DHARM
PROPOSITIONAL LOGIC 139
N(y) : ,, “y is a number”
Then we translate the statement straightforward by,
(∀x) [N(x) → ((∃y) N(y) ∧ G(x, y))]
- “Gentleman prefers honesty to deceit.
Let, G(x) : “x is gentleman”
H(x) : “x is honest”
D(x) : “x is deceit”
And P(x, y, z) : “x prefers y over z”
Then, using the universal quantifiers the predicate expression of the statement will be,
(∀z) (∀y) (∀x) [(G(x) ∧ H(x) ∧ D(x)) → P(x, y, z)
- “There are no Holy Ganges in Purva”
Using the symbol expressions,
H(x) : “x is Holy”
G(x) : “x is Ganges”
P(x) : “x is Purva”
Then the statement can be expressed by equivalent expressions like as,
~ (∃x) [H(x) ∧ G(x) ∧ P(x)]; (there exists some x for that x is holy and x is
ganges and x is purva is not true)
or, (∀x) [(P(x) ∧ G(x)) → ~ H(x)]; (for all x, if x is purva and x is ganges then x is
not holy)
or, (∀x) [(H(x) ∧ G(x)) → ~ P(x)]; (for all x, if x is holy and x is ganges then x is not
purva)
or, (∀x) [(H(x) ∧∧∧∧∧ P(x)) → ~ G(x)]; (for all x, if x is holy and x is purva then x is not
ganges)
or, (∀x) [ ~ H(x) ∨ ~ G(x)) ∨ ~ P(x)]; (for all x, x is not holy or x is not ganges or x is not
purva)
In order to determine the truth values of the statements involving universal and/or
existential quantifier/s, one may be able to persuade the truth values of the statement func-
tions. Since statement functions don’t have the truth values, and when the name of the indi-
viduals is substituted in place of variables then the statement have a truth value. Of course,
we can determine the truth value of the statement on the basis of the domain set D.
For example, D = {1, 2, 3, 4}
· Then, truth value of the predicate expression
(∀x) (∃y) [E(y) ∧ G(y, x)] : where E(y) stands “y is a even number” and G(y, x)
stands “y ≥ x”
will be true; because for all numbers of the set D, we can find at least a number greater than or
equal to that number.
· Truth value of the predicate expression
(∀x) (∃y) [~ E(y) ∧ G(y, x)] : where ~ E(y) stands “y is not a even number” and
G(x, y) stands “y ≥ x”
will be false; because for the number 4 there is no odd number in the set which is greater than
or equal to it.
