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What are the types of logical statements?
Logical connectives are used to build logical statements. There are five types of logical connectives used to build logical statements, which can be analysed using a truth table.
- Logical Negation
Every logical statement has a certain truth value assigned to it. The negation operator is used to reverse the truth value of a given logical statement and output its opposite value. The truth table of a given pair of logical statements can help us identify if the two statements are logical negations of each other.
- Logical Conjunction
Logical conjunction is used to combine two statements when we want to create a statement such that the truth value of the statement is true only when the truth value of the two components of the statements are also true.
- Logical Disjunction
Unlike conjunction, disjunction is used when we want to create a compound statement such that the truth value of the compound statement is true when the truth value of either of the two components of the compound statement is also true.
- Logical Implication
When the truth value of one statement is dependent on the truth value of another statement then a logical implication is realised between the two statements. The pair of statements are also called hypothesis and conclusion pairs. For example if statement P is a Conclusion and a statement Q is the hypothesis then the truth value of P will decide the truth value of the compound statement formed by combining P and Q.
- Logical Biconditional
When the truth values of two statements are related to each other in such a manner such that both of them will be true concurrently or else the truth value will be false, then the two statements are said to be connected to each other as biconditionals.
Truth table for Logical reasoning
Truth tables have been in use since late 1893, and while there is no clear answer as to who invented the truth tables, Ludwig Wittgenstein is credited with popularising the truth table and is also often credited with inventing the truth tables too.
In its essence a truth table is a mathematical tool that is specifically applied in the domain of logic when the task at hand has connections with boolean algebra or functions for propositional calculus. A truth table is formed for a given logical expression when its output has to be evaluated given a certain combination of its input arguments.
In terms of a function, a truth table can be expressed as the combination of all the functional values produced by a logical expression for a given set of combinations of functional arguments submitted as input. A popular usage of the truth table is to prove that a given propositional expression is logically valid; that is, for all available input values, there is a valid output produced by the expression.
A truth table consists of rows and columns: there are n number of columns for n number of input variables and one column for the output variable. There are r number of rows for r possible combinations of the input variables. Every row in a truth table is distinct from other rows.
Example of a truth table in logical reasoning
Let us understand the usage of a truth table for logical reasoning with a few examples:
Consider two statements P and Q such that:
P: It is morning.
Q: It is evening.
The truth table for the P and Q can be created as
P | Q |
T | F |
F | T |
From this table we can observe that if statement P is true then Q is false, and if statement Q is true then statement P is always false. Hence statement Q is said to be the negation of statement P and vice versa.
Consider a compound statement A:
A: A driver must have a licence and know how to drive a car.
This is a compound statement made of two statements, B and C, which are;
B: A driver must have a licence
C: A driver must know how to drive a car
When we create a truth table for the two statements we observe:
C | B | C→B |
T | T | T |
F | T | T |
T | F | F |
F | F | F |
In this example, the first row tells that if a person knows how to drive a car and has a licence then it’s implied that they have a licence.
The second row tells that if a person has a licence, it can be implied that they have a licence even though the person does not know how to drive a car.
In the third row, the person only knows how to drive a car, then it cannot be implied that the person has a licence.
In the fourth row the person does not have a licence or know how to drive a car: hence, it can be implied that they do not have a licence. This form of logical relation is a logical implication.
Conclusion
Logical statements are the key to understanding various communications and reasoning. One strong tool used for analysing the logical reasoning of a logical statement is the truth table. It gives a tabular view of all the possible inputs and the possible outcomes for a given logical statement or reasoning. There are five types of logical statements, negation, conjunction, disjunction, implication and biconditional, which can be analysed using a truth table.
