Mathematical Foundation of Computer Science

DHARM

PROPOSITIONAL LOGIC 135

That is,

1. X 1 : True


2. X 2 : True


3. X 3 : True


4. X 4 : True


5. X 5 : True /∴ ~ T : False
   Therefore argument is invalid for following interpretation,
   T : True
   M : False (since T is true; so ~ T → M will be True only when M is false)
   H : True (since T is true; so ~ H → ~ T will be true only when H is true)
   J : false (since H is true; so J → ~ H will be true only when J is false)
   Example 5.28. Prove formula (P ∨ ~ P) is a tautology.
   Sol. We will see that by assuming negation of the formula we find a contradiction, which
   concludes that X will be tautology.
   Using ATM, expand the tableaux by assuming ~ (P ∨ ~ P) will be labeled at root.
   ~ (P∨~ P)

~ P

~ ~ P

P

×

So, we find a closed path, therefore tableaux is closed. Hence formula is a tautology.

5.7 Predicate Logic................................................................................................................

So far our discussion of symbolic logic and inference theory are concern statements and the

propositional variables are the basic units which are silent about the analysis of the state-

ments. In order to investigate the property in common between the constitute statements; the

concept of a predicate is introduced. The predicate is the property of the statement and the

logic based upon the analysis of the predicate of the statement is called predicate logic. Con-

sider an argument,

1. All human are mortal : A


2. John is a human : J

/∴ John is mortal : M
If we express these statements by symbols, then the symbols do not expose any common
feature of these statements. Therefore, particular to this symbolic representation inference
theory doesn’t derive the conclusion from these statements. But in course, conclusion appears
unthinkingly true. The reason for such deficiency is the fact that the statement “All human are
mortal” can’t be analyzed to say anything about an individual or person. If the statement is
slices from its property “are mortal” to the part “All human” then it might be possible to con-
sider any particular human.