B tech. Discrete mathematics (I. T & Comp. Science Engg.) Syllabus

Rules of Inference for Propositional logic
We can always use a truth table to show that an argument form is valid. Arguments b a s e d on t a u t o l o g i e s r e p r e s e n t u n i v e r s a l l y c o r r e c t method of r e a s o n i n g . Their validity depends only on the form of statements involved and not on the truth values of the variables they contain such arguments are called rules of inference.
These rules of inference can be used as building blocks to construct more complicated valid argument forms
e.g.

Let P: “You have a current password” Q: “You can log onto the network”.
Then, the argument involving the propositions,

“If you have a current password, then you can log onto the network”.

“You have a current password” therefore: You can log onto the network” has the form …

.

P  Q P

^ Q

Where ^ is the symbol that denotes ‘therefore we know that when P & Q ^{are proposition variables, the statement} ((P ^ Q) ^ P) ^ Q ^{is a tautology}

.

So, this is valid argument and hence is a rule of inference, called modus ponens or the law of detachment.

(Modus ponens is Latin for mode that affirms)

The most important rules of inference for propositional logic are as follows…..

Example16:
Test the validity of the following arguments :

1. 
   If milk is black then every crow is white.
   
2. 
   If every crow is white then it has 4 legs.
   
3. 
   If every crow has 4 legs then every Buffalo is white and brisk.
   
4. 
   The milk is black.
   
5. 
   So, every Buffalo is white.
   

Solution :
Let P : The milk is black

Q : Every crow is white

R : Every crow has four legs. S : Every Buffalo is white

T : Every Buffalo is brisk The given premises are

1. 
   P  Q
   
2. 
   Q  R
   
3. 
   R  S  T
   
4. 
   P
   

The conclusion is S. The following steps checks the validity of argument.

1. 
   P  Q premise (1)
   
2. 
   Q  R Premise (2)
   
3. 
   P  R line 1. and 2. Hypothetical syllogism (H.S.)
   

4. 
   R  S  T
   
5. 
   P  S  T
   

Premise (iii)

Line 3. and 4.. H.S.

6. 
   P Premise (iv)
   
7. 
   S  T L^{ine 5, 6 modus ponens}
   
8. 
   S Line 7, simplification
   

 The argument is valid
Example17 :

Consider the following argument and determine whether it is valid or not. Either I will get good marks or I will not graduate. If I did not graduate I will go to USA. I get good marks. Thus, I would not go to USA.

Solution :

Let P : I will get good marks.

Q : I will graduate. R : I will go to USA

The given premises are

1. 
   P V  Q
   
2. 
    Q  R
   
3. 
   P
   

The conclusion is  R.