Logic is an important ingredient in both critical thinking and mathematical reasoning. While its roots are deep in philosophy, logic has wide applications in the area of mathematics in particular and in almost every aspect of human life. So it is important to learn some important laws of mathematical reasoning and established logical rules.

 

First, let us understand some important logical terms through examples:

 

Look at the following passage:

 

All humans are mortal. D. J. Trump is a human. Therefore, D. J. Trump is mortal.

 

In above passage, there are 3 sentences, or in logical terms, propositions. They each make a statement that is either true or false. This passage is trying to build an argument, which has 2 components: the premises (All humans are mortal. and D. J. Trump is a human.), and the conclusion (D. J. Trump is mortal).

 

Another thing that we know form above passage is that it tries to do the reasoning from one generic premise (All humans are mortal), and moves to a more specific premise (D. J. Trump is a human), and then reaches a logical conclusion (D. J. Trump is mortal). This logical process is called deductive reasoning or simply, deduction.

 

Now let us look at another example.

 

1 is a prime number. 3 is a prime number. 5 is a prime number. 7 is a prime number. Therefore, all odd integers between 0 and 8 are prime numbers.

 

In above passage, the writer is also trying to make an argument. But this time, the argument moves from specific premises (1 is a prime number. 3 is a prime number. 5 is a prime number. 7 is a prime number.) to a more generic conclusion (all odd integers between 0 and 8 are prime numbers). This logical process is called inductive reasoning or simply induction.

 

Both deduction and induction are commonly used in mathematical and everyday life reasoning.

 

Now let us look at another example:

 

If it snows, then the school will be cancelled.

 

First, we recognize that it is a conditional statement that consists of two parts, a hypothesis in the if clause, and a conclusion in the then clause.

 

From the above conditional statement, we can switch the hypothesis and the conclusion, and we will get the converse of the statement:

 

If the school is cancelled, it must be snowing.

 

We know that this is not necessarily true, as the cancelling of the school may be due to different reasons.

 

From the original conditional statement, we can also put the negative word to both the hypothesis and the conclusion, and we will get the inverse of the statement:

 

If it does not snow, then the school will not be cancelled.

 

We know that this is also not necessarily true, as the school can be cancelled even if it is not snowing.

 

Now, let us switch the hypothesis and conclusion, and put the negative word to both the hypothesis and the conclusion, and we will get the contrapositive of the original conditional statement:

 

If the school is not cancelled, it must not be snowing.

 

Now this is true, given the original conditional statement is true. A conditional statement is logically equivalent to its contrapositive statement.