What is a Proposition?

Any statement that is either true or false, is termed as proposition.

There are two laws regarding proposition as follows:

  • Law of excluded middle

    This law states that a proposition is required to be either true or false but not both.

  • Law of contradiction

    True = not False
False = not True

  • Types of proposition:

    • Atomic Proposition : Any proposition which cannot be further divded is termed as atomic proposition.

    • Compound Proposition : One or more atomic proposition combined to form a compound proposition using connectors/operators (i.e ~, ^, v, ->, <->).

    • Note:- Both atomic and compound Proposition's are generally termed as Premises.

    • Tautology: A propositional function which is true in every possible case. A tautology is also termed as Valid statement.

    • Contradiction : A propositional function which is false in every possible case.

    • Contingency : A propositional function which is some time true and sometime false.

Satisfiability

A statement or a propsoition is said to be satisfiable if its truth table has atleast one true value, otherwise statement is unsatisfiable.

Clearly, Tautology and Contingency are satisfiable whereas Contradiction is unsatisfiable.

What is Argument?

Collection of Premises on the basis of which we derive a conclusion is termed as Argument.

Connectors / Operators

  • Negation (~) : It is a NOT of digital electronics.
P ~P
True False
False True
  • Disjunction (v) : It is a OR of digital electronics.
P Q P v Q
True True True
True False True
False True True
False False False
  • Conjunction (^) : It is AND of Digital Electronics
P Q P ^ Q
True True True
True False False
False True False
False False False

  • Implication (->) : It is also an binary operator. P -> Q, is read as If P then Q.

    • With truth table as follows:
P Q P -> Q
True True True
True False False
False True True
False False True

  • P -> Q = ~p v Q | If P then Q is equal to Not P OR Q

  • Bidirectional operator (<->) :

    • If sigifies if and only if (i.e iff)
    • Work as Ex-NOR of digital electronics
    P Q P <-> Q
    True True True
    True False Flase
    False True False
    False False True

Precedence of Operators

Operators Name Precedence
~ Negation 1
^ Conjunction 2
v Disjunctin 3
-> Implication 4
<-> Biconditional 5

Converse, Inverse & Contrapositive

Statement If p, then q. If two angles are Congurent, then they have same measure.
Converse If q, then P. If two angles have same measure, then they are congurent.
Inverse If not p, then not q. If two angles are not Congurent, then they don't have same measure.
Contrapositive If not q, then not p If two angles don't have same measure, then they are not congurent.

Quantifiers

First we have to understand the concept of Predicates. One of the downside of propositonal-logic is that we cannot restrict the domain of the subject, predicate’s help us to do so.

Predicate are wriitten as P(x), where P is the logic and x is the subject.

Predicates are used alongside quantifiers to express the extent to which a predicate is true over a range(domain) of elements.

There are two types of quantifiers:

  1. Universal Quantifier
  2. Existential Quantifier

Some Properties of Quantifiers