Priest was one of the editors of a wonderful collection of essays, some assailing and defending the Law of Non-contradiction. This traditional axiom of logic states that, in essence, "a statement and its denial cannot be true at the same time." In other words, I cannot affirm the sentence "This is both a cat and also in no way a cat."
In the end, I remain unconvinced that dialetheias are possible, or at least in a sense of being conceivable, but it's a fascinating debate to be had, and I may one day be convinced.
And it's a very profitable discussion, lest one think it's just language games and talking in circles. The idea that the Law of Non-contradiction can be questioned, amended, or removed speaks to the very core of logic, the sort of reasoning used everyday by everyone.
I.
In the opening essay, Priest aims to describe a few possible settings where contradictions can arise and be tolerated. One of these I find very unconvincing.
He sets up a situation where a person has written a book.
This seems to be the lesson of the 'paradox of the preface'. You write a (non-fictional) book on some topic - history, karate, cooking. You research it as thoroughly as possible. The evidence for the claims in your book, a1,...,an, is as convincing as empirically possible. Hence, you endorse them - rationally. None the less, as you are well aware, there is independent inductive evidence of a very strong kind that virtually all substantial factual books that have been written contain some false claims. Hence, you also believe ~(a1,...,an) - rationally. However, you do not believe (a1,...,an) Λ ~(a1,...,an), a simple contradiction, even though this is a logical consequence of your beliefs. Rational belief is not, therefore, closed under logical consequence.
I think Priest misses something key here. If a person has thoroughly researched her book, and is convinced of the claims a1,...,an, then she likely won't grant ~(a1,...,an). However, if she is wise she'll realize ◊(a1,...,an), that is, it's possible that of her claims are true. In fact, she thinks it very, very likely. But since we're talking induction and empirical evidence, she admits the outside chance that she's wrong. So, likewise, at most, she'll grant ◊(a1,...,an), that is, it is possible that not all claims in her book are correct.
This hardly counts as a contradiction; these are subcontraries. They could both be true. If I am told that there's a dog outside my door, it's possible that it's brown and it's also possible that it's not brown.
II.
One interesting point of the discussion is that it so frequently turns to the Liar Paradox.
(1) This sentence is false.
This is an elementary paradox, but one that has weathered several millenia. I for one suspect that there are some who simply don't want it to be solved, but regardless, the book mentioned above opened my eyes to a curious expansion: the Revenge Liars. These statements take the arguments made and generate new paradoxes.
Today one came to me that I an confident has been thought of before, and likely laid to rest. I would like to muse about it myself before I hunt for what wiser folks have said. This Revenge came to me when contemplating the answer I initially preferred in regards to the Liar Paradox: that such a statement is meaningless, nonsensical.
We all recognize that the following sentence (if it even counts as a sentence) is nonsensical:
(2) Swam greenish features knowingly maddened of golden phantoms.
One answer to the Liar Paradox states that that (1) is simply a well-disguised nonsense sentence. After all, it fails to assert anything. If what it is asserts is the case, then it must not be the case, and vice versa. This, I think, is in line with Wittgenstein's take of the situation, that such a sentence is essentially meaningless. It may seem grammatically well-formed, having a subject, verb, and predicate. In the end, however, it fails to assert anything that can be seen as true or false. So the following sentence came to me:
(3) This statement is meaningless.
If (3) is meaningless, we run into a problem. Because its lack of meaning is what it asserts. If it has no meaning, the claim made by the sentence is therefore true. But if the statement is true, it's asserting something. So it can't be meaningless. But what it asserts is precisely that it has no meaning.
If we initially take it as true, we run into the same problem. If true, then the statement is meaningless. But how can it be true if it lacks meaning?
So perhaps (3) is false. This doesn't seem to help things. For if "This statement is meaningless" is wrong, then it must have meaning. But what meaning?
(4) Crows have feathers.
(4) has meaning because we can conceive its contents. We can imagine what it would be like it is was false; we can imagine what it would be like if it was true.
So, how can (3) be false? Can a statement be false if we cannot even conceive of how things would settle if it were true? If it is false that "This sentence is meaningless," and if "This sentence is meaningless" corresponds to "This sentences has no meaning," then "This sentence has meaning." Perhaps this sentence is itself paradoxical.
(5) This sentence has meaning.
Assuming there's nothing enthymemetic (it is a word) at play, and this statement is referring to itself, then what do we have? It asserts it has meaning, but there's nothing meaningful being said beyond the claim that meaning is present. Here I must end my rambling, because I haven't read enough on self-reference and meaning.
The answer seems to be that (3) is logically false, that is, it is necessarily false. Examples of logically false statements include (for the non-dialetheist):
(6) A & ~A
There is no situation where this could be true. It is always false. Likewise, (3) can never be true, nor can it be meaningless. It must be false, and false alone. But that brings around to the idea that "This sentence is meaningless" is false, leading to the conclusion that "This sentence is meaningful (not meaningless)" is true. Which is essentially (5).
I assume work on meaning and reference will aid me here.
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