Proposition :-
A simple proposition ‘P’ is a linguistic statement contained within a universe of elements ‘x’ that can be identified as being a collection of elements in ‘x’ that are simple true or simply false.
Truthness or Veracity of proposition P :-
Truthness (or Veracity) is denoted by T(P) and it can be assigned a binary truth value.
Let P and Q be two simple propositions defined on same universe of discourse, these simple propositions can be combined using the following 5 logical connectives:
Disjunction (V) :-
Let the two logical propositions be
- P : truth that x ∈ A
- Q : truth that x ∈ B
T(P) = 1 when x ∈ A otherwise T(P) = 0
T(Q) = 1 when x ∈ B otherwise T(Q) = 0
P V Q = x ∈ A or x ∈ B
hence T (P V Q) = T(P) V T(Q) = max [ T(P), T(Q)]
Conjunction :-
P / Q = x ∈ A and x ∈ B
hence T(P / Q) = min [ T(P), T(Q)]
Negations :-
If T(P) = 0 then T(Pc) = 1
T(P) = 1 then T(Pc) = 0
Implication (→) :-
P → Q = x ∉ A or x ∈ B
T(P → Q) is true ∀ cases except when,
first proposition T(P) = 1 is true and
second proposition T(Q) = 0 is false
Hence T(P → Q) = T (Pc V Q)
The logical connective application that is P → Q (P implies Q) is true in all cases except when P is true and Q i false.
In P → Q the simple propositions P and Q are called as below:
- P is called hypothesis or anticident.
- Q is called consequence or conclusion.
So finally, “A true anticident cannot imply a false consequence.”
or it can also be said as – “A true hypothesis cannot imply a false conclusion.”
Equivalence (↔) :-
When P ↔ Q it is called compound proposition.
T(P ↔ Q) = 1 for T(P) = T(Q)
T(P ↔ Q) = 0 for T(P) ≠ T(Q)
| T(P) | T(Q) | T(P V Q) | T(P / Q) | T (Pc | T(P → Q) | T(P ↔ Q) |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | 1 | 1 |