## The Fundamental Laws of Logic

The Fundamental Laws of Logic or the Classic Laws of Thought may be redundant in some respects; the introduction of propositional logic and fuzzy logic has meant that some of the axioms (some disrupted to be axioms) these laws are not always applicable when discussing situations regarding logic like the construction of mathematical proofs. The three laws are still used in most systems of logic, and consist of the following:

Let $P$ be any proposition, then the law of contradiction states that:

$\forall P, \sim (P \wedge \sim P)$

The above statement states that it is impossible for both P and the negation of P to both be true at the same time. For example, an even number can’t also be odd (or non-even), it is either even or odd.

Law of Excluded Middle:

Let $P$ be any proposition, then the law of excluded middle states that:

$P \vee \sim P$

The mentioned statement asserts that the proposition P is either true or it is false, it is not possible for both P and the negation of P to be both true.

Principle of Identity:

Let X be some proposition or variable, then the principle of identity asserts that:

$\forall X, X = X$

The principle of identity implies that for all instances of the object named X, each instance is exactly the same as each other.

Extensions to the Fundamental Laws of Logic

There have been some additions by Logicians to the traditional laws of thought, Leibniz added two of these laws, which are as follows:

Principle of Sufficient Reason:

The principle of sufficient reason is simple and self explanatory within the name; it states that nothing is without reason, and thus every event has a cause. This forms the basis of our rational thinking and of scientific method.

In some aspects, the principle of sufficient reason can be used to contest the notion of free will; the paradox of free will indicates that while we are given free will, we are not truly free since God created us and the rules which are used to control the game known as life.

The principle additionally shows us that for something to not exist, then there must be a reason for that object’s non existence; the principle also has given rise to the philosophical concept of Necessitarianism.

Identity of Indiscernibles:

This is slightly more interesting than Leibniz’s other principle, it implies that no two distinct objects are exactly identical, even though that it may appear to be that way. This statement can be summarised in the following formula:

$\forall P (Px \leftrightarrow Py) \rightarrow x = y$, where P denotes the property.

The above principle can be counter intuitive in some circumstances, especially when considering hypothetical situations. For instance, what if the property which distinguishes the two objects is shared by both of the objects?

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In the future, I’m intending to write some more philosophical posts regarding Logic and Mathematics, however, due to other commitments it may be a while before anything else is written on this blog.