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(Philosophy novice here and not sure how best to phrase this question; if it's unclear please point out the problems).

Imagine this conversation:

Alice: I believe that X.

Bob: Do you also believe Y? (Alice says yes) But that means you believe in Z.

Alice: That's true, but I still don't believe in Z.

Alice's position is clearly inconsistent. The question is, so what? From what I have seen in philosophy, people simply assume that things must be consistent. I'm wondering what happens if we accept inconsistency.

Based on what I've seen, if we accept inconsistency then everything becomes possible. However this seems to be a mathematical proof that isn't directly translate-able to real life. For example we know quantum mechanics is inconsistent with general relativity, yet the world goes on as normal (and things don't start falling upwards).

The other possibility is that, if we accept inconsistency, then logic and rational discourse becomes impossible. By accepting both X and ~Z, Alice is saying she cannot be reasoned with. But this doesn't seem to work either. Most people don't make it past PhilosophyExperiment's Battleground God unscathed, yet we are still able as a society to agree on certain things (such as "murder is morally wrong"). Descartes famously claimed that God can make 2x4 not equal 8, which most people would call inconsistent, and yet most people would also not say Descartes is impossible to reason with either.

Why is it common to assume that things must be consistent (indeed, proof by contradiction is commonly used in mathematics)? Does inconsistency lead to disaster? If so, why are we able to hold inconsistent beliefs and still go about our lives without problems?

  • 2
    Everything becomes possible is called trivialism, and it follows from inconsistency under the classical logical law of explosion. However, classical logic is a very crude approximation of how we reason, under more nuanced ones the law of explosion fails, and trivialism does not follow. They are called paraconsistent logics. There is even inconsistent mathematics built on them. – Conifold Mar 25 at 4:35
  • If we allow logical inconsistency in our philosophy then we won't be able to understand it. It's a high price to pay. – PeterJ Mar 25 at 10:27
  • Individuals, society are largely in the excluded middle. This is the norm! – Gordon Mar 25 at 16:23
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We are asked to imagine the following conversation:

Alice: I believe that X.

Bob: Do you also believe Y? (Alice says yes) But that means you believe in Z.

Alice: That's true, but I still don't believe in Z.

At this point Alice looks like she is inconsistent in her beliefs. She believes in X. She believes in Y. She believes there is a warrant justifying Z if X and Y are true.

What would Alice's next move be to maintain consistency and continue disbelief in Z? She could do any of three things.

  1. She could reconsider her belief of X, rejecting X entirely or revising it so the warrant justifying Z is invalid.
  2. She could reconsider her belief of Y, rejecting Y or revising it so the warrant justifying Z is invalid.
  3. She could reject the warrant justifying Z given X and Y.

If she does any of these she would continue to be consistent and not believe in Z. She needs to be given the opportunity to change her mind, not only about Z, but also about X or Y or the warrant justifying Z from X and Y.

One way to challenge the warrant justifying Z from X and Y would be the open-world assumption. Douglas Walton (page 406) describes this as meaning:

...there is the possibility of new information affecting the reasoning.


Let's consider the questions:

Why is it common to assume that things must be consistent (indeed, proof by contradiction is commonly used in mathematics)? Does inconsistency lead to disaster? If so, why are we able to hold inconsistent beliefs and still go about our lives without problems?

Given the open-world assumption we can often claim that we do not know all the information. This would permit beliefs that are only apparently inconsistent.


Walton, D. N. (1990). What is reasoning? What is an argument?. The Journal of Philosophy, 87(8), 399-419. (Available at Walton's website: https://www.dougwalton.ca/papers.htm)

  • There are people who realize it's a paradox and hold fast in their inconsistent beliefs though, e.g. Descartes on the omnipotence paradox (he once said that God could make 2 x 4 not equal 8). People don't generally dismiss him as unreasonable, either. – Allure Mar 25 at 23:19
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    @Allure The views might be inconsistent with the closed-world assumption. However, with an open-world assumption there might be something one does not yet know or understand about what one knows that makes what is apparently inconsistent consistent after all. In particular propositions about God may at best be viewed as metaphors. Nor may they have the bivalence property, that is, they may not be either true or false. This is not just something theists can use, but atheists can as well when faced with theistic arguments claiming atheists are inconsistent. – Frank Hubeny Mar 26 at 0:18
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This answer has four parts. First, I'll analyze the specific example you give. Then I'll connect this to Alice's behavior, and follow up by talking about objections to paradox in general. I'll end with a bit of a digression which I think is nonetheless relevant enough to include.


You write

"we know quantum mechanics is inconsistent with general relativity."

This isn't really true - we know that our current theories about these situations are inconsistent. So we aren't satisfied with them. But they're clearly "not totally false" (they each yield super-accurate predictions in their realms of relevance).

This is an important observation - it tells us that mere truth and falsity aren't the end of the story. We know "QM(now) + GR(now) is false," but we also know (OK FINE - rather, have a high-confidence belief on the grounds of evidence so far) "QM(now) and GR(now) are each close to true."

The key point here can be phrased as:

We are often faced with the task of arguing from approximately-true claims.

(At least, when trying to reason about objects presented to us non-axiomatically - such as the world around us!)

In such contexts it is not a good idea to allow proof by contradiction! However, I'm sure we've all met people who simply refused to acknowledge the obvious seriousness of a contradiction they've been presented with. This raises the issue of where to draw the line (or more sensibly, how to understand the gradation) and I think this is a serious and interesting problem for philosophy.


OK, so now back to the first question: what's up with Alice?

Well, I would argue that your sample conversation does not provide sufficient information to determine whether Alice is acting properly or not. At some level, Alice needs to justify the current pragmatic acceptability of "Y and Z" - that is, she needs to demonstrate that Y and Z are each "mostly true" (under current knowledge, anyways). The specific example you gave (QM and GR) is one where I would argue such a justification is present; there are also plenty where it isn't.


Of course, this is all predicated on the assumption that contradictions are bad in the first place. This amounts to the idea that our imperfect knowledge of the world notwithstanding, there is a "fact of the matter" according to which every (sensible and exact) statement is either true or false. Some people reject this, either by accepting paradoxes or doubting the existence/meaningfulness of a "true reality", but this is the vantage point from which we oppose contradiction. And I would lump this in with Alice above (call it "meta-Alice") and say (putting everything together now):

Any claim that a specific apparent paradox may be tolerated requires justification, either of the applicability of a meta-principle or of the "mostly truth" of the principles involved.


I can't help but mention another specific example of reaction to paradox: does the problem of theodicy constitute a compelling argument against the existence of god(s)?

Those who argue "no" (for what it's worth, I would argue "yes") either argue against one of the premises or arguing that it is a "mystery" which must be accepted out of faith. The former amounts to trying to avoid paradox, so that's not really what we're looking at here; it's responses of the latter type which are more interesting for this question.

Do "mystery arguments" fit the paradigm of acceptable paradox-tolerance I've described above? Ignoring whether those arguments work, I would say that they do fit that paradigm - the key point is that experience argues for the premises of theodicy and the argue-er takes faith to be a justification of "mostly truth." Whether or not that is correct - and I'd argue it isn't - it's certainly true that the shape of this argument fits the pattern I've described.

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Tldr: logic and mathematics very purpose is to infer new statements from known ones while maintaining consistency. By not caring about it, you expose yourself to severe, possibly letal disasters. Except if you are doing metaphysics, then anything goes ;-p

Long version: in Critique of Pure Reason Kant establishes the difference between Synthetic statements, that introduce new information, and Analytical statements, that merely establish the equivalence of two statements.

For example, in "Socrates is a man(1), all men are mortal(2), therefore Socrates is mortal(3)", 1 and 2 are synthetic. They are information you got from some experience in the world. The whole sentence is analytic, as its result 3 was already contained in 1 and 2.

You can think of it like a game of 20 questions: if your partner is thinking of Socrates and you ask "is it a man?", the response gives you new, valuable information. But then asking "is he mortal, then?" would be a question wasted, because assuming you know that all men are mortal, you already have the answer to this question.

Consistency is the whole business of analytical statements, and logic and math, which are all about them. We want them to reliably establish the validity of a statement provided we are sure that its premises are true.

Now, why should you care? Well, because you only want to drive a car that can brake efficiently, fly a plane that can land, etc. Consistency is of crucial importance for science and technology. It allows you to make predictions and be confident that, given your car brakes apply such pressure and such friction, they can stop your car under a given delay, and this makes the difference between life and death.

Let's plug this in your example:

Alice: I believe that [my parachute is made of mint gelly] (picture her about to jump from the plane)

Bob: Do you also believe [mint belly is not a proper material for a parachute]? (Alice says yes) But that means you believe in [crashing on the ground and die].

Alice: That's true, but I still [believe it is safe].

By not caring about consistency, Alice puts herself in great danger. Her sacrifice for the sole purpose of demonstrating the value of consistent logic shall not be forgotten.

On the other hand, inconsistency can have less immediate consequences, allowing people to maintain them for a while. It often so happen because both side of the inconsistency are agreeable to us. For example, "I am a good person who cares about environmental issues" and "I have 3 cars and fly overseas every other weekend". But as long as the inconsistency is about tangible, verifiable things, people can be brought to their senses.

Metaphysical inconsistencies are the worst, for people who hold them will never be exposed to the bad consequences of their contradictory belief, at least in the physical world. I guess this is what you are referring to in your question. Alice can definitely hold inconsistent metaphysical views all her life.

Alice: I believe that God is almighty.

Bob: Do you also believe he lets child rapists molest kids? (Alice says yes) But that means you believe God is an uncaring asshole.

Alice: That's true, but I still believe He is all loving and compassionate.

  • On the other hand, you should be aware that "Parachute use did not reduce death or major traumatic injury when jumping from aircraft in the first randomized evaluation of this intervention." per bmj.com/content/363/bmj.k5094. Completely useless of course; it's a satire paper. But it does introduce some important questions about hidden assumptions. – Josiah Mar 26 at 7:01
  • @Josiah: I see. My main concern with this study is that it wasn't done with a double blind protocol. I highly suspect the placebo effect introduces a bias in the data. – armand Mar 26 at 7:38

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