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Simplicity is the ultimate sophistication.” — Leonardo da Vinci
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Saturday 3 June 2017

Nature of logical statements and postulates.

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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