## Trivialism

• Trivialism (from Latin trivialis, meaning "found everywhere") is the logical theory that all statements (also known as propositions) are true and that all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true. In accordance to this, a trivialist is a person who believes everything is true.

In classical logic, trivialism is in direct violation of Aristotle's law of noncontradiction. In philosophy, trivialism may be 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.

Theoretical arguments and anecdotes have been offered for trivialism to contrast it with theories such as modal realism (possibilism), dialetheism and paraconsistent logics.

Trivialism, as a term, is derived from the Latin word trivialis, meaning something that can be found everywhere. From this, "" was used to suggest something was introductory or simple. In logic, from this meaning, a "trivial" theory is something regarded as defective in the face of complex phenomenon that needs to be completely represented. Thus, literally, the trivialist theory is something expressed in the simplest possible way.

In symbolic logic, trivialism may be expressed as the following:

The above would be read as "given any proposition, it is a true proposition" through universal quantification (∀).

${\displaystyle \forall pTp}$
${\displaystyle p\leftrightarrow Tp}$
(T0) Minimal trivialism: At some world, all propositions are true and have a designated value.
(T1) Pluralist trivialism: In some worlds, all propositions are true and have a designated value.
(T2) Actualist trivialism: In the actual world, all propositions are true and have a designated value.
(T3) Absolute trivialism: In all worlds, all propositions are true and have a designated value.
(AT0) Actualist minimal anti-trivialism: In the actual world, some propositions do not have a value of true or false.
(AT1) Actualist absolute anti-trivialism: In the actual world, all propositions do not have a value of true or false.
(AT2) Minimal anti-trivialism: In some worlds, some propositions do not have a value of true or false.
(AT3) Pointed anti-trivialism (or minimal logical nihilism): In some worlds, every proposition does not have a value of true or false.
(AT4) Distributed anti-trivialism: In every world, some propositions do not have a value of true or false.
(AT5) Strong anti-trivialism: Some propositions do not have a value of true or false in every world.
(AT6) Super anti-trivialism (or moderate logical nihilism): All propositions do not have a value of true or false at some world.
(AT7) Absolute anti-trivialism (or maximal logical nihilism): All propositions do not have a value of true or false in every world.
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