First order predicate logic(fopl)
- 2. OBJECTIVE INTODUCTION DIFFERENCE BETWEEN PROPOSITIONAL LOGIC & FIRST ORDER LOGIC PARTS OF FIRST ORDER LOGIC QUANTIFIERS What are Rules of Inference for? PROS & CONS OF FOPL REFERENCES
- 3. INTRODUCTION First-order logic—also known as first-order predicate calculus and predicate logic—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 sentences 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" and there exists is a quantifier while X is a variable.
- 4. PL(Propositional Logic) • It uses prepositions in which complete sentence is denoted by symbol. • PL can’t represent individual entities. Eg: Meena is short • It can’t express generalization, specialization or pattern. Eg: Triangles have 3 sides. D I F F E R E N C E B E T W E E N P R O P O S I T I O N A L L O G I C & F I R S T O R D E R L O G I C FOL(First Order Logic) • FOL was predicated which involve constants, variables, functions, relations. • FOL can represent individual properties. Eg: short(Meena). • It can express generalization, specialization or pattern. Eg: No._of_sides(triangle 3)
- 5. PARTS OF FIRST ORDE R L OGIC There are two key parts of first-order logic. The syntax determines which collections of symbols are legal expressions in first-order logic, while the semantics determine the meanings behind these expressions.
- 6. SYNTAX OF FIRST-ORDER LOGIC: BASIC ELEMENTS Note: The equality predicate is always in the vocabulary. It is written in infix notation
- 7. Atomic sentence predicate ( term1; : : : ; termn ) or term1 = term2 Term function ( term1; : : : ; termn ) or constant or variable
- 8. S Y N TA X O F F I R S T - O R D E R L O G I C : A T O M I C S E N T E N C E S Example
- 9. S Y N TA X O F F I R S T - O R D E R L O G I C : C O M P L E X S E N T E N C E S Built from atomic sentences using connectives
- 10. SEMANTICS A predicate logic expression gets a meaning through a configuration, i.e. the specification of 1. a non-empty domain, 2. an interpretation that gives • for every constant an element of that domain, • for every function symbol with arity n some concrete n-ary function on the domain, and • for every predicate symbol with arity n some concrete n-ary relation on the domain, and 3. an assignment for the free variables in the expression.
- 11. SE MANTICS OF TE RMS & FORMUL AS
- 13. QUANTIFIERS The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier. Universal Quantifier Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀. ∀xP(x) is read as for every value of x, P(x) is true. Example − "Man is mortal" can be transformed into the propositional form ∀xP(x) where P(x) is the predicate which denotes x is mortal and the universe of discourse is all men.
- 14. Existential Quantifier Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃. ∃xP(x) is read as for some values of x, P(x) is true. Example − "Some people are dishonest" can be transformed into the propositional form ∃xP(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people. Nested Quantifiers If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier. Example ∀ a∃bP(x,y) where P(a,b) denotes a+b=0 ∀ a∀b∀cP(a,b,c) where P(a,b)P(a,b) denotes a+(b+c)=(a+b)+c Note − ∀a∃bP(x,y)≠∃a∀bP(x,y) To deduce new statements from the statements whose truth that we already know, Rules of Inference are used.
- 15. WHAT ARE RUL E S OF INFE RE NCE FOR? Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements. An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “∴∴”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.
- 16. TABL E FOR RUL E S OF INFE RE NCE
- 17. Addition: If P is a premise, we can use Addiction rule to derive P∨Q.
- 21. PROS & CONS OF FOPL PROS: Propositional logic is declarative: pieces of syntax correspond to facts Propositional logic allows partial / disjunctive / negated information (unlike most data structures and databases) Propositional logic is compositional: meaning of B1;1 ^P1;2 is derived from meaning of B1;1 and of P1;2 Meaning in propositional logic is context-independent (unlike natural language, where meaning depends on context) CONS: Propositional logic has very limited expressive power (unlike natural language) Example: Cannot say “pits cause breezes in adjacent squares” except by writing one sentence for each square
- 22. APPLICATIONS Predicate logic is useful in its own right as a tool for presenting arguments rigorously & determining their validity. The original purpose of predicate logic was to provide a formal procedure for proving mathematical theorems. The main work done by predicate logic is as the framework for the formulation of theories.
- 23. REFERENCES https://en.wikipedia.org/wiki/Geographic_information_system https://www.youtube.com/watch?v=pcV2lL6yNZ8 https://www.tutorialspoint.com/discrete_mathematics/operators_ and_postulates.htm