Trivialism

 Trivialism in symbolic logic; Read as "given any proposition, it is a true proposition."
Stylistic origins Analytic philosophy, Philosophy of logic, Non-classical logic, Paraconsistent logic Theory of truth 1980s - 2000s Academic journals, Books Graham Priest Dialetheism, Non-trivialism (Anti-trivialism), Philosophical skepticism, Nihilism, Modal realism

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.[1][2]

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

The vast majority of philosophers deny trivialism and can be considered non-trivialists or anti-trivialists. Public advocates of the theory are nearly non-existent due to it being considered an absurdity (reductio ad absurdum). It is generally agreed that trivialism as a committed philosophy cannot be genuinely applied in life.

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

Religions and theories, such as Jainism, may be claimed to be advocates of trivialism.

Overview

Etymology

Trivialism, as a term, is derived from the Latin word trivialis, meaning something that can be found everywhere. From this, "trivial" 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.[3]

Theory

In symbolic logic, trivialism may be expressed as the following:[4]

$\forall pTp$

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:

$p \leftrightarrow Tp$

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 provably true may be considered inconsistent with the assumed theory.[4]

Taxonomy of trivialisms

Luis Estrada-González in "Models of Possiblism 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:

(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.[3]

Arguments against trivialism

The consensus among the majority of philosophers is descriptively a denial of trivialism, termed as non-trivialism or anti-trivialism.[3] This is due to it being unable to produce a sound argument through the principle of explosion and it being considered an absurdity (reductio ad absurdum).[2][4]

Aristotle

Aristotle's law of noncontradiction and other arguments are 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."[3]

Priest

It is implicitly claimed by Graham Priest, a professor of philosophy, that a position for trivialism is unsubstantial: "...a substantial case can be made for [dialetheism]; belief in [trivialism], though, would appear to be grounds for certifiable insanity."[5]

He has coined his rejection of trivialism "the law of non-triviality" as a replacement for the law of non-contradiction in paraconsistent logic and dialetheism.[6]

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 Plentitude of Truth" from the following:

"...

1. Possibilism is true [premise]
2. If possibilism is true, then there is a world (either possible or impossible or both), w, in which trivialism is true [premise]
3. w is a possible world [premise]
4. It is true in w that w is identical to the actual world, A [2]
5. If it is true that there is a world, w, and w is a possible world, and it is true in w that w is identical to A, then trivialism is true [premise]
6. Trivialism is true [1 - 5]"[2][4]

Above, possibilism (modal realism; related to possible worlds) is the barely accepted theory that every proposition is possible. With this assumed to be true, trivialism can be assumed to be true as well according to Kabay.

The liar's paradox, Curry's paradox alongside the principle of explosion all can be asserted as valid and not required to be resolved and used to defend trivialism.[2][4]

Philosophical implications

Comparison to skepticism

In Paul Kabay's comparison of trivialism to schools of philosophical skepticism (in "On the Plentitude of Truth") -- such as Epicureanism, Stoicism and Pyrrhonism—who seek to attain a form of ataraxia, or a lack of anxiety; 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—that is claimed to be accepted as true and made peaceful.

Kabay compares the figurative pyyrhonist skeptic to the figurative trivialist and claims the skeptic will attempt to attain a lack of anxiety through a lack of belief, while the trivialist will attempt to attain a lack of anxiety through an abundance of belief.

In this case — and according to independent claims by Graham Priest — trivialism is considered the complete opposite of skepticism.[2][4][7]

Impossibility of action

It is claimed by both Priest and Kabay that it is impossible for a trivialist to truly choose, thus being an impossible state. Priest argues this by the following in Doubt Truth to Be a Liar: "One cannot intend to act in such a way as to bring about some state of affairs, s, if one believes s already to hold. Conversely, if one acts with the purpose of bringing s about, one cannot believe that s already obtains."[6][8]

Jainism

It is held that the Kevala Jnana (an ideal of the soul) of Jainism is a true state of trivialism — as often presented — through the ideal that it is all-knowing and believes in everything. It is claimed that Jainism is implicitly committed to trivialism.[8]

Azzouni

Jody Azzouni is a purported advocate of trivialism in his article The Strengthened Liar by claiming that natural language is trivial and inconsistent through the existence of the liar paradox ("This sentence is false"), and claiming that natural language has developed without central direction. It is heavily implied by Azzouni that every sentence in any natural language is true.[9][10][11]

Anaxagoras

The Greek philosopher Anaxagoras is suggested as a possible trivialist by Graham Priest in his 2005 book Doubt Truth to Be a Liar. Priest writes, "He held that, at least at one time, everything was all mixed up so that no predicate applied to any one thing more than a contrary predicate."[6]

Anti-trivialism

Absolute anti-trivialism (or maximal logical nihilism) in symbolic logic; Read as "given any proposition, it is neither a true or false proposition."

Luis Estrada-González in "Models of Possiblism and Trivialism" lists eight types of anti-trivialism (or non-trivialism) through the use of possible worlds:

(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.[3]

References

1. ^ Priest, Graham (2007). "Paraconsistency and Dialetheism". In Gabbay, Dov M.; Woods, John. The Many Valued and Nonmonotonic Turn in Logic. Elsevier. p. 131. ISBN 978-0-444-51623-7.
2. Paul Kabay (2010). On the Plenitude of Truth. A Defense of Trivialism. Lambert Academic Publishing. ISBN 978-3-8383-5102-5.
3. Estrada-González, Luis. "Models of Possibilism and Trivialism". Logic and Logical Philosophy 21: 175–205.
4. Kabay, Paul. "A defense of trivialism". PhD thesis, School of Philosophy, Anthropology, and Social Inquiry. The University of Melbourne, Research Collections (UMER). p. 29. Retrieved 20 May 2014.
5. ^ Priest, Graham (1999). "Perceiving contradictions". Australasian Journal of Philosophy: Volume 77, Issue 4, p. 443.
6. ^ a b c Priest, Graham (2008). Doubt truth to be a liar (1st pbk. ed. ed.). Oxford: Oxford University Press. pp. 69–71. ISBN 0199238510.
7. ^ Priest, G. (2000). "Could everything be true?". Australasian Journal of Philosophy 78 (2): 189–195. doi:10.1080/00048400012349471. edit
8. ^ a b Kabay, Paul. "Interpreting the Divyadhvani: On Why the Digambara Sect Is Right about the Nature of the Kevalin". The Australasian Philosophy of Religion Association Conference. Retrieved 23 May 2014.
9. ^ Kabay, Paul. "A defense of trivialism". PhD thesis, School of Philosophy, Anthropology, and Social Inquiry. The University of Melbourne, Research Collections (UMER). p. 42. Retrieved 21 May 2014. "...According to Azzouni, natural language is trivial, that is to say, every sentence in natural language is true...And, of course, trivialism follows straightforwardly from the triviality of natural language: after all, 'trivialism is true' is a sentence in natural language..."
10. ^ Bueno, O. V. (2007). "Troubles with Trivialism". Inquiry 50 (6): 655–667. doi:10.1080/00201740701698670. edit
11. ^ Azzouni, Jody (2003). "The Strengthened Liar, the Expressive Strength of Natural Languages, and Regimentation". Philosophical Forum 34 (3-4): 342. Retrieved 21 May 2014.