People often think that vagueness is bad, a kind of darkness that can never be fully dispelled, while distinction is hailed as the clarifying answer to vagueness. Here is how the reverse is also true: Vagueness is the light and distinction a darkness.
The distinction I pick is not random, but an important part of all other kinds of logical distinction—the distinction between the “if, then”: “→” and the “conclusion” symbol: ⊢. ⊢ is ambiguous, however, and can mean other things such as assertion that a proposition is true and not just being named, or to assert in a metalanguage that the following is a theorem in the object language. Used in our sense here, the good property of the “→” is “true” and the good property of the “⊢” is “sound”. The distinction goes back to Aristotle. The main point is that if we do away with this distinction, call these two symbols the same, an interesting insight can be made—that a sound argument:
Can be represented without the ⊢ as follows: [A AND (A→B)] is logically equivalent to [A AND B], so that the conclusion [A AND B]→B is merely a deduction of A from [A AND B]. Allowing a vagueness between → and ⊢ reveals what logical deduction is—it is a cut from a larger whole, e.g. logical deduction is the act of drawing a distinction from the larger [A AND B]. With the introduction of the distinction between → and ⊢ this is concealed:
cannot be collapsed into [A AND B]→B. As promised, vagueness reveals and distinction conceals, but not just any concealment, here we have a concealment which allows distinction to reveal, since this distinction is at the root of any further logical distinction.
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