# Principle of explosion

From RationalWiki

Part of the series on Logic

Key articles

General logic

Bad logic
v - t - e

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.

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.

## Tacit applications

The Bible is known to contain numerous inconsistencies. Although for obvious reasons the principle of explosion is not used to draw conclusions from them, the many inconsistencies allow a Bible quote to be furnished to support just about any position imaginable, across the spectra of politics, economics, and emotions.

For example, quotes supporting a speech on wrath and judgment can be quote-mined from the Old Testament, quotes supporting a speech on love and mercy from the New.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.94826340675354, "perplexity_flag": "middle"}