# Trivialism

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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 with this, a trivialist is a person who believes everything is true. [1] [2]

In logic, the term statement is variously understood to mean either:

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "One cannot say of something that it is and that it is not in the same respect and at the same time."

## Contents

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

Classical logic is the intensively studied and most widely used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.

Aristotle was a Greek philosopher during the Classical period in Ancient Greece, the founder of the Lyceum and the Peripatetic school of philosophy and Aristotelian tradition. Along with his teacher Plato, he has been called the "Father of Western Philosophy". His writings cover many subjects – including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him, and it was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry. As a result, his philosophy has exerted a unique influence on almost every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion.

In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "A is B" and "A is not B" are mutually exclusive. Formally this is expressed as the tautology ~(p & ~p).

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

Modal realism is the view propounded by David Kellogg Lewis that all possible worlds are real in the same way as is the actual world: they are "of a kind with this world of ours." It is based on the following tenets: possible worlds exist; possible worlds are not different in kind from the actual world; possible worlds are irreducible entities; the term actual in actual world is indexical, i.e. any subject can declare their world to be the actual one, much as they label the place they are "here" and the time they are "now".

Dialetheism is the view that there are statements which are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetheia, or nondualisms.

## 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 a complex phenomenon that needs to be completely represented. Thus, literally, the trivialist theory is something expressed in the simplest possible way. [3]

Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets and ultimately from the Phoenician alphabet.

Logic is the systematic study of the form of valid inference, and the most general laws of truth. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. In ordinary discourse, inferences may be signified by words such as therefore, thus, hence, ergo, and so on.

A phenomenon is any thing which manifests itself. Phenomena are often, but not always, understood as "things that appear" or "experiences" for a sentient being, or in principle may be so.

### Theory

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

${\displaystyle \forall pTp}$

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

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

A claim of trivialism may always apply its fundamental truth, otherwise known as a truth predicate:

${\displaystyle p\leftrightarrow Tp}$

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 probably 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 have a designated value.
• (T1) Pluralist trivialism: In some worlds, all propositions have a designated value.
• (T2) Actualist trivialism: In the actual world, all propositions have a designated value.
• (T3) Absolute trivialism: In all worlds, all propositions 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

Graham Priest considers trivialism untenable: "a substantial case can be made for [dialetheism]; belief in [trivialism], though, would appear to be grounds for certifiable insanity". [5]

He formulated 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 Plenitude 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, and 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 Plenitude of Truth")—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. [2] [4] [7] However, insofar as the trivialist affirms all states of affairs as universally true, the Pyrrhonist neither affirms nor denies the truth (or falsity) of such affairs. [8]

### Impossibility of action

It is asserted by both Priest and Kabay that it is impossible for a trivialist to truly choose and thus act. 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] [9] Ironically, due to their suspension of determination upon striking equipollence between claims, the Pyrrhonist has also remained subject to apraxia charges. [10] [11] [12]

Paul Kabay, an Australian philosopher, in his book A Defense of Trivialism has argued that various philosophers in history have held views resembling trivialism, although he stops short of calling them trivialists. He mentions various pre-Socratic Greek philosophers as philosophers holding views resembling trivialism. He mentions that Aristotle in his book Metaphysics appears to suggest that Heraclitus and Anaxagoras advocated trivialism. He quotes Anaxagoras as saying that all things are one. Kabay also suggests Heraclitus' ideas are similar to trivialism because Heraclitus believed in a union of opposites, shown in such quotes as "the way up and down is the same". [13] Kabay also mentions a fifteen century Roman Catholic cardinal Nicholas of Cusa, stating that what Cusa wrote in De Docta Ignorantia is interpreted as stating that God contained every fact, which Kabay argues would result in trivialism, but Kabay admits that mainstream Cusa scholars would not agree with interpreting Cusa as a trivialist. [14] Kabay also mentions Spinoza as a philosopher whose views resemble trivialism. Kabay argues Spinoza was a trivialist because Spinoza believed everything was made of one substance which had infinite attributes. [15] Kabay also mentions Hegel as a philosophers whose views resemble trivialism, quoting Hegel as stating in The Science of Logic "everything is inherently contradictory." [16]

### 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. [17] [18] [19]

### 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

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]

## Related Research Articles

In logic, the law of excluded middle states that for any proposition, either that proposition is true or its negation is true. It is one of the so called three laws of thought, along with the law of noncontradiction, and the law of identity. The law of excluded middle is logically equivalent to the law of noncontradiction by De Morgan's laws. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan's laws.

In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that he or she is lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.

In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. It is a particular kind of the more general form of argument known as reductio ad absurdum.

In logic, negation, also called the logical complement, is an operation that takes a proposition to another proposition "not ", written , which is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .

In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0, O, and the up tack symbol ⊥.

Logically possible refers to a proposition which can be the logical consequence of another, based on the axioms of a given system of logic. The logical possibility of a proposition will depend on the system of logic being considered, rather than on the violation of any single rule. Some systems of logic restrict inferences from inconsistent propositions or even allow for true contradictions. Other logical systems have more than two truth-values instead of a binary of such values. However, when talking about logical possibility it is often assumed that the system in question is classical propositional logic. Similarly, the criterion for logical possibility is often based on whether or not a proposition is contradictory and as such is often thought of as the broadest type of possibility.

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.

The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.

In contemporary analytic philosophy, actualism is the view that everything there is is actual. Another phrasing of the thesis is that the domain of unrestricted quantification ranges over all and only actual existents.

Non-classical logics are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.

Graham Priest is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne where he was Boyce Gibson Professor of Philosophy and also at the University of St Andrews.

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic.

Jc Beall is an American philosopher, currently the Board of Trustees Distinguished Professor of Philosophy at University of Connecticut,.

The following is a list of works by philosopher Graham Priest.

## References

1. Priest, Graham (2007). "Paraconsistency and Dialetheism". In Gabbay, Dov M.; Woods, John (eds.). 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. 77 (4): 443. doi:10.1080/00048409912349211.
6. Priest, Graham (2008). Doubt truth to be a liar (1st pbk. ed.). Oxford: Oxford University Press. pp. 69–71. ISBN   978-0199238514.
7. Priest, G. (2000). "Could everything be true?". Australasian Journal of Philosophy. 78 (2): 189–195. doi:10.1080/00048400012349471.
8. Empiricus, S. (2000). Sextus Empiricus: Outlines of Scepticism. Cambridge University Press. "Suspension of judgement is a standstill of intellect, owing to which we neither reject nor posit anything" (p. 5).
9. 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.
10. Comesaña, J. (2012). Can Contemporary Semantics Help the Pyrrhonian Get a Life?. In Pyrrhonism in Ancient, Modern, and Contemporary Philosophy (pp. 217-240). Springer Netherlands.
11. Wieland, J. W. (2012) Can Pyrrhonists act normally? Philosophical Explorations: An International Journal for the Philosophy of Mind and Action 15(3):277-289.
12. Burnyeat, M. (1980). Can the sceptic live his scepticism?. In M. Schofield, M. Burnyeat, and J. Barnes (eds.), Doubt and Dogmatism (pp. 20-53). Cambridge University Press.
13. Kabay, P.D. (2008) A Defense of Trivialism. Phd Thesis, School of Philosophy, Anthropology, and Social Inquiry, The University of Melbourne pages 32–35
14. Kabay, pages 36–37
15. Kabay, pages 37–40
16. Kabay, pages 40–41. A Defense of Trivialism
17. 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...
18. Bueno, O. V. (2007). "Troubles with Trivialism". Inquiry. 50 (6): 655–667. doi:10.1080/00201740701698670.
19. Azzouni, Jody (2003). "The Strengthened Liar, the Expressive Strength of Natural Languages, and Regimentation". The Philosophical Forum . 34 (3–4): 342. Retrieved 21 May 2014.