Simple Proposition – Its Truthness and Logical Connectives

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:

1. disjunction P V Q
2. conjunction P / Q
3. negation P^c
4. Implication P → Q
5. equivalence P ↔ Q

Disjunction (V) :-

Let the two logical propositions be

1. P : truth that x ∈ A
2. 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(P^c) = 1
   T(P) = 1 then T(P^c) = 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 (P^c 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)