Among Hegel fans and post modernists I hear this misconception about identity and difference: a thing is defined by what it is not, and therefore every identity is essentially its opposite. Their conclusion: A is ~A.
Now, it's one of the simplest proofs in logic that if a theory contains a contradiction like both A and ~A are true, it's easy to prove everything and anything (see the proof at the bottom). Assuming both A and ~A are false won't help the post modernists out of this consequence since "everything is false" is a liar paradox -- true if false, false if true -- so its negation "something is true" (not everything is false) must be true, even if "something is true" were the only truth.
Let all that aside. I want to point out that semantically, the members of A and ~A are mutually exclusive. They are different individuals. This is the difference between syntax and semantics. One is an axiomatic system, the other is a world of things or observable patterns that can be culled together as properties aka categories or similarities, whether inhering in reality or merely cognized (see information faster than light and reality versus science, math and thought).
Even the set of all things can be defined by ~A, the empty set since everything includes everything except nothing. (Yet notice that this isn't even so of symbols and language which can represent nothing.) Syntax is a bit like the laws of physics which don't in themselves distinguish the direction of time, except for the intersection of entropy and probability. Syntax alone doesn't distinguish the properties of the members of the set denoted by its symbols. Only semantics does, as it looks at the things, independently of their place in the axiomatic system being applied. To say that a thing is what it is not - because it is defined by what it isn't -- is just another post modernist gloss, in this case a lack of understanding of the logical difference between logical syntax and semantics. Once again, linguistics - a science, I hasten to add, not a piece of masturbatory philosophy - linguistic science clarifies the misunderstandings of the glib.
Post modernism targets a straw man of science, as if science were a body of knowledge and not a process -- an ongoing investigation into data including exploration and questioning of its own theories. Science begins -- only a beginning -- with defining categories of things and their properties. Say, start with biological organisms: explore and learn about that until you get to viruses and you realize that the categorical definitions don't work perfectly and need tweaking or deeper insight. One day whales are fish, then we discover they're mammals. That's a meaningful, useful and productive exploration of the category and its excluded Other. To say that there's no meaning misses the point. The meaning in science is quite substantial, but it's conditional and hypothetical. This is a tough position: it implies that what you understand today and live with and by, will be false some day in the future. That's the real aporia of history, not the synchronic aporia touted by post modernists.
Synchronic meanings -- meanings relative to a specific time and social structure of other meanings -- provide a useful, productive system for action and understanding, if we're always aware that at the margins its categories are vague, and only conditionally probable, not absolutely true. Without the synchronic system, there's no movement in any direction. I suppose there could be random drift, and random drift happens even within a synchronic categorical system, and pursuing the process of investigation has its own failures -- trendy but bad ideas, misleading paradigms. Sometimes drift is necessary just to open up new ideas. Post modernism seems to me to be one of these misdirected paradigms. It's clever, and its cleverness attracts the respect-seekers (look how clever am I!), but not much more value than that. Scientific questioning, when not used as a bludgeon, seems vastly more productive. Just look at the sciences and its favorite child technology. What has post modernism done for us? Lots of academic self-congratulations, publications and positions, not much knowledge. Questioning our assumptions? Sorry, that's what science is all about.
Here's the proof that if A and ~A are both true, everything must be true:
1. Premises: A, ~A are both true.
2. Since A is true according to (1) then "A or B" is true. 3. If ~A is true, then A is false. (Law of contradictories). 4. So if "A or B" is true by (2), and A is false by (3), then B must be true.
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