Retortion and Non-Contradiction in Aristotle, Metaphysics, Gamma 3, 4

Retortion is the philosophical procedure whereby one seeks to establish a thesis by uncovering a performative inconsistency in anyone who attempts to deny it. It is something like that benign form of ad hominem in which person A points out to person B that some proposition p that B maintains is inconsistent with some other proposition q that B maintains. "How can you maintain that p when your acceptance of p is logically ruled out by your acceptance of q? You are contradicting yourself!" This objection is to the man, or rather, to the man's doxastic system; it has no tendency to show that p is false. It shows merely that not all of B's beliefs can be true. But if the homo in question is Everyman, or every mind, then the objection gains in interest. Suppose there is a proposition that it is impossible for anyone (any rational agent) to deny; the question arises whether the undeniability or ineluctability of this proposition is a reason to consider it to be true. Does undeniability establish objective truth? Consider

Clearly, (1) cannot be asserted if it is true. For if anyone were to assert (1), his act of asserting it would falsify it. The performance of anyone who asserts (1) 'contradicts' the content of (1). We therefore speak of performative inconsistency. There is a 'contradiction' between the speech act of asserting and the content asserted. (I use sneer quotes here because performative inconsistency is not the same as logical consistency: the former is a relation between a speech act (which is not a proposition) and a proposition whereas the latter is a relation between a proposition and a proposition.) But does this performative inconsistency show that the negation of (1), namely,

~1. There are assertions

is true? (~1) is of course true. But can it be objectively established as true by retortion, i.e., by the 'retort' made to anyone who asserts (1) that the asserting of (1) is inconsistent with (1)'s truth? Does the undeniability of (~1) establish it as objectively true?

Not as far as I can see. If by 'objectively true' we mean true whether or not any subjects exist, then it seems that (~1) is true only if there are subjects who make assertions. In this case, then, undeniability does not entail objective truth. It is also quite clear that undeniability does not entail necessary truth. (~1) is contingently true: true in some, but not all, possible worlds.

Now let's consider a juicier example, the Law of Non-Contradiction which, 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.

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 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?

In Metaphysics Gamma, 3, 4, Aristotle can be read as using retortion, or proof by refutation, 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.

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?

I am not clear about this, or about the value of retortion as a philosophical procedure. Which is why I blog on.