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CHAPTER 5 : First-order Logic, WAN NUR AFIQAH BINTI ZULKEFLI…
CHAPTER 5 : First-order Logic
Definition
Also known as predicate logic, quantificational logic, and first-order predicate calculus - is a collection of formal systems used in mathematics, philosophy, linguistics and computer science
First-order logic uses quantified variables over non-logical objects and allows the use of sentence that contain variables
So, that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man, where "there exists" is a quantifier, while x is a variable.
This distinguishes it from propositional logic, which does not use quantifiers or relations, in this sense, propositional logic is the foundation of first-order logic.
propositional logic - statement either true/false
propositional logic example :
if the moon is made of cheese than basketball are
round and if the spiders have eight legs then sam
walk with a limp.
P-->Q
First-order Logic
Much more powerful than the propositional (Boolean) logic
Greater expressive power than propositional logic
Allows for facts, objects and relations
Pros and Cons of Propositional Logic
Pros
declarative : pieces of syntax correspond to facts
Allow for partial/disjunctive/negated information (unlike most data structures and DB
Context independent : unlike natural language, where the meaning depends on the context
Propositional logic is compositional
Cons
has a very limited expressive power
lack of syntax
Components of First-order Logic
Term
Constant, eg : Red
function of constant, eg : Color(Block1)
Atomic Sentence
eg: Brother(John,Richard)
eg: Married (Mother(John),Father(John))
Predicate relating objects (no variable)
Complex Sentence
Atomic sentences + logical connectives
eg : Brother(John,Richard) ^-Brother(John,Father(John)
Quantifiers
Each quantifier defines a variable for the duration of the following expression, and indicates the truth of the epression
Existential quantifier "there exists"
The expression is true for at least one value of the variable
Universal quantifier "for all"
The expression is true for every possible value of the variable
8 Logics in general
Properties of Quantifiers
:
Truth in First-Order Logic
Sentence are true with respet to amodel and an interpretation
Model contains >=1 object (domain elements) and relations among them
Interpretation specifies referents for
constant symbols -> objects
function symbols -> functional relations
predicate symbols -> relations
WAN NUR AFIQAH BINTI ZULKEFLI (D20191089434)
