒(p∧ ~ p); where a, b, f, p are all different simple sentences and “^, ∨, ⇒” are all symbols of different operators.
So now that we know about the OR operator, let us read more about the other operators.
AND Operator
This logical operator is used to join two or more simple statements to form a compound statement. The joining of two statements using the “AND” operator is also known as a conjunction.
The symbol that is used to denote the “AND” operator is ^
Thus, if a and b are two simple statements, then the conjunction of these statements will be denoted symbolically as “a^b” and it will be read as “a AND b”.
Truth Table for AND
The conjunction truth table is created when two statements a and b are combined using the AND operator. The conjunction value is denoted by a^b and it can only be true if both the inputs are true.
The truth table for AND is given as:
a | b | a^b |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Thus, in an AND truth table, the output can be true only if both the inputs are true.
IF…THEN Operator
This is known as a conditional operator. If two simple sentences are combined with “IF..THEN”, it gives a compound statement known as a conditional or implicational statement.
Suppose we have two simple sentences a and b, then the “IF…THEN” Operator will be denoted by “a → b” or “a ⇒b”. These symbols are read as “a implies b”, where a is the antecedent and b is the consequent.
Truth Table for IF…THEN
The truth table for conditional statements is as follows:
a | b | a ⇒ b |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Thus, for a truth table for a conditional statement, the output only depends on the consequent (that is b). If b is true, then the output is true and vice versa. The output is also true if both the inputs are false.
Only in one condition, the output comes out as false when the consequent is false while the antecedent is true.
IF AND ONLY IF (IFF)
The “IF AND ONLY IF (IFF)” Operator is used in combining two statements to give a biconditional statement.
Suppose a and b are two statements, then a biconditional statement is denoted by “a ⇔ b” or “a b”
Truth Table for biconditional statements: IF AND ONLY IF (IFF)
The biconditional statements are denoted by a ⇔ b
Biconditional statements are formed by the conjunction of two conditional statements, where one conditional statement is the converse of the other. Thus, if we have two statements, a and b, then,
a ⇔ b = (a ⇒ b ) ^ (b ⇒ a)
The truth table for biconditional statements is as follows:
a | b | a ⇒ b | b ⇒ a | a ⇔ b |
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | F |
F | F | T | T | T |
Thus, in biconditional operators, the values can be true only if both the inputs are true, or if both the inputs are false.
Conclusion
The key takeaway from this Tautology study material is that it is a compound statement that is always true. A compound statement is made up of two or more simple sentences and operators, but despite any combination of the individual statements, a tautology is always true.
This study material notes on tautology also consist of the different truth tables made up of different logical operators and the condition required to get the output as true.