On contradictions

By Walter Carnielli
(Source: The Paraconsistency Webgroup)
Dear Friends,

Please see below some comments on Dick's views expressed in
"On contradictions".
I would like to encourage you all to participate in the
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ON CONTRADICTIONS

I agree with (what I think was) the conclusion of Fred's talk that we don't have any good arguments for the law of non-contradiction. It's too basic--either we accept it or we don't. Any argument for it we've seen or can imagine uses that law either explicitly or implicitly.

OK, but is not the situation the same for many other laws too? For basic laws concerning natural numbers? After all, when you start enumerating any kind of arguments about numbers, you are already using numbers. Or the grammarians using established grammar to explain grammatical rules.

However, I now think that we do accept some contradictions as true in our daily lives.

EXAMPLE 1: (The example of the candle in the auditorium, drawn >beautifully by Alex on p. 17 of CT) In the middle of the room we accept
"It's dark and it's not dark."

EXAMPLE 2: Talking to a lawyer about the other side's position on a case, the lawyer says and we accept, "He's right and he's not right."

The standard response to Example 1 is to say, "Oh, that's just because the predicate 'is dark' is vague--too vague in this case to make a claim. Once we make it precise, we can see that there's no violation of the law of non-contradiction." But that's just to say that we are throwing it out because it doesn't obey the law of contradiction, even though in ordinary life we understand it well enough and assent to it. It's not that we can't come up with any examples where the law of non-contradiction is violated. Rather, we ensure that the law of non-contradiction is not violated by throwing out any possible counterexamples.

For Example 2, we might also reply, "What do you mean?" Even though we understand what is said, we want claims that won't violate the law of non-contradiction. The response might be, "He's right in some respects and wrong in some other respects," which relieves us of the distaste for accepting a contradiction.

I believe that there are different levels of contradictions.
If we accept that, it would be easier to discuss. First, it seems to be obvious that information can be contradictory.
In the Examples 1 and 2 above, what we have "prima facie" are just contradictory information, and the cited reaction is just attempts to restore consistency. We can blame vagueness ("Oh, that's just because the predicate 'is dark' is vague...") or blame our confusion about different approaches to the question ("He's right in some respects and wrong in some other respects").
However this does not mean that the facts are not contradictory.
If facts were contradictory, information about them would be too. But not necessarily vice-versa.
So contradictory information is not a problem to accept, and even people working on databases and computer systems not only agree with that, but also look for inference mechanisms which could help manipulating contradictions while preserving other logical functions. This is precisely the point where "paraconsisntent logics" are becoming useful (for example, I am working on database computer models using paraconsistent logic in the purely operative sense, that is, ontologically and metaphysically free). But again, that does not mean that there are not ontological and metaphysical problems connected to the law of non-contradiction. This just means that at the information level we do not need to worry about them.
But then comes a second level. Action can be contradictory.
This is a more complicated form, because may involve intention.
One may think and say something, and act contrarily.
I suspect that in this level, existence of contradictions can be connected to the "acts of total freedom" that Sartre and the existentialists talked in the 30's. Is there an action which is totally free, or is any action, including the ones which seem to be contradictory, explainable somehow? And finally a third level (and there may be more levels): is there a really contradictory object in the world (like, say, a cosmic black-hole which really defies all natural laws, including the law of non-contradiction?

"But what do you mean?" as a response to perfectly clear sentences such as these seems to me to be an expression of metaphysical unease in accepting contradictions. We cannot cite the lack of examples of true contradictions as evidence for the law of non-contradictions, when the lack is imposed by us due to our adherence to that law. Citing vagueness as a reason not to accept Example 1, requiring more precision in both examples, comes from our unwillingness--on reflection--to accept contradictions.

The law of non-contradiction in the broad sense that I set out in Propositional Logics--all reasoning is based on dividing up all propositions into two mutually exclusive classes, ones we accept and ones we reject--seems absolutely fundamental to reasoning in which we reflect on our reasoning. But it is not inescapable, as these examples and many Zen masters have tried to show.

I totally agree with the idea, specially to what concerns Zen masters. But the question of dividing up all propositions into two mutually exclusive classes, ones we accept and ones we reject, does not seem to me as "absolutely fundamental". This seems to be more a simplified hypothesis, coming from our narrow mathematical views. I just discovered, contrary to what Dick defended in our first Arf Meeting, that THERE ARE logics where division of truth-values into distinguished and non-distinguished does not work.
Systems of this sort can be given by adopting notions of consequence operation which assume less than Tarski did in his proposals for general concepts of deduction (Grzegorsz Malinowski have proposed the so-called q-consequence operator with this characteristics, in a paper I can inform in other occasion).
But now we have to face another problem: is anything we invent in pure Mathematics somehow real? What about non-Euclidean geometry? If we invented non-standard models for geometry, are there two distinct lines m and n parallel to the line p at the point Q? Well, there are, but we are forced to see lines in a different perspective. Now, for the sake of non-Euclidean geometry, arcs in a circle can be considered lines...
Summing up, I believe we should not confuse different levels of existence, and not discount the influence of new mathematical ideas in the realm of philosophy. They may make us look to the reality with different eyes.