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I mean, let's take an example of science. Is there have any theory that was contradictory in nature but proved correct in observations or practical?

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    If you ask for "correct in observations or practical" this fact implies an empirical theory. In this case "contradictory" is difficult to check: usually empirical theories are not formalizez. Jul 25, 2022 at 13:45
  • In empirical theories, the paradigmatic case is that of Newtonian mechanics: in the light of Relativity we have to consider it "falsified" but it is still very useful in many "practical" applications. Jul 25, 2022 at 13:46
  • A contradictory theory will correctly predict everything because for every true/false experiment, it will predict both true and false, and one of those is always right. Of course, the other is always wrong, so the theory will also incorrectly predict everything. It's hard to see how such a theory could be practical. Jul 25, 2022 at 14:15
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    There can be situations where you have theories A and B that are successful in different domains, but if you assume both theories are exact descriptions of the way nature works and look at the combined theory A+B, it leads to contradictions--this is true of general relativity and quantum field theor, see arxiv.org/abs/1001.1205 (note that the contradictions would only become non-negligible and measurable at the Planck scale of very high energy densities or very short intervals of time and distance, which we don't have the ability to test in a lab with current technology).
    – Hypnosifl
    Jul 25, 2022 at 19:42
  • I thought that all theories were practically correct yet contradictory in principle? "In theory, there is no difference between theory and practice. In practice, there is."
    – Scott Rowe
    Aug 25, 2022 at 1:09

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Consider the 'Grandfather Paradox', and Relativity. Closed Timelike Curves seem to be possible according to the math of Relativity, but we don't see evidence of chronology violations - so much so that we have the Chronology Protection Conhecture. Discussed in more detail here Paradoxes never exist in nature, so why does the grandfather's paradox make sense in physics?

Contradictions and paradoxes really show contradictions between inferences of our premises, and that can mean gaps where an otherwise good theory fails - and identify areas and phenomena for new scientific work. In the case of Relativity we think it is not a complete theory, and other apparent paradoxes about black holes also seem to be accounted for by this, naked singularities and cosmic censorship hypothesis, blackhole information paradox and Hawking Radiation.

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What If theory can be proved practically correct even if being contradictory to itself in theory? Can it be possible?

No, this is not possible.

Theories are checked against the facts of observation. Most precisely, you compare to observations whatever you can infer logically from the axioms of the theory.

The impossibility here comes from your assumption that the theory is self-contradictory, which I have to guess means that some two axioms are contradictory to each other. It is a basic principle of deductive logic that contradictions are false, so the conjunction of all the axioms of your theory will be false because it contains a contradiction.

Given this, it is not possible to infer anything from the axioms taken together.

You could draw inferences from various subsets of the axioms not containing the two mutually contradictory axioms but this would be fallacious. Whatever conclusion you infer logically has to be consistent with all premises, and this is not possible if any two premises are mutually contradictory.

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  • Thanks for answering. Let me present you a example like saying, Nothing is absolute in this world except the change. Is it self contradictory?
    – Schnoz
    Jul 25, 2022 at 17:01
  • @Schnoz. The example does not appear contradictory. It is saying that change is always present. There is nothing internally contradictory about that statement. Jul 25, 2022 at 17:57
  • @Schnoz The notion of self-contradictory can be somewhat misleading. Formally, a statement is self-contradictory if it contains a contradiction. Informally, it may be said to be self-contradictory if is contains opposite words. Opposite words are for example day/night, black/white, good/bad etc. The problem with conflating opposition and contradiction is that we often don't agree on whether two words are contradictory, which makes self-contradiction something in the eyes of the beholder. So, it is going to depend on what exactly you mean by "absolute" and "change", and "in this world" etc. Jul 26, 2022 at 13:14
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    @Schnoz: "There is nothing permanent except change" -Heraclitus
    – CriglCragl
    Aug 24, 2022 at 22:58

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