What does "contradictions trivialize truth" mean?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.