What is the Law of Excluded Middle?

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The law of excluded middle, like the other two above laws, is also a fundamental law in the sense that every good argument must conform to this law. It asserts that everything is either or not A, where A stands for any quality. A is not an exhaust the entire discourse.

In other words, a thing can be either A or not-A but it cannot be neither. Hence there cannot be an intermediary between contradictory properties. Let a stand for “to be good” and B stands for “not to be good”. Then ‘either A or B’ will belong to everything.

In the propositional interpretation the law of excluded middle asserts that every proposition is either true or false. Symbolical it can be formalized as (p V ~ p) is always true for any proposition P. (Read ‘p V ~ p’ as ‘p < not p’). For any proposition p, p admits either the truth value T (for truth) or F (for falsehood P cannot be neither i.e. there is nothing in between T and F. The third or intermediary value between T and F is excluded.

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These three laws in their propositional interpretation are all tautologies and hence are logically true. Since they are all tautologies, they are equivalent to each other. For that matter they all are equivalent to any of the tautologies. But logicians consider these laws as having a very special status. They are basic or fundamental principles to which any good or correct argument conform.

Further, these laws cannot be proved. Because, to prove them amounts to constructing valid arguments in which each such law must occur as conclusion. Since any valid argument, in general, must conform to these laws, the proofs of such law (if any) as a form of an argument must also conform to these three laws. This means the proof of these laws would involve the fallacy of petition principia (i.e. the fallacy of assuming what we wish to prove).

Thus, we say that these laws are presuppositions of any good argument. Further, the fundamental re of these laws can be seen in relation to the construction of truth tables. These laws provide the necessary instructions for assigning truth values to the propositions in classical logic.

Each proposition is assigned the value T or F (in accord with the law of excluded middle) but not both accord with the law of contradiction) and distinct occurrences of the same variable always receive the same truth value through out the expression (in accord with the law of identify).

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So these laws are fundamental, self evident and unavoidable for providing consistent arguments in field of human knowledge.

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