Semantics of Quantifiers

Semantics of Quantifiers 

Universal Quantifiers 

• If all the assignments of the variable symbol r in w are true then the value of (∀r)w(r) is true. 

Example 

Fig: Discs World Situation

• Now if we have to know the truth value of [∀x][On(x, Q) ⊃ᆨClear(Q)] from above Fig. 2.4.3. Then we have to substitute x as P, Q, R and floor and check. 
• If we substitute x as P then for the statement to be true P, Q should be in relation. On and should not be in relation Clear. 
• Now it is true for x = P as R is not in relation Clear. 
• As in each interpretation R is in relation Clear, the truth value of the Statement is true. 

Existential Quantifiers 

    If atleast one of the assignments of the variable r is statement (Ǝr) w(r) has truth value as true then it is true.

Useful Equivalences 

If the semantics of the quantifiers are given, the equivalences similar to DeMorgan's law can be established. They are, 

• ᆨ(∀r)w(r) ≡ (Ǝr) ᆨw(r)
• ᆨ(Ǝr)w(r) ≡ ( ∀r) ᆨw(r). 
• (∀r)w(r) ≡ ( ∀η) w(η)

Rules of Inference

• The rules of inferences .which are generalized by propositional calculus can be used with predicate calculus. 
• The rules of inferences include modus ponens, ⴷ introduction, ∨ introduction, ⴷ elimination, ᆨelimination and resolution. 
• The other rules added in predicate calculus are:

• Universal Instantiation (UI): If in a universal quantified wff like (∀r) w(r) a constant ∝ is substituted at all the occurrences of r in w it is called as universal instantiation. 
• Example: From (∀y) Q(y, f(y), A) 
• we infer Q(A, f(A), A). 
• Existential Generalization (EG): If we have a wff w(∝) where ∝ is a constant symbol inferring (Ǝr) w(r) is known as existential generalization where w(r) is got by replacing all the occurrences of ∝ with r throughout w. 
• Example: From (∀y) Q(B, g(B), y) 
• we infer (Ǝx)(∀y) Q(x, g(x), Y) 
• These two are important rules of inference.