Law of Thought - Rationale

Rationale

That everything be 'the same with itself and different from another' (law of identity) is the self-evident first principle upon which all symbolic communication systems (languages) are founded, for it governs the use of those symbols (names, words, pictograms, etc.) which denote the various individual concepts within a language, so as to eliminate ambiguity in the conveyance of those concepts between the users of the language. Such a principle (law) is necessary because symbolic designators have no inherent meaning of their own, but derive their meaning from the language users themselves, who associate each symbol with an individual concept in a manner that has been conventionally prescribed within their linguistic group. The degree to which this law must be obeyed depends upon the kind of language that one is utilizing. In a natural language there is considerable tolerance for violations since there are other means whereby one can determine which of a number of different concepts one is intended to call to mind by the use of a given symbol, such as the context in which the symbol is used. However, in the language of mathematics or formal logic, there is no such tolerance. If, for example, the symbol “+” were allowed to denote both the function of addition and some other mathematical function, then we would be unable to evaluate the truth value of a proposition such as, “2+2=4”, for the truth of such a proposition would be contingent upon which of the possible functions the symbol “+” was intended to denote. The same is true of symbols such as '2' and '4', for if these symbols did not denote conventionally prescribed quantities, then one could not attribute proper meaning to them, and the proposition would be rendered unintelligible.

The law of non-contradiction and the law of excluded middle are not separate laws per se, but correlates of the law of identity. That is to say, they are two interdependent and complementary principles that inhere naturally (implicitly) within the law of identity, as its essential nature. To understand how these supplementary laws relate to the law of identity, one must recognize the dichotomizing nature of the law of identity. By this I mean that whenever we 'identify' a thing as belonging to a certain class or instance of a class, we intellectually set that thing apart from all the other things in existence which are 'not' of that same class or instance of a class. In other words, the proposition, “A is A and A is not ~A” (law of identity) intellectually partitions a universe of discourse (the domain of all things)into exactly two subsets, A and ~A, and thus gives rise to a dichotomy. As with all dichotomies, A and ~A must then be 'mutually exclusive' and 'jointly exhaustive' with respect to that universe of discourse. In other words, 'no one thing can simultaneously be a member of both A and ~A' (law of non-contradiction), whilst 'every single thing must be a member of either A or ~A' (law of excluded middle).

What's more, since we cannot think without that we make use of some form of language (symbolic communication), for thinking entails the manipulation and amalgamation of simpler concepts in order to form more complex ones, and therefore, we must have a means of distinguishing these different concepts. It follows then that the first principle of language (law of identity) is also rightfully called the first principle of thought, and by extension, the first principle reason (rational thought).