1. Rules of inference for quantifiers
Aniqa riasat
Sammon azam
Hafsa anjum
Rubba rehman
Mahnoor arif
Momina hassan
Aaima Mansoor
Haroon Iqbal
Fatima ehsan
Arsalan somroo

2. Inference Act of drawing a conclusion.
Can be interpreted as a function which:
takes premises
Analyzes their syntax
Returns a conclusion (or conclusions)

3. Quantifiers
A type of determiner that indicates quantity and expresses the meaning of the words “all” and “some”.
The two most important quantifiers are:
Universal quantifier, “For all”.
Symbol: ∀
∀ x P(x) asserts that P(x) is true for every x in the domain.
Existential quantifier, “There exists”.
Symbol: Ǝ
Ǝ x P(x) asserts that P(x) is true for some x in the domain.

4. Universal Quantifier Example:
For all x P(x) is read as “For all x, P(x)” or “For every x, P(x)”.
The truth value depends not only on P, but also on the domain U.
Example:
Let P(x) denote x > 0.
If U is the integers then ∀ x P(x) is false.
If U is the positive integer then ∀ x P(x) is true

5. Existential Quantifier
There exist x P(x) is read as “For some x, P(x)” or “There is an x such that,
P(x)”, or “For at least one x, P(x)”.
The truth value depends not only on P, but also on the domain U.
Example:
Let P(x) denote x < 0.
If U is the integers then Ǝ x P(x) is true.
If U is the positive integers then Ǝ x P(x) is false.

6. Translating English to Logic
Translate the following sentence into predicate logic:
“Every student in this class has taken a course in Java.”
Solution:
First decide on the domain U.
Solution 1:
If U is all students in this class, define a propositional function J(x) denoting
“x has taken a course in Java” and translate as ∀ x J(x).
Solution 2:
But if U is all people, also define a propositional function S(x) denoting
“x is a student in this class” and translate as ∀ x (S(x) J(x)).

7. Universal Instantiation
All humans have two legs. John Smith is a human. Therefore, John Smith has two legs.
H(x) is “x is a human.”
L(x) is “x has two legs.”
j is John Smith, a element of the universe of discourse.
∀ x (H(x) L(x)) Premise.
H(j) L(j) Universal instantiation from 1.
H(j) Premise.
∴ L(j)

8. Universal Generalization
P(c) for an arbitrary c
∴ ∀ x P(x)
We must:
Define the universe of discourse.
Then, we must show that P(c) is true for an arbitrary, and not a specific, element c of the universe of discourse.

9. Existential Instantiation
Ǝ x P(x)
∴ P(c) for some element c
The rule that allows us to conclude that there is an element c in the universe of discourse for which P(c) is true if we know that Ǝ xP(x) is true.
We can not select an arbitrary value of c here, but rather it must be a c for which P(c) is true

10. Existential Generalization
P(c) for some element c ∴ Ǝ x P(x) If we know one element c in the universe of discourse for which P(c) is true, therefore we know that Ǝ x P(x) is true.

11. Examples:
Every student is ambitious. No one who is ambitious will be lazy. Therefore, there will be no lazy students. Student = “S(x)” Ambitious = “A(x)” Lazy = “L(x)” Every student is ambitious = ∀ x (S(x) A(x)) No one who is ambitious will be lazy = ∀ x (A(x) ~L(x)) Therefore, there will be no lazy students = ∀ x (S(x) ~L(x))

12. Examples:
Some students do work hard. Anyone who works hard will pass. Therefore, there are students who will pass. Students = S(x) Hard work = H(x) Pass = P(x) Some students do hard work = Ǝ (S(x) ^ H(x)) Anyone who work hard will pass = ∀ (H(x) P(x)) Therefore, there are students who will pass = Ǝ (S(x) ^ P(x))