Principle of explosion

The principle of explosion is a logical rule of inference. According to the rule, from a set of premises in which a sentence $$A$$ and its negation $$\neg A$$ are both true (i.e., a contradiction is true), any sentence $$B$$ may be inferred. It is also known by its Latin name ex contradictione quodlibet, meaning from a contradiction anything follows, or ECQ for short. Since a contradiction is always false, another Latin term is ex falso quodlibet. In layman's terms, if you start with two contradictory premises, you can actually deduce literally anything.

Classical logic accepts the principle of explosion; but in paraconsistent logic it is rejected. It is also rejected in relevance logic, since relevance logic is based on the competing principle that the premises must be relevant to the conclusion. (All relevance logics are paraconsistent, but not all paraconsistent logics are relevant.)

Rule
Formally, the rule is stated as follows. For arbitrary sentences $$A$$ and $$B$$:

$$\left\{A,\neg A\right\}\vDash B$$

Informally, the rule is applied thus: Suppose that one has $$A$$ and $$\neg A$$ for premises and wishes to prove $$B$$. One then may employ reductio ad absurdum, assuming to the contrary $$\neg B$$ and then bringing down the premises $$A$$ and $$\neg A$$. This is, of course, a contradiction, meaning that $$\neg B$$ may be concluded to be false, i.e., $$B$$ is true.

Following the logic
Here is a "Plain English" attempt to apply this rule and follow the logic up to a conclusion.

Assume two contradictory premises: A) 'All ice cream is frozen.'; B) 'Not all ice cream is frozen.'

Now, just to show that it's possible, say one wants to use those two premises to prove that: C) 'Words don't exist'.

To do so, construct a disjunction out of A and C:

'All ice cream is frozen or words don't exist.'

This statement appears to be perfectly acceptable here because it holds true under any of these three circumstances:
 * 1) All ice cream is frozen.
 * 2) Words don't exist.
 * 3) All ice cream is frozen and words don't exist.
 * (Of which at least the first one is true because it was assumed as a premise.)

Now use that disjunction for a disjunctive syllogism:

This also appears to be perfectly acceptable here because if it is said that at least one of A or C are true, then when it turns out A is not true (which is B, which has been accepted as a premise), at least it can be held that C is true.

So now it has been proven that words don't exist…

So what is this sentence, then?!

Solutions
Either reject that reasoning behind the principle of explosion (say, by rejecting that disjunction or that disjunctive syllogism) — or — just simply say no to assuming two contradictory premises (otherwise one would have to claim that one accepts yet more contradictions, i.e. one accepts both A) 'Words don't exist.' and B) 'Words do exist.' as premises … which somehow allows for sentences to exist as well as not exist.)

Alternatively, understand that while the syllogism is logically valid, because one of the premises must be false the syllogism is not logically sound and therefore the conclusion cannot be drawn.

Tacit applications
The Bible contains numerous inconsistencies. Although — for obvious reasons — theologians (professional and amateur) do not use the principle of explosion to draw conclusions from such inconsistencies, the many cases allow them to find a Bible quote to support just about any position imaginable, across the spectra of politics, economics, and emotions.

For example, one can quote-mine passages supporting wrath and judgment from the Old Testament, and quotes supporting love and mercy from the New.