Syntax and Semantics
∽ Negation
& Conjunction
v Disjunction
→ if then
↔ iff
If P and Q are formulas then the following are Formulas :
∽P
∽P V Q
P → Q
P → Q
P ↔ Q
Truth Table for sentences
|
P |
Q |
∽P |
∽P V Q |
P → Q |
Q → P |
P ↔ Q |
|
T |
T |
F |
T |
T |
T |
T |
|
T |
F |
F |
F |
F |
T |
F |
|
F |
T |
T |
T |
T |
F |
F |
|
F |
F |
T |
T |
T |
T |
T |
Rule 1: Conditional elimination: P → Q = ∽P V Q
Rule 2: Bi-Conditional elimination : P ↔ Q = ( P → Q) & (Q → P)
Rule 3: Inference Rule: Given a set of sentences , S={ s1 , s2, s3, ... ,sn}
s is proved . (Conclusion) written S /- s.
Rule 4: Modus Ponens : 
Rule 5: Chain Rule :
Rule 6: Conjunction : From P and From Q infer P & Q
Rule 7: Transposition: From P → Q infer ∽P → ∽ Q
Syntex of FOPL:
- Connective: There are five connective symbols. ∽, V, → , &, ↔
- Quantifier: Two quantifier symbols are : ∃ (existential ) ∀ (universal)
- Constants: constants are fixed value terms that belong to a given domain. They are denotedas numbers, words, and small letters near the start of the alphabet . Ex- house, a, b, 2
- Variables: These are terms that are assume different values over a given domain. They are denoted by words, small letters near the end of the alphabet.
- Function: Function symbol denotes relation defined on a domain , they map n elements to a single element of the domain . Ex- f,g,h.. An array function is written as f( t1, t2, ... tn) where ti are constants, variables and functions a zero array function is a constent.
- Predicate: Pradicate symbol denotes relations or functional mapping . From the elements of a domain d to the values true and false . Capital Letters and Capitalize words are used to represent predicate. Predicate may have n terms as arguments P (t1, t2, ..., tn) . A zero array predicate is a proposition.
NOTE: Constant variables and functions are referred to as 'Terms' and predicates are different are referred as 'Atoms'.
Prob- a) All employees earning rs 10000 or more per year pay taxes.
b) Some employees are sick today
c) No employee earns more than the president.
Solution- Knowlwdge Base : E(x) → Earn
P(x) → for x is a president
I(x) → for the income of x.
GE( U, V)
S(x) → for x is sick today
T(x) → for the income of x.
a) ∀ x (( E(x) & GE ( i (x), 10000)) c T(x)
b) ∃ x (E (x) → S (x))
c)∀ x ( (E(x) &P(y) → ∽GE( i(x), i(y))
All topics:
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