Tranlating to Predicate Logic

To translate statements stated in English using a given set of predicate symbols, we  first restate English proposition using the predicates, connectives, and quantifiers such that it preserve its original meaning. Then replace the English phrases with the corresponding symbols
.
Example 1: Given the sentence "Not every integer is even".
Now let the predicate "E(x)" represent x is even, and that the universe is the set of integers, First restate it as "It is not the case that every integer is even"
Then "it is not the case" can be represented by the connective "", "every object x in the universe" by " x", and "x is even" by E(x).
Thus altogether wff becomes x E(x).

Example 2

Take universe of discourse a set of all students of Kathmandu College.

P(x) represents:  x takes Discrete Mathematics class.

Here universal quantification is ∀x  P(x),  which represent the English sentence “all students of Kathmandu college take Discrete Mathematics class”,  and now it is a proposition.

The universal quantification is conjunction of all the propositions that are obtained by assigning the value of the variable in the predicate. Going back to above example if universe of discourse is a set {Ram, Shyam, Hari, Sita} then the truth value of the universal quantification is given by P(ram) ∧ P(Shyam) ∧ P(Hari) ∧ P(Sita) i.e. it is true only if all the atomic propositions are true.

Existential Quantifier

Universal quantifier, denoted by ∃, is used for existential quantification. The existential quantification of P(x), denoted by ∃x P(x), is a proposition “P(x) is true for some values of x in the universe of discourse”. The other forms of representation include “there exists x such that P(x) is true” or “P(x) is true for at least one x”.

Example 3

For the  same  problem  given  in  universal  quantification  ∃x  P(x)  is  a  proposition  is represent like “ some students of Kathmandu College take Mathematics class”.

The existential quantification is the disjunction of all the propositions that are obtained by assigning the values of the variable from the universe of discourse. So the above example is equivalent to P(Ram) ∨  P(Shyam)∨P(Hari)∨ P(Sita), where all the instances of variable are as in example of universal quantification. Here if at least one of the students takes graphics class then the existential quantification results true.

Translating the Sentences into Logical Expression

Example 4

Translate “not every integer is even” where the universe of discourse is set of integers.

Solution

Let E(x) denotes x is even.

Then ¬∀xE(x) represents the above statement “not every integer is even”

Example 5

Translate “every man is mortal”

Let M(x) denote x is mortal, where x is from set of man (here universe of discourse is all man)

Then, ∀x M(x) represent that “for all x ,  x is mortal.”