A proposition is a statement that can either be true or false and cannot hold a neutral or indeterminate value. For example, the statement "2 + 3 = 5" is a true proposition because the mathematical expression evaluates to the correct result of 5.
A contradiction is a proposition that is always false, regardless of the truth and false values of the individual propositions that produce the result. This concept is useful in mathematics and sciences to help us identify errors, inconsistencies, and conflicts in our data.
Let's discuss some examples of contradiction: one and two propositions.
Let's understand contradiction with an easy example involving just one proposition: m. In the following table, we calculate the
Note: Here T refers to true while F refers to false.
m | ∼m | m ∧∼m |
T | F | F |
F | T | F |
We conclude that m ∧∼m is a contradiction as it results in all false outputs.
Let's dive deep into the contradiction concept now involving two propositions: m and n. We want to calculate ∼m ∧ (m ∧ n), so we find ∼m and m ∧ n individually first. Then, we take the AND between them to get our desired output.
m | n | ∼m | m ∧ n | ∼m ∧ (m ∧ n) |
T | T | F | T | F |
T | F | F | F | F |
F | T | T | F | F |
F | F | T | F | F |
We conclude that ∼m ∧ (m ∧ n) is a contradiction as it results in all false outputs.
Contradictions play an important role in various areas of science. They help us find errors, maintain accuracy, and improve the quality of our systems. Contradictions can help us find potential errors and shortcomings in our system so we can fix and improve it accordingly.
Let's test what we have learned so far.
Quiz on Contradiction
In a truth table, a contradiction occurs when:
All the propositions are true.
All the propositions are false.
Some propositions are true, and some are false.
None of the above.
Unlock your potential: Truth table series, all in one place!
To continue your exploration of truth tables, check out our series of Answers below:
How to construct truth table
Learn how to create truth tables to evaluate logical expressions.
Contradiction truth table
Understand how a contradiction truth table is used to show statements that are always false.
What is the contingency truth table?
Explore contingency truth tables that represent statements with mixed truth values.
Tautology truth table
Master the concept of tautology truth tables to represent statements that are always true.
What is an implication truth table?
Learn how to construct truth tables for logical implication operations.
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