I assume some knowledge of propositional logic, along with logical notation, in this post.
In a previous post, I explained why bad things would follow from a genuine logical contradiction in classical logic.
That bad thing, namely, is that a contradiction allows you to prove any proposition to be true. A contradiction, if realised, would make it true that the moon is made of cheese, that 1 = 42 and that the entire universe is a tiny golden pyramid with fluffy green eyes.
It's called the Principle of Explosion, and it isn't a very intuitive outcome of our usual models of logic.
Ex contradictione quodlibet. From a contradiction, anything follows.
Why? Roughly, if we assume "P" is true, then "P or Q" is true. But since we know P isn't true, taken with "P or Q", we deduce Q. And Q can be any arbitrary, trivial proposition. More on this here.
Paraconsistent logics avoid the principle of explosion
Systems of logic are paraconsistent if they are not explosive. Explosive logics are those systems where a contradiction makes any arbitrary proposition true.
So whilst inconsistency in classical logic is trivial to reason about, in paraconsistent logic contradictions are potentially useful, and can be reasoned about.
The fact that most propositions are not true – the moon is not made of cheese, 1 is not equal to 42 and the entire universe is not a tiny golden pyramid with fluffy green eyes – must make us suspicious that either there can never be a genuine contraditiction (but as we will see, there are reasons why we might not want to allow this) or that there is something wrong with classical logic.
So, let's look at some reasons why we might want to adopt paraconsistent logic.
Motivation 1: The Liar Paradox
Liar: "This sentence is not true."
The sentence above is a problem for classical logic, for if the sentence "Liar is false" is true, Liar must be false; yet if the sentence "Liar is true" is false, Liar must be true. Classical logic takes us to a murky place where you can prove that if Liar is true, Liar must be false. We get a contradiction, and therefore, an explosion.
This is a good example of a true contradiction which might motivate paraconsistent logic.
Motivation 2: Non-trivial scientific inconsistency
There are a number of theories which are inconsistent but non-trivial. An easy one to get our heads around is wave-particle duality, or the idea that particles or quantum objects might correctly be described both as particles or as waves.
So this, amongst many other examples, gives us a motivation to want a logical system that can cope with the contradiction without exploding into triviality.
Motivation 2: Artificial intelligence
Computers store and operate upon information, and infer from that information. It's common for that information to be inconsistent in places. Now, while systems exist to try to remove inconsistent information, they're of limited use and are limited in ability.
Computer scientists can find a paraconsistent logic helpful to deal with those contradictions when they arise.
Motivation 4: Vagueness
Eating one sweet is not greedy.
If eating one sweet is not greedy, eating two sweets is not greedy.
If eating two sweets is not greedy, eating three sweets is not greedy.
...If eating 999 sweet is not greedy, eating 1000 sweets is not greedy.
At some point, this logic yields something most would presume to be false. And the logic runs in the other direction, too. We can start with the assumption that eating a million sweets is greedy, and end up saying eating 0 sweets is greedy. We end up admitting that eating 1 sweet is both greedy and not greedy. It's known as the Sorites paradox.
But we manage to use vague predicates successfully in ordinary discourse all the time – so that should give us a hunch that we can solve this one without having to admit that vagueness leads to explosion.
An example – many-valued logics
Because paraconsistent logic is defined by what it is not – namely, it's not explosive – there is huge divergence in the range of approaches to paraconsistent logic which have little in common other than that they're not explosive.
One example of a paraconsistent approach is many-valued logic. It can circumvent the explosion by allowing that sentences don't need to be true or false. We could, for example, allow a third value. This is perhaps the simplest way of generating a paraconsistent logic.
One of the oldest systems of logic that works like this is the logic of paradox. The valid sentences of the logic of paradox coincide with the valid setences of classical logic – so tautologies, like ¬(A ∧ ¬A) are valid – but valid arguments do differ radically. In particular, ex contradictione quodlibet is invalid. It's a true theorem, but can't be used to argue anything. So (A ∧ ¬A) → B. It gets worse. Not even modus ponens is not valid..! Some variations on this circumvent these problems (for example, they allow modus ponens), and you can find out more about this here.
A couple of other solutions are Strong Kleene logic and Łukasiewicz logic.
Strong Kleene has a third value which is, roughly, 'unknown' – or neither true nor false. It was motivated by the exploit of science, to provide a value for matters still under investivation. But it suffers from the feature that no sentence is valid in Strong Kleene logic. Not even the law of identity, P → P. In classical logic, you can always infer 'P' from 'P' because a proof beginning with 'P' can conclude 'P' immediately, but this isn't the case in Strong Kleene logic.
Łukasiewicz logic was introduced to make P → P valid and to deal with the open future, where the third value means roughly 'ungrounded', or not corresponding to reality (P → P has the value 'ungrounded' in this system). It tries to tackle vagueness by letting vague cases be 'ungrounded' and is prima facie successful in doing so – although there are possible rebuttals. The problem with this one is that future-directed tautologies end up with the value 'ungrounded'. For example, we might think that the sentence "Either I will live to be 100, or I won't" should be straightforwardly true (this is a future-excluded middle sentence), but Łukasiewicz logic doesn't achieve that.
There are other solutions
Many, many other systems of paraconsistent logic exist, and the conversation is still very much live. For example, there are systems which assign truth values relative to the person who is making the assertion, systems which allow us to measure the level of consistency of a set of premises and systems which isolate contradictions from the rest of the system. You can find out more about those logics here on the SEP.