# What does "contradictions trivialize truth" mean?

I hope there's no regulations against posting a series of consecutive questions.

In the past decade or so I've read around 3-4 books on logic; they were 101's (introductory books) so more about application than theory.

Several methods of argumentation were taught as special, 1) the conditional proof and 2) reductio ad absurdum. A reductio argument has as its conclusion a contradiction which then falsifies one/some/all of the premises(?).

On the matter of contradictions (p ∧ ¬p), I did some reading, mostly from Wikipedia, and the claim is "contradictions trivialize truth". I know of trivialism per which all statements are true and I can see how we get there via ex contradictione sequitur quodlibet but I have an issue.

What does it mean exactly, to trivialize truth?

My own take on this is that it is negation that is being trivialized in the sense that it's an empty operator, "doesn't do anything at all" as p is true AND also ¬p is true.

• Everything becomes true, so truth becomes a trivial notion. Commented Aug 12 at 7:10
• Contradictions falsify the conjunction of the premises, not any one or some in particular. "Trivializing truth" means that every sentence is deduced to be both true and false. Hence the system with a contradiction cannot play the usual discriminating role of classifying sentences into two disjoint groups, true and false ones. Commented Aug 12 at 7:13
• It is basically a wring saying: what is true is true and what is false is false. A system that proves a contradiction will prove everything, both true and false claims. Thus, what is "trivial" is the system that fails wrt its basic purpose: to prove/discover truth. Commented Aug 12 at 7:14
• "Trivial" as in "not interesting" or "too simple to be worth talking about". This is standard usage in mathematics, see en.wikipedia.org/wiki/Triviality_(mathematics) Commented Aug 12 at 7:33
• Stringing together propositions on paper or in the head does not guarantee that the result is imaginable or makes any sense. Paper can suffer anything. Truth tables are of no help when it comes to imagining a world. You have to imagine a room where you are and are not. At the same time. Try it. This is why people puzzle over Schrödinger's cat, although it is not quite the same. Commented Aug 12 at 7:58

Several direct, technical explications can be the given for the mentioned phrase, particularly, within the context of paraconsistent systems of logic, because the fundamental motivation of those systems is to allow a contradiction to hold, that is, both P and ¬P hold for some proposition P, but without "trivialising truth" (i.e., all propositions become true).

I shall bring forward an indirect, conceptual context to see the phrase under another, broader light. Trivialising truth, whether by contradiction or by some other notion, can be regarded as an extreme form of relativism. A concise definition of this generic position is offered in Emrys Westacott's IEP article on relativism as

Although there are many different kinds of relativism, they all have two features in common.

(1) They all assert that one thing (e.g. moral values, beauty, knowledge, taste, or meaning) is relative to some particular framework or standpoint (e.g. the individual subject, a culture, an era, a language, or a conceptual scheme).

(2) They all deny that any standpoint is uniquely privileged over all others.

So, for example, suppose a social norm may allow, and even, encourage a behaviour in one society, while an antagonistic norm of another society may reject and strongly discourage such a behaviour. The relativist viewpoint, taken typically, tend to uphold that both attitudes should be accepted right, for each case has its own validating conditions, thus, render any moral evaluation insignificant.

Allowing contradictions in a logical system can be seen as an abstractive (hence, converting the sides to polar opposites) reflection of a relativist position in this sense: P is true by its truth conditions, and ¬P is also true by its own truth conditions. Thus, the notion truth is deflated to a trivial status with no essentially discriminating power; there remains nothing of interest to inspect and assess whether some assertion is true or false.

• What is trivialized is the distinction between truth and falsity, aye? Damn! I like my answer, but I daresay I'd recommend yours for acceptance (it's way clearer and way more direct). Commented Aug 12 at 15:18
• Thank you for your kind words. It can be phrased as you say. When the distinction is trivialised, the notion becomes bereft of its definitive element and it is trivialised, too. Commented Aug 12 at 17:33
• Superb answer. Truth and falsity are antithetical dualistic concepts that help us as @Conifold stated to separate reality from illusion. If contradictions are allowed we end up losing that discriminatory power, undermining as it were the very purpose of truth (includes falsum). Good that you brought up relativism. From a Jain anekantavada (no one-sidedness) POV, claims are always prefixed with syad (to remind us that the claim is (only) a perspective. Commented Aug 12 at 18:36
• What if the truth is that everything is true. In this case are we still trivializing the truth? The model (trivialism) is accurately applicable to reality then, si? Also can you opine on my take that it is negation that's being trivialized, it serves no purpose? Commented Aug 12 at 18:39
• Thank you for your appreciation. Your last question opens up many ramifications; it may be appropriate to move back and forth between premisses and consequences to accurately stabilise thought in such cases. For an instance, we can consider that of Leibniz's thesis of best of all possible worlds. From a critic's viewpoint, yes, Leibniz's optimism trivialises the truth. The questions on triviality arise not from that negation fails in its function (it fulfils its function), but from deductive and semantic principles adopted. You may wish to see the views on paraconsistency and dialethism. Commented Aug 12 at 22:01

It trivializes truth because anything follows from the assumption, p and not p, in a trivial way.

So, now the proof of any statement becomes a trivial proof, no matter how complicated the statement is. Also, that the statement can be deduced at all is also trivial, since all statements can be deduced.

Assume p and ¬p, we prove q

Proof:

Assume p

Assume ¬p

We then have ( the details on how, depend on the exact specifications of the logic we are working in)

p → (¬q → p) This form is often just taken as an axiom.

by Modus Ponens

¬q → p

by Modus Tollens

¬¬q

by double negation syllogism

q

Remark: I would say that trivial here, refers to the fact that the result follows from a simple argument- which holds regardless of the complexity of q.

It also refers to how, the question of "is there a proof of q"- is trivially answered in the affirmative.

It's not really that "truth" is trivialized. It is that proofs are trivialized.

To say that truth is trivialized, we need to add the further assumption that things we are able to prove, are true.

• The wording is quite clear. Although I agree with your, proving (anything) is a trivial exercise, it's truth that's trivialized. Perhaps the point is dualistic in nature, having to do with the contradistinction true vs. false. It's non liquet Commented Aug 12 at 18:27

If we use the word "trivial" in the mathematics/logic context to refer to the "simpler" or "weaker" versions of something (per the phrase "trivial object" as in category theory), yet why would we say that inferring every version of something would be trivial? In fact, there might be a gap in the terminology here (c.f. lumping "degenerate" cases in with excessive/"obgenerate" ones), but on the other hand, consider the question of proof objects. If we use a set of premises to get a conclusion, we can measure the difficulty of the inference in terms of the number and interior complexity of the premises and the attendant inference rules.

But so that's why a contradiction, in explosive logics, trivializes things: the bare premise A&~A is very simple, and it makes deriving whichever random B into such an easy exercise that proofs, from contradictory premises, are what are so trivial, here. Or, to keep up with a logical explosion's shockwave, if we can infer any random B, C, etc. from A&~A, then if we frame A as a question, "A?" then since we are also claiming ~A, we can reply to, "A?" with anything, and again it is trivially easy to solve that given problem.

• In what sense can truth be trivial in a logical system? Commented Aug 12 at 18:29
• @Hudjefa I think we would say, "It is trivially true that..." and so we then would say, "If A*&~*A, it trivially is true that B..." and B can be anything. Interestingly, a logical explosion would presumably include the proposition, "Nothing is trivial at all," along with, "Everything whatsoever is trivial," so on another level, you would have that trivialism = nontrivialism, there. Commented Aug 12 at 18:41
• Yes, but "it is trivially true ..." is not a judgment on the veracity/facticity of a claim but is a comment on how easily the claim can be proven. Something may be proved easily but be of great signficance. Commented Aug 12 at 18:44
• @Hudjefa there do seem to be two senses of triviality at issue. For example, maybe we can differentiate between quantitative and qualitative triviality, which can come apart. Also, even if it is trivial that X, it might not be trivial that it is trivial that X. Commented Aug 12 at 18:55
• (cont.) seem like trivial properties for an object to have. If every assertion or proposition is true, and if every a/p has this property necessarily (as by a logical explosion, say), then being true becomes a trivial property. Note that a logical explosion can be seen as trivializing falsity just as well (if every false sentence corresponds to a true assertion that something else is false). Commented Aug 13 at 5:42