Nature of logical statements and postulates.

Postulates are usually propositions that do not have a definite proof. They are usually statements concluded from an experiment. A Logical statement is defined as an assertion obtained by the application of

*rules*that are defined to be correct. In the context of this article, term ‘*rule’*is used interchangeably in place of the term ‘*logical*-*statement’*and the term*rule-set*is always used interchangeably with the phrase ‘*a set of*logical statements*’.*And by definition, a logical statement is a statement derived from*fixed principles*that are known to be correct. Contrary to the popular belief that every statement can be proved, in reality,*not all statements can be proven*. This is because, in the end, every rule is constructed by fixed rules and to prove the ‘fixed rules’ correctness, we will need*more ‘*fixed rules’.**Definition 1:**

**Let’s define an**

*abstract operation*$Y = C(X)$, where $X$ is a rule-set and the notation $C(X)$ represents the rule-set*reasoned*from the rule-set $X$. The expression $C(X)$ does not provide any information of the kind of reasoning that’s done on the*rule-set*$X$ to obtain the reasoned*rule-set*$Y$.**Postulate 1:**

**For some rule-set $X$,**

**$C(X) = C(C(… C(X) ...))$ or $C \circ C \circ … \circ X = C (X)$.**

Discussion :

It is obvious to conclude the correctness of the above-presented statement as the symbol $C$ represents an ambiguous operation that relates the statement given as a parameter to obtain another statement derivable from the rule-set provided as a parameter. Therefore, writing $C(X) = C \circ C \circ … \circ C(X)$ can be considered to be correct.

**Theorem 1 :**

**For any rule-set $X_2$ derived from the rule-set $X_1$ , there does not exist a rule-set $X_3$ such that $X_3$ can be derived from $X_2$ or a combination of $X_1$ and $X_2$, but not from $X_1$.**

Proof :

Let’s assume the following statement to be true.

$X_3 \neq C(X_1), X_2 = C(X_1), X_3 = C(X_2)$. … (1)

But from the expressions $X_2 = C(X_1), X_3 = C(X_2)$ we can write,

$X_3 = C ( C(X_1)) \implies X_3 = C (X_1)$ ( from

*postulate 1**)*… (2)
The statement (2) contradicts the statement (1); therefore, we can conclude that the statement (1) is incorrect which implies $X_3 = C(X_1)$ to be true.

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