This document introduces predicate logic and its key concepts. It discusses that predicate logic uses predicate symbols to represent relations between objects, with the number of objects defining a predicate's degree. Predicate logic formulas can be joined using logical connectives like negation, conjunction, disjunction, and implication. Quantifiers like universal and existential are used to express properties of object collections. The syntax of predicate logic is inspired by Prolog and uses terms formed from constants, variables, and function symbols, along with quantifiers and sentences. Some useful equivalences for eliminating and distributing quantifiers are also presented. The document concludes by assigning students to describe predicate logic and discuss useful formulas for designing facts.

1. Introduction of Predicate Logics
Prof. Neeraj Bhargava
Kapil Chauhan
Department of Computer Science
School of Engineering & Systems Sciences
MDS University, Ajmer

2. Predicate Symbols and Signatures
 Predicate symbols represent relations between zero or more
objects
 The number of objects define a predicate degree
 Examples:
 Likes(george, kate)
 Likes(x,x)
 Likes(joe, kate)
 Friends (father_of(david), father_of(andrew))
 Signature: A signature is a collection of constants, function
symbols and predicate symbols with specified arguments.

3. Connectives
 FOL formulas are joined together by logical operators to
form more complex formulas (just like in propositional
logic)
 The basic logical operators are the same as in propositional
logic as well:
 Negation: ¬p („it is not the case that p“)
 Conjunction: p ∧ q („p and q“)
 Disjunction: p ∨ q („p or q“)
 Implication: p → q („p implies q“ or “q if p“)
 Equivalence: p ↔ q („p if and only if q“)

4. Quantifiers
 Two quantifiers: Universal (∀) and Existential (∃)
 Allow us to express properties of collections of objects instead of
enumerating objects by name
 Apply to sentence containing variable
 Universal ∀: true for all substitutions for the variable
 “for all”: ∀<variables> <sentence>
 Existential ∃: true for at least one substitution for the variable
 “there exists”: ∃<variables> <sentence>
 Examples:
 ∃ x: Mother(art) = x
 ∀ x ∀ y: Mother(x) = Mother(y) → Sibling(x,y)

5. Predicate logic, syntax
 The syntax in Luger is inspired by Prolog
 Propositions are replaced by –
 Predicate symbols (with arity) (p, q, r, …)
 – Terms, made of
 Constants (a, b, c, …)
 Variables (x, y, z, …)
 Function symbols (with arity) (f, g, h, …)
 Quantifiers ∀ (for all) and ∃ (exists)
 If ϕ is a sentence and x a variable, then ∀x:ϕ and ∃x:ϕ
are sentences.

6. Some useful equivalence
 Elimination: ∃x:p(x) ≡ ∀x:¬p(x)
also: ¬∃x:p(x) ≡ ∀x:¬p(x), ¬∀x:p(x) ≡ ∃x:¬p(x),
• • Names of variables can be changed:
∃x:p(x) ≡ ∃y:p(y), ∀x:p(x) ≡ ∀y:p(y)
• • ∀x:(p(x) ∧ q(x)) ≡ ∀x:p(x) ∧ ∀x:q(x) - not for ∨!
• • ∃x:(p(x) ∨ q(x)) ≡ ∃x:p(x) ∨ ∃x:q(x) - not for ∧!

7. Assignment
 Describe the predicate logic in logic of AI and discuss
the useful formulas to design facts.