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I have seen some arguments for why should we accept law of non contradiction, and it seems to works in almost all areas. But some argument for it is like an argument for principle "nothing comes from nothing", such as if we don't accept principle then everything should comes from nothing, and everything should becomes true. Or there is not really difference between something and nothing if such transition can happens, and there will not really difference between truth and false if there is such transition without law. And almost for all practical purposes, both law and principle seems to follow.

But as I read, we may not know that "nothing comes from nothing" as priori. Or can we know? And if we can't know that "nothing comes from nothing" as priori, can it be similarly that we can't know law of non contradiction a priori?

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I don't believe we can know a priori that there are no true contradictions, and the existence of the dialethists suggests we do not have to assume it. But I know of no examples so the idea seems ad hoc. Examples cited by philosophers (Priest, Melhuish et al) and physicists (Heisenberg, Redhead et al) depend on an unnecessary and I'd say incorrect application of logic.

If the universe is reasonable then there are no true contradictions but we cannot know a priori it is reasonable. Hence all the arguments. From analysis we cannot even know that ex nihilo creation is impossible, but as it would mean the universe is incomprehensible it seems safe to assume we can trust our reason, and our reason rejects contradictions.

There is a deeper technical point here about logic and the impossibility of Reality breaking Aristotles rules, but this would be a topic too far for an answer here.

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  • "Ex Nihilo appearance" is postulated for the universe in a variety of Cosmology models in physics, and these models are not rejected as logically invalid. Ex Nihilo creation should therefore be presumed as locally possible. – Dcleve Jan 31 at 17:13
  • @Dcleve - I would rather believe these theories disregard logic since they are certainly not consistent with it. But usually on analysis they turn out to be not ex nihilo after all. – user20253 Feb 1 at 11:46
  • @PeterJ "I would rather believe these theories disregard logic since they are certainly not consistent with it." How so? (I disagree with this claim ...) – Noah Schweber Feb 1 at 16:43
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    @NoahSchweber - The absurdity of ex nihilo theories does not seem to be a matter of opinion to me. All the so-call ex nihilo theories I've seen are not quite ex nihilo. If you know of an exception I'd be interested to study it. . – user20253 Feb 2 at 11:53
  • @PeterJ -- "Ex nihilo" classically meant that new matter or space came to exist -- violating the assumed conservation of both. A God was needed to create new matter or space ex nihilo, as matter by itself could only transform. All of the examples I cited in my answer are ex nihilo. One can redefine "ex nihilo" to allow violation of all conservation laws but require violating causation, but then the classic definition is also not "ex nihilo". Also, modern physics violates causation too, so this doesn't help unless you also redefine causation. Note Hawking violated both in my cited example. – Dcleve Feb 3 at 18:05
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A proposiiton can be known, ether a priori, or a posteriori.

In case a law is known a posteriori, this requires induction ( since a law is a universal statement, and no universal stement can be known by a single observation).

So, can one know by induction that " for all proposition P , ~ (P& ~P)?

Can one know by induction that " for all object A, " it is false that A is not identical to A"?

Induction would only make the alledged law probable; but a law of logic is supposed to be absolutely universal ( true in all possible cases).

We couuld know a posteriori that the " law" is false, in case we actually encounted a contradiction in the real world, or grasped conceptually a contradiction in some possible world ( or scenario). But the attempt at knowing the truth of the law a posteriori seems doomed to fail.

All this tends to show that the law of non-contradiction can only be known a priori.

Another option is to say that this law is not actually known, but is laid down as a linguistic convention, a " rule of the game".

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Various paradoxes make it clear that the law of non-contradiction not intuitively clear enough to be an a priori truth.

Before we get to the vagaries of 'something' and 'nothing' and their 'thingness', let's start with actual, countable things.

The sorites paradox is one way to see the weakness of negation as an a priori concept. Given the statements of the form "X grains of sand are enough to constitute a heap." and "X grains of sand are not enough to constitute a heap." For which X does the former become true and the latter false? If you can't answer that, can you really state that either A or not A is always true?

Negation can be defined via the principle of contraction, making it true in a formal sense. But in any reality beyond empty formalism, 'not A' is not, in fact, entirely clear. (Because it is an idealization of something post-hoc.)

Because the notion of negation is not clear, but rather flexible and incomplete, any absolute sense of 'nothing' is virtually meaningless. We can come up with conventions around what 'nothing' means in various contexts, but there is no 'nothing'. Anything we can conceive of or speak about is not 'nothing', it is only 'some state that is left when I drive out all of the relevant things from my mind.' "Nothing plus various laws and expectations" is not nothing, and there is a basic structure and various expectations that we cannot purge from our descriptions. Raymond Smullyan has delved into various notions of nothing in innumerable cute ways that make this point pretty clear.

But perhaps more relevantly, psychotherapists see in 'primary process thinking' how our more basic forms of cognition do not understand negation. Talking about not wanting something or not being afraid of something, if that 'something' is specific enough, makes us more likely to suffer from the lack of it or to actually feel the fear it. If negation were some kind of really basic concept, and not a posteriori adaptation to the world, our simpler ideas would handle it better than our sophisticated ones. Instead, it is the other way around.

So when you talk about creation from nothing, you really are not saying anything, unless you qualify the variety of nothing. There are concepts of nothing that will give rise to something else, like the modern 'virtual particles' view of empty space that is really made up of things and their opposites perfectly overlapping, which theoretically leads to the Big Bang out of its very nature. And there are concepts of nothing that will not become something else, like the Newtonian picture of absolute empty space that our quantum one replaced.

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You are grappling with the idea of logical and truth pluralism. This is a question that philosophy has been trying to come to terms with for a century and a half, with much of that time spent with loads of philosophers denying the issue, which has slowed the development of an answer.

The consensus view in the mid 1800s is there is one math, and one logic, and they are both true by necessity. Kant articulated this view.

However, non-Euclidean geometry was developed and shown to be self-consistent, and as this was Kant's go-to example of necessity -- the necessity view of math was refuted. Necessarians tried to hold on to "at least the world is Euclidean", which would make math true a posteriori, rather than a priori (this is a peculiar position for math, but straws were being grasped). But then Einstein blew that weak fall-back out of the water too. The consensus among mathematicians is that math is real (abstract object plationism), but that there are LOTS of maths, and the choice of what particular math to apply is a formalism, no specific form is "necessary". This is mathematical pluralism. The preference for a particular math is PRAGMATIC, not based on logic.

Physicists embraced this approach to math, and as logic and math are basically the same category of system, one should expect pluralism to apply to logic as well. And a century ago, Quantum Mechanics was developed with that as an assumption. Quantum math does not follow classical logic. The double slit experiment, the Heisenberg uncertainty principle, entanglement, Bells Inequality -- these are all based on non-classical logic.

As you note, this brings up the question of ultimate causes, and the validity of "nothing comes from nothing". And sure enough, physicists, who embraced plural logic pretty early, propose "nothing from nothing" events. Hoyle's "Steady State" universe existed forever, but the MATTER in it -- came from nothing. He proposed that there was a small rate of spontaneous proton formation in empty space. And Hoyle is hardly alone. Guth's proposal of "inflation", in which mass and space just form spontaneously due to an "inflation field" is assumed in pretty much all subsequent cosmology. Susskind's Cosmic Landscape expands this to the spontaneous formation of an infinite multiverse, not just our universe. And while Guth and Susskind did not give up on causation (they start with a seed "field") Hawking did -- in A Brief history of Time he proposes that the universe is "a closed shape in spacetime", IE nothing before (no cause) and nothing after (no consequence).

Logicians have been slow to accept this thinking. But over the last several decades, logical pluralism has become the consensus. A useful discussion is here: https://arxiv.org/abs/0705.1367 (the PDF download is free).

As you note -- this brings TRUTH into question. Many people, including philosophers, resist the radical consequences of pluralistic logic, because they don't see any other way to get to "truth". But there is an alternative -- it is to approach truth pragmatically. Formal logic is "very useful" therefore one should generally accept what it demonstrates, as tentatively true, unless one has significant rationale or justifications to think otherwise. Truth is uncertain, and the means to find it is uncertain, but we have a lot of tools which are highly useful, and generally bring us to "good enough" approximations of truth.

So -- your top question -- the answer is "no". We cannot know that non-contradiction is true a priori, and based on the intrinsically pluralistic nature of logic, we can actually know it is NOT always true, and know this a priori.

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  • I cannot agree that most mathematicians see it as formalism. Real Analysts have a real model of real things in mind. Algebraists think of the class of groups is a genuine thing. Ordinal theory happens in some universe that spans all formalisms. They have a kind of localized Platonism in mind and they use its full power. Really, Goedel's theorem blows Hilbert's program of formalism out of the water. The standard mathematical position is not formalism, it is a hypocrisy meant to placate logicians: "Platonism all week and Formalism on Sundays." (en.wikipedia.org/wiki/Reuben_Hersh) – hide_in_plain_sight Feb 3 at 18:17
  • I may have phrased that sentence poorly. I know mathematicians are primarily small p platonists, but are also pluralists. The choice of what math to apply to the physical world then becomes a formalism, although the math itself is real in an abstract space. Do you have a suggested rewrite? – Dcleve Feb 3 at 18:23
  • It is hard, because mathematicians think and claim that they think this, but it does not describe how they act. But it flows into your next generalization. Sagan militated for including math on Voyager, pi in particular, because of its universality... Quantum math uses classical analysis for everything, even when the experiment described defy traditional logic... Feynman integrals go backward in time rather than question calculus. Etc. There is something oversimplified here. It fits a mythology rather than behavioral data. (I will stop here, until I have something more positive to say.) – hide_in_plain_sight Feb 3 at 18:33
  • @hide_in_plain_sight -- I have done a minor rewrite, does this correct the point you were concerned about? – Dcleve Feb 3 at 18:38
  • I would like to see an example of the LNC being broken by the world. I have never seen one. I would also note that your description of the ex nihilo theories do not make them seem ex nihilo to me. There is usually space, a field or some sort of potential involved. Of course, there is nothing to stop us conjecturing it all comes from nothing, but abandoning ordinary logic in order to make sense of the idea looks like cheating and seems unlikely to lead to a comprehensible theory. But the issue is examples. In what circumstances is the LNC broken? . . – user20253 Feb 4 at 12:35

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