Lecture 1: LOGIC: Boolean Algebra for class XII Computer Science Students

Date: 23.06.12

Introduction to Logic

The main ingredient in the study of logic is the principles and method used to distinguish between arguments that are valid and those that are not. Logic deals with reasoning and the ability to deduce or come to some reasonable conclusions. In everyday life we guess what is going to happen on the basis of past experiences; “It looks like its going to rain” we say meaning that it may rain today. If we wait around long enough then it may rain. This is an example of inductive reasoning. 

In mathematics we can discover whether or not a guess is correct by checking if our conclusions can be deduced from results already known. This is called deductive reasoning. (This portion of tutorial is taken from|yale.edu)

 Inductive Reasoning v/s Deductive Reasoning

Example of Inductive Reasoning:

All of the swans that all living beings have ever seen are white.

Therefore, all swans are white. 

(example taken from|wikipedia.org)

This type of reasoning is based on general conceptions and tries to draw conclusions on the basis of previously laid down concepts and theories. In the above example, ALL LIVING BEINGS HAVE SEEN EVER SWANS AND FOUND THAT ALL SWANS ARE WHITE, THAT MEANS VARIOUS DIFFERENT EXAMPLES AND INSTANCES (STORIES, EVENTS) ARE BEING RECORDED IN HISTORY WHERE LIVING BEINGS HAVE SEEN WHITE SWANS. SO WE CONCLUDED OUR RESULT THAT ALL SWANS ARE WHITE.

Example of Deductive Reasoning:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

This type of reasoning in contrast to inductive reasoning is not based on any general principle, rather it takes various examples and then mathematically tried to deduct accurate solutions. 'Mathematical induction' is a type of deductive reasoning, which can be easily seen in the above example, THE FIRST PREMISE STATES THAT ALL OBJECTS CLASSIFIED AS "MEN" HAVE THE ATTRIBUTE "MORTAL". THE SECOND PREMISE STATES THAT "SOCRATES" IS CLASSIFIED AS A "MAN" – A MEMBER OF THE SET "MEN". THE CONCLUSION THEN STATES THAT "SOCRATES" MUST BE "MORTAL" BECAUSE HE INHERITS THIS ATTRIBUTE FROM HIS CLASSIFICATION AS A "MAN".

(example and explanation taken from|wikipedia.org) 

Statements

The starting point of logic is a statement. A statement in the technical sense is declarative and is either true or false, but cannot be both simultaneously. In logic it is irrelevant whether a statement is true or false, the important thing is that it should be definitely one or the other. Logic statements must be either true or false.

A Statement: is a declarative sentence which is either true or false.
Examples of declarative statements:

(a) New Haven is a city in Connecticut.

(b) The month of June has thirty days.

(c) The moon is made of red cheese.

(d) Tomorrow is Saturday.

The following are not statements:

(a) Come to our party!

(b) Is your homework done?

(c) Close the door when you leave.

(d) Good by dear.

Those are not good statements because they cannot be considered true or false.

The basic type of sentence in logic is called a simple statement. A simple statement is one that has only one thought with no connecting word.

Examples of simple statements

(a) Three is a counting number.

(b) Ann is early for class

If we take a simple statement and join them with a connecting word such as and, or, if . . . then, not, if and only if, we form a new sentence called a complex or compound statement.

Compound Statements: are formed from the combination of two or more simple statements.
(a) Ann is early for class and she has her note books.

(b) Three is a counting number and is also a odd number.
(This portion of tutorial is taken from|yale.edu)

Next topic: 24.06.12: Lecture 2: Types of Compound statements and their connectives