DHARM
PROPOSITIONAL LOGIC 143
This rule is called universal instantiation and is denoted by UI in the inference theory.
Therefore, according to rule UI, whenever universal quantifier exists it will drop with intro-
ducing other variable say j in place of x.
For example, consider the argument:
“All human are mortal. John is human. Therefore, John is mortal”.
The argument can be translated into predicate premises and conclusion e.g.
- (∀x) (H(x) → M(x))
- H(j)/∴ M(j)
- H(j) → M(j) 1, UI
- M(j) 3 & 2, MP
Hence, argument is valid.
Rule II. Universal Generalization (UG)
:
n.A
:
/∴ (∀x) Axy
Let A be any predicate formula then it can conclude to (∀x) Axy i.e., whenever y occur-
ring put x provided following restriction,
· y is an arbitrary selected variable.
· A is not in the scope of any assumption, it contains free y.
e.g., :
:
( .........) free occurrence of y
:
:
n.A
n + m. (∀x) Axy since it violate second restriction, hence wrong.
Above rule is called universal generalization and is denoted as UG. UG will permit to
add the universal quantifier in the conclusion and variable x is replaced by an arbitrary se-
lected variable y.
Consider an argument,
I. “No mortal are perfect. All human are mortal. Therefore, no human are perfect”.
(where we symbolize M(x): “x is mortal”; P(x): “x is perfect”; H(x): “x is human”)
Thus, we express the corresponding predicate premises and conclusion as,
- (∀x) (M(x) → ~ P(x))
- (∀x) (H(x) → M(x)) /∴ (∀x) (H(x) → ~ P(x))
- M(y) → ~P(y) 1, UI
- H(y) → M(y) 2, UI
- H(y) → ~ P(y) 4 & 3, HS
- (∀x) (H(x) → ~P(x)) 5, UG
