# What is an example of a true contradiction in a paraconsistent logic?

While reading the Wikipedia article on trivialism I noticed the following:

In classical logic, trivialism is in direct violation of Aristotle's law of noncontradiction. In philosophy, trivialism is considered by some to be the complete opposite of skepticism. Paraconsistent logics may use "the law of non-triviality" to abstain from trivialism in logical practices that involve true contradictions. [my emphasis]

I was trying to come up with an example of a true contradiction in the context of a paraconsistent logic so that it would not become a form of trivialism, but I was not able to do so.

Hence the question: What is an example of a true contradiction in a paraconsistent logic?

Wikipedia contributors. (2019, September 9). Trivialism. In Wikipedia, The Free Encyclopedia. Retrieved 13:23, September 25, 2019, from https://en.wikipedia.org/w/index.php?title=Trivialism&oldid=914757562

Long comment (but I'm not sure to fully understand your question...)

Some definitions from Walter Carnielli & M.E. Coniglio, Paraconsistent Logic : Consistency, Contradiction and Negation (Springer, 2016) :

For a language with the negation symbol, we say that a set T of formulas is :

Contradictory - if and only if there is a proposition α in the language of T such that T proves α and T proves ∼α.

Trivial - if and only if for any proposition α in the language of T , T proves α;

Explosive - if and only if T trivializes when exposed to any pair of contradictory formulas—i.e.:

T ∪ {α,∼α} ⊢ β, for all α and β.

We have also in place two different but classically equivalent notions of consistency :

i. S is consistent if and only if there is a formula β such that S ⊢ β;

ii. S is consistent if and only if there is no formula α such that S ⊢ α and S ⊢ ∼α.

What (i) says is that S is non-trivial; and (ii) says that S is non-contradictory. In classical logic both are provably equivalent.

So, a theory whose underlying logic is classical is contradictory if and only if it is trivial. But this is the case precisely because such a theory is explosive, since the principle of explosion holds in classical logic.

The obvious move in order to deal with contradictions is, thus, to reject the unrestricted validity of the principle of explosion. This is a necessary condition if we want a contradictory but not-trivial theory.

In a nutshell, paraconsistent logic does not change the meaning of "contradiction" : a pair of formulas α and ∼α.

What changes is the way to "manage" it.

• It seems this way me also. The question could be just, 'Is there an example of a true contradiction', and it's an interesting one. – user20253 Sep 25 '19 at 13:53
• This really is a long comment and not an answer. It does not contain an example. – user9166 Sep 26 '19 at 0:12

Kant's Antinomies are a good starting point:

• Time began. But before any given time, there was another time.

• Any space or time can be divided in two. But Zeno's paradox is not true -- we witness motion.

• Physical beings make choices. But physical things obey hard-and-fast rules. And beings are things.

• Anything that exists could be otherwise. But something is absolutely necessary or there would be no reason for anything to be any way in particular.

Math is another good source:

• Any definition determines a collection. But the collection of all things that don't contain themselves cannot be a collection.

• The Axiom of Choice makes sense. But it allows every space to be broken down into infinitely many non-overlapping spaces of the same measure as the original.

• The rational numbers are dense on the real line. But they have measure zero.

Basically, we like to imagine logic is complete, but when we actually apply it to anything thoroughly enough, we find paradoxes. Most useful domains of application are to some degree paraconsistent.

• @user9166 I am baffled as to the relevance of your examples to the question. Not one of them is expressed as a formal contradiction. Did anyone actually proved that these were actual contradictions, or is it a case of just saying so and getting away with it? – Speakpigeon Apr 20 at 10:05

If we maintain strict concept definitions we will see that people who opt for true contradictions are violating the law of non contradiction in concept.

First the proper definition of a deductive reasoning term contradiction. A contradiction is an inconsistent relationship between two or more propositions where the truth value of one of the propositions MUST BE TRUE while the one of the others MUST be false. That is strictly that Both propositions cannot be true and both propositions cannot be false.

In contrast Mathematical logic uses the same term contradiction differently. To math people a contradiction is any premise A and the occurrence of premise not A. Furthermore contradiction is also defined in Mathematical logic as any proposition that is always false. So all triangles have four sides will be a contradiction eventhough there is no comparison to another proposition present. This is inconsistent of math people who say logic is only about form. To say a triangle has four sides is a contradiction is a CONTENT matter and not of logical FORM.

Mathematical logic has eliminated other inconsistent relationships such as contrary, sub contrary, sub alternation, etc. In this way what a math person call contradictory may not fit the original definition. Math some how changed the meaning and people need to understand the same word is being used DIFFERENTLY. I suppose contrary propositions would be called contradictory by math people in error: "all black holes are formed from dying and collapsed stars" compared to the proposition "No black holes are formed from dying and collapsed stars." These are not contradictory propositions to each other. It turns out both are false.

The law of non contradiction is often read super literally by people outside of philosophy. What the laws implies or expresses is that no two propositions can have the same context, domain and hold more than one truth value simultaneously. In order to pull off a true contradiction allegedly, one needs to change the context the word is used, change the domain of discourse or change other parameters that were not mentioned in the original proposition. This mainly means equivocation of some kind or changing of the topic in order to make the same proposition have multiple truth values.

Stick to the proper definitions of the concept and we see the three fundamental logic laws hold.