PREDICATE CALCULUS
Motivation
- The limitations of proposition calculus led to the motivation of predicate calculus.
- The limitations were using propositional calculus we cannot represent some facts and atoms in the propositional calculus are strings with no internal structure.
- A language is needed to state name of objects and names of propositions.
- Example: Proposition such as ON_P__Q ⊃ D | Clear_Q has to be expressed for every block. Instead a language is needed to express like On(x, y) ⊃ ᆨClear(y) generally for all blocks.
- So a language called first-order predicate calculus is used to which have the features that are needed.
- It has constants like object constants, function constants, relation constants and other constructs.
The Language and its Syntax
COMPONENTS
- The object constants are strings of alphanumeric characters. They start with a capital letter or a number.
- Example : A, Pq, 155, Kiran.
- The functional constants are strings of alphanumeric characters. They start with lower case letters. An infinite set of functional constants of all arities will be present. Where arity comes from the 'ary' suffix in binary(arity = 2), teritary (arity = 3) etc. The functional constants are superscripted with their respective arities.
- Example : brotherOF', times2.
- Relational constants are strings of alphanumeric characters which start with capital letter and superscripted by their arity.
- .Example: C163, Smalls1, Y124.
- The connectives that are used in propositional calculus like ∧, ∨, ⊃ and separators like ',' '.' and delimiters (, ), [, ] are used here.
TERMS
- Term specifies an object constant.
- A term can also be a functional constant of arity n, which is followed by η terms separated by commas in parenthesis. This is known as a functional expression. The parenthesis is omitted by a functional constant of arity 0.
- Example : brotherOf (Smith, Jain), Sam.
WFFS
- Atoms: If is also known as Atom formula. An atom or atom formula is a relational constant followed by terms separated by commas in parenthesis. A atom of arity 0 does not use the parenthesis.
- Example: Lessthan(2, 5), Q(1,2), A
- Propositional wffs: If the way of forming an expression in predicate calculus is similar to that of propositional calculus then that expression is called the propositional wff.
- Example: [Lessthan(2, 5) ∨ Greaterthan(5, 4)] ∨ ᆨBrother(John, Bill) ∨ Q.
