Trivialism
Trivialism is the logical theory that all statements are true and, consequently, that all contradictions of the form "p and not p" are true. In accordance with this, a trivialist is a person who believes everything is true.
In classical logic, trivialism follows from a violation of the law of noncontradiction due to the principle of explosion. 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.
Theoretical arguments and anecdotes have been offered for trivialism to contrast it with theories such as modal realism, dialetheism and paraconsistent logics.
Overview
Etymology
Trivialism, as a term, is derived from the Latin word trivialis, meaning commonplace, in turn derived from the trivium, the three introductory educational topics expected to be learned by all freemen. In logic, from this meaning, a "trivial" theory is something regarded as defective in the face of a complex phenomenon that needs to be completely represented. Thus, literally, the trivialist theory is something expressed in the simplest possible way.Theory
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.
A claim of trivialism may always apply its fundamental truth, otherwise known as a truth predicate:
The above would be read as a "proposition if and only if a true proposition", meaning that all propositions are believed to be inherently proven as true. Without consistent use of this concept, a claim of advocating trivialism may not be seen as genuine and complete trivialism; as to claim a proposition is true but deny it as probably true may be considered inconsistent with the assumed theory.
Taxonomy of trivialisms
Luis Estrada-González in "Models of Possibilism and Trivialism" lists four types of trivialism through the concept of possible worlds, with a "world" being a possibility and "the actual world" being reality. It is theorized a trivialist simply designates a value to all propositions in equivalence to seeing all propositions and their negations as true. This taxonomy is used to demonstrate the different strengths and plausibility of trivialism in this context:- Minimal trivialism: At some world, all propositions have a designated value.
- Pluralist trivialism: In some worlds, all propositions have a designated value.
- Actualist trivialism: In the actual world, all propositions have a designated value.
- Absolute trivialism: In all worlds, all propositions have a designated value.
Arguments against trivialism
Aristotle
's defense of the law of noncontradiction is sometimes considered to be against trivialism. Luis Estrada-González in "Models of Possiblism and Trivialism" has interpreted Aristotle's Metaphysics Book IV as such: "A family of arguments between 1008a26 and 1007b12 of the form 'If trivialism is right, then X is the case, but if X is the case then all things are one. But it is impossible that all things are one, so trivialism is impossible.' ... these Aristotelian considerations are the seeds of virtually all subsequent suspicions against trivialism: Trivialism has to be rejected because it identifies what should not be identified, and is undesirable from a logical point of view because it identifies what is not identical, namely, truth and falsehood."This view however is contentious. Aristotle said the law of non-contradiction was indemonstrable; that trying to prove the law of non-contradiction shows "want of education." Others note for Aristotle the law of noncontradiction is a principle of metaphysics rather than one of logic. Aristotle notes his logic would still work even if the law of non contradiction were false. This seems to mean Aristotle's logic denies explosion and so is paraconsistent rather than classical.
Avicenna's commentary on the Metaphysics similarly says the law of non-contradiction is indemonstrable; that it "and their like are among the things that do not require our elaboration." However, like Aristotle, he then seems to go on to give a demonstration against those who deny the law, albeit a facetious one: "he must be subjected to the conflagration of fire, since 'fire' and 'not fire' are one. Pain must be inflicted on him through beating, since 'pain' and 'no pain' are one. And he must be denied food and drink, since eating and drinking and the abstention from both are one ."
Priest
considers trivialism untenable: "a substantial case can be made for dialetheism; belief in , though, would appear to be grounds for certifiable insanity".He formulated the "law of non-triviality" as a replacement for the law of non-contradiction in paraconsistent logic and dialetheism.
Arguments for trivialism
There are theoretical arguments for trivialism argued from the position of a devil's advocate:Argument from possibilism
Paul Kabay has argued for trivialism in "On the Plenitude of Truth" from the following:Above, possibilism is the oft-debated theory that every proposition is possible. With this assumed to be true, trivialism can be assumed to be true as well according to Kabay.
Paradoxes
The liar's paradox, Curry's paradox, and the principle of explosion all can be asserted as valid and not required to be resolved and used to defend trivialism.Philosophical implications
Comparison to skepticism
In Paul Kabay's comparison of trivialism to schools of philosophical skepticism —such as Pyrrhonism—who seek to attain a form of ataraxia, or state of imperturbability; it is purported the figurative trivialist inherently attains this state. This is claimed to be justified by the figurative trivialist seeing every state of affairs being true, even in a state of anxiety. Once universally accepted as true, the trivialist is free from any further anxieties regarding whether any state of affairs is true.Kabay compares the Pyrrhonian skeptic to the figurative trivialist and claims that as the skeptic reportedly attains a state of imperturbability through a suspension of belief, the trivialist may attain such a state through an abundance of belief.
In this case—and according to independent claims by Graham Priest—trivialism is considered the complete opposite of skepticism. However, insofar as the trivialist affirms all states of affairs as universally true, the Pyrrhonist neither affirms nor denies the truth of such affairs.