A local philosophy professor writes,
I often find myself among what might be called postmodern philosophers. They are willing to say things like "I don't accept the law of non-contradiction." Does this seem to be sufficient enough to say that further conversation is not possible?
In general, yes. Life is short, philosophy is long, and fools are many. One shouldn't waste precious time debating with mush-heads, including many in POMO precincts. That being said, there are some discussions about LNC that I would engage in.
If a student sincerely wants to learn about LNC, then I would surely talk to him.
If a person doubts the truth of LNC, or wants to know how we know it to be true, then I would talk to him.
Also worthwhile are discussions with serious and well-informed people about the 'reach' of such logical principles as LNC. The following sort of discussion I would take to be highly profitable:
Are the 'laws of thought' 'laws of reality' as well? Since such laws are necessities of thought, the question can also be put by asking whether or not the necessities of thought are also necessities of being. It is surely not self-evident that principles that govern how we must think if we are to make sense to ourselves and to others must also apply to mind-independent reality. One cannot invoke self-evidence since such philosophers as Nagarjuna and Hegel and Nietzsche have denied (in different ways) that the laws of thought apply to the real. (See here.)
As I read Aristotle, he too was aware of a possible 'gap' between thought and reality.
The Law of Non-Contradiction, in its property version, can be put like this:
LNC. (F)(x)~(Fx & ~Fx)
which is to say: for any property F-ness, and any object x, it is not the case that x is F and x is not F. For example, nothing is both red and non-red.
This is subject to the usual three qualifications: an object cannot be F and not F (i) at the same time, (ii) in the same respect, and (iii) in the same sense. Thus a ball could be both red and non-red at different times, or red and non-red in respect of different hemispheres, or in different senses: Jack can be both red and non-red at the same time if 'red' in its first occurrence refers to a color, and in its second occurrence to a political affiliation. One can be a redskin without being a commie.
Now Aristotle was quite clear that first principles like (LNC) are non-demonstrable. They are so basic that they cannot be proven. Since a proof cannot be circular, (LNC) cannot be derived from itself or from any logically equivalent proposition. To use (LNC) to prove (LNC) would be to beg the question. It is also clear that no proof can have infinitely many inferential steps. So what justifies (LNC)? Is it perhaps unjustifiable, a dogmatic posit? Is it a groundless assumption?
One might just announce that (LNC) is (objectively) self-evident, that it is self-justifying, that it 'glows by its own epistemic light.' But then how respond to someone like Heraclitus who sincerely maintains that it is not self-evident? If a proposition is subjectively self-evident, self-evident to one, it does not follow that it is objectively self-evident, self-evident in itself.
At Metaphysics Gamma, 3, 4, Aristotle can be read as using retortion to establish (LNC). Since he cannot, on pain of begging the question, resort to a direct proof in the case of this most fundamental of all principles, "the surest principle of all," (1005b10) he must try to show that anyone who denies (LNC) falls into performative inconsistency. As I read Aristotle, the key idea is that (LNC) is " a principle one must have to understand anything whatever. . . ." (1005b15) It is a principle that governs all understanding, all definite and determinate speech. So it is at least a transcendental principle in a roughly Kantian sense of 'transcendental.'
As such, (LNC) seems to function as a semantic constraint: one cannot mean anything definite or make any definite judgment unless one abides by, and thus presupposes, the principle that no subject of discourse both has and does not have a property at the same time and in the same respect. To counter the (LNC)-denier, Aristotle simply demands that the man say something, that he express the same idea to himself and to another, "for this much is necessary if there is to be any proposition (legein, dicere) at all." (1006a20) If the (LNC)-denier says nothing, then "he is no better than a plant" (1006a15) and one can ignore him. But if he says anything definite at all, then he makes use of (LNC). For suppose he asserts 'The arrow is at rest.' He thereby commits himself to 'It is not the case that the arrow is not at rest.' If he asserts both 'The arrow is at rest' and 'The arrow is not at rest,' then, far from making two assertions, he does not even make one. He expresses no definite thought since he violates a principle observance of which is necessary for making sense.
The idea here is that he who asserts something contradictory asserts nothing at all: a necessary condition of there being a definite thought, a definite proposition, is that (LNC) be satisfied. The retortion might be spelled out as follows. The denier states
2. (LNC) is false.
But in making this definite statement, a statement that opposes what the (LNC)-affirmer states, the (LNC) denier commits himself to
3. It is not the case that (LNC) is not false.
But the commitment to (3) is tantamount to an acceptance of (LNC). So the denier's performance — his stating of (2) — 'contradicts' the content of (2).
But what exactly does the retortion show? Does it show that (LNC) is true of reality, or does it show merely that it is true of thought-contents? Is it an ontological principle or is it merely a law of thought, a principle that governs how we must think if we are to make sense to ourselves and others? Is it an ontological principle or merely a transcendental one? Is it perhaps true of only phenomenal reality but not of noumenal reality?