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Topic To Be Covered:
First Order Logic(Part-02)
Jagdamba Education Society's
SND College of Engineering & Research Centre
Department of Computer Engineering
SUBJECT: Artificial Intelligence & Robotics
Lecture No-10(UNIT-03)
Logic & Reasoning
Prof.Dhakane Vikas N

First Order Logic
Inference in First-Order Logic
 Inference in First-Order Logic is used to deduce new facts or sentences
from existing sentences.
 understanding the FOL inference rule, let's understand some basic
terminologies used in FOL.
I. Substitution:
 Substitution is a fundamental operation performed on terms and
formulas.
 It occurs in all inference systems in first-order logic.
 The substitution is complex in the presence of quantifiers in FOL.
 If we write F[a/x], so it refers to substitute a constant "a" in place of
variable "x".

First Order Logic
Inference in First-Order Logic
II. Equality:
 First-Order logic does not only use predicate and terms for making atomic
sentences but also uses another way, which is equality in FOL.
 For this, we can use equality symbols which specify that the two terms
refer to the same object.
Example: Brother (John) = Smith.
 As in the above example, the object referred by the Brother (John) is
similar to the object referred by Smith.
 The equality symbol can also be used with negation to represent that two
terms are not the same objects.
Example: ￢(x=y) which is equivalent to x ≠y.

First Order Logic
FOL inference rules for quantifier:
As propositional logic we also have inference rules in first-order logic, so
following are some basic inference rules in FOL:
1. Universal Generalization
2. Universal Instantiation
3. Existential introduction
4. Existential Instantiation

First Order Logic
FOL inference rules for quantifier:
1. Universal Generalization:
 Universal generalization is a valid inference rule which states that if
premise P(c) is true for any arbitrary element c in the universe of
discourse, then we can have a conclusion as ∀ x P(x).
 It can be represented as:
 This rule can be used if we want to show that every element has a similar
property.
 In this rule, x must not appear as a free variable.
 Example: Let's represent, P(c): "A byte contains 8 bits", so for ∀ x P(x)
"All bytes contain 8 bits.", it will also be true.

First Order Logic
2. Universal Instantiation:
 Universal instantiation is also called as universal elimination or UI is a
valid inference rule.
 As per UI, we can infer any sentence obtained by substituting a ground
term for the variable.
 The UI rule state that we can infer any sentence P(c) by substituting a
ground term c (a constant within domain x) from ∀ x P(x) for any object
in the universe of discourse.
Example:1.
IF "Every person like ice-cream"=> ∀x P(x) so we can infer that
∀x:person(x) ->Likes(x,icecream)
So from this information, we can infer following statements using Universal
Instantiation:
person(john) ->Likes(john,icecream)

First Order Logic
2. Universal Instantiation:
Example:2
Let's take a famous example,
"All kings who are greedy are Evil." So let our knowledge base contains this
detail as in the form of FOL:
∀x king(x) ∧ greedy (x) → Evil (x),
So from this information, we can infer any of the following statements using
Universal Instantiation:
King(John) ∧ Greedy (John) → Evil (John),
King(Richard) ∧ Greedy (Richard) → Evil (Richard),

First Order Logic
3. . Existential introduction
 An existential introduction is also known as an existential generalization,
which is a valid inference rule in first-order logic.
 This rule states that if there is some element c in the universe of discourse
which has a property P, then we can infer that there exists something in
the universe which has the property P.
Example: Let's say that,
"Priyanka got good marks in English."
"Therefore, someone got good marks in English."

First Order Logic
4 . Existential Instantiation:
 Existential instantiation is also called as Existential Elimination, which is
a valid inference rule in first-order logic.
 This rule states that one can infer P(c) from the formula given in the form
of ∃x P(x) for a new constant symbol c.
 The restriction with this rule is that c used in the rule must be a new term
for which P(c ) is true.

First Order Logic
4 . Existential Instantiation:
 Example:
Sentence: There is at least one crown on the head of John.
 From the given sentence: ∃x Crown(x) ∧ On Head(x, John),
 So we can infer: Crown(K) ∧ On Head( K, John), as long as K does not
appear in the knowledge base.
 The above used K is a constant symbol, which is called Skolem constant.

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Ai lecture 10(unit03)

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