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Something that leads to a contradiction can not be true, and if so, the opposite must be true.

In fact, this principle is used in proof by contradiction. but, is this principle always true?

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    Yes, the rule is: if from P a contradiction follows, then not-P. Dec 11, 2022 at 11:27
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    It is true unless it leads to a contradiction. Then the universe explodes.
    – Scott Rowe
    Dec 11, 2022 at 16:12

4 Answers 4

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Although the LNC has been accepted as true by most analysts over most of recorded history, there are dissenters, i.e. dialethists. For an academic overview, see "Paraconsistent Logic" as well as both the IEP and SEP articles on inconsistency-friendly mathematics.

Now paraconsistent logics are not automatically LNC-violators; they're based more on suspension of the argument for the LNC based on "logical explosions." An early example is relevance logic, which waives the kind of disjunctive inference that yields the "irrelevant" deduction of arbitrary conclusions from inconsistent premises. One can be a relevance logician and accept the LNC for some other reason. Still, though, paraconsistent logic is most useful for dialethists because it gives them formal space in which to avoid trivialism.

Another "way out" is to have multiple flavors of negation. That negation semantics are LNC-theoretic is arguably the main claim, alongside the explosion argument scheme, by virtue of which the LNC has been defended. I.e., to many of us, it seems as if there is a crucial sense of the word not and its cognates (e.g. prefixes like un-) which makes holding a pair of incompatible assertions to be true sound like a misuse of the word not. However, closer inspection at least raises the question of whether this must always be so. For example, one might also suppose that not-semantics must conform to double-negation elimination, but intuitionistic logic says otherwise. And paracompleteness can be regarded as the "dual" of paraconsistency. So by a quirky back-door approach, one can get a different sense of negation semantics in paraconsistent logic via the alternative such semantics in paracomplete (incl. intuitionistic) logic.

Moreover, consider the difference between empty nodes and fully empty graphs in graph theory. I.e., there is a difference between a graph with however many nodes but no paths, and one with no nodes at all (and one supposes that there is also the degenerate case of a graph with no nodes but to which paths are nevertheless somehow attributed). Perhaps fully empty graphs are akin to C. S. Pierce's empty spaces:

  1. An empty space is a well-formed diagram.
  1. If D is an empty space, then it is translated into ⊤.

And "⊤" being the symbol for the Truth Value, it might seem to follow that a fully empty graph doesn't represent the negation of any specific truth. Be that as it may, otherwise, there seems to be a graph-theoretic distinction in negation semantics at least as far as empty cases of the node/path pairing is concerned.

Also consider that there are received distinctions between predicative and propositional negation, e.g. it is possible for a horse to be red and not red, as long as the scope of "not red" is not "all of the horse" (it has red parts as well as not-red parts); and then between absence and opposition (passive and active negation). Furthermore, 0/0 is an indeterminate expression (per its evaluation), whereas n/0 is undefined: the former can be evaluated to any value of n, whereas the latter has no evaluation. One can subtract zero from itself however times one likes and get zero as a result; one can't subtract zero from some n any number of times and get zero.

Or then consider the theory of the negative hyperoperator sequence. If we try to define xa y as x-a y, two main issues arise. First, we lose out on zeration-as-succession for the ascension operator scheme, since zero is neither positive nor negative in the required manner (or perhaps might be thought of as both). Moreover, trying to compose the negative hyperoperator sequence under the same rubric as the positive one, viz. xa x = xa+1 2, does not go through, regardless of whether the succession of the operators goes forward as usual or backwards from a = 0 in the descension hypersequence. I.e., imagine that 1 ↑-1 1, which would be 1 - 1 = 0, were composed as usual; the result would be 1 ↑0 2, or “the second successor of 1,” which is 3; or if 1 ↑-1 1 = 1 ↑-2 2, we would have ½, which is not 0 either. So the negative hyperoperator sequence, or rather the introduction of the descension operator scheme, is not reducible to negatively signing the ascension schematics.

The “upshot,” then, is that the minus sign as such, and the downward arrow, are not only different glyphs, but seem to have internally distinct negation semantics. This allows us to intuitively grasp how it is that the imaginary unit can be productively composed with itself to form a minus-signed number: i is accessed by a negative hyperoperator, an even-root operator, and hence is first a matter of the down-arrow glyphset, and then it yields a minus-signed number, modulo the separate negation semantics for the minus-sign.

Accordingly, proof-by-contradiction might have to be limited to the context of the kind of negation/contradictions that obey "normal" such semantics, but one might be able to infer a different flavor of contradiction from some premise and this wouldn't tell against the premise (at least not in the same way).

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    "What part of NOT don't you understand?" I guess... Wow.
    – Scott Rowe
    Dec 11, 2022 at 16:17
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    @ScottRowe this is gonna sound ridiculous but a lot of my obsession with the topic came from trying to define a god-of-destruction in a fantasy story I was writing 🤣 Dec 11, 2022 at 16:45
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    A nice, fun answer. Relevance logics typically don't allow you to move from A → ⊥ to ¬A. Usually they treat negation as a primitive modal operator. And of course intuitionistic logic allows A → ⊥ therefore ¬A, but not ¬A → ⊥ therefore A.
    – Bumble
    Dec 12, 2022 at 15:55
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    @KristianBerry. So a contradiction is problematic (ex falso quodlibet)and if one is to allow it, we tweak the system to prevent $\bot \to p$ where p is any proposition - possible by e.g. prohibiting disjunction introduction aka addition. The rest of your post regarding graphs (nodes & paths) went over my head, but still, intersting. Gracias for the answer and the links. Have a nice day.
    – Hudjefa
    Jan 3, 2023 at 4:35
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    @AgentSmith my understanding of EFQ is that it's a syntactic effect (of disjunctive syllogism?), but one question I've had about it is whether we might curtail it by claiming that finite theories don't have "enough" possible sentences available to them, for a proper explosion. So firstly, a difference between explosivity = one random sentence inferrable, vs. all random sentences inferrable. (Granted, perhaps we could construe the all-consuming blob as a very large conjunction, externally consolidated "as if" it were a single sentence. But I need a better theory of conjunction, anyway...) Jan 3, 2023 at 5:56
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The very short answer

Yes, for most applications this is a very useful working assumption. Treat it as true.

Longer answer

No. What you are using is classical logic, which only allows two "truth states", true or false, and assumes the Law of Non-Contradiction, and the Principle of the Excluded Middle (things can only be True or False).

IN CLASSICAL LOGIC, your proposition is always true. But not all aspects of our universe seem to follow classical logic, and we have shown that there are infinite other logics, many of which do not require the Law of Non-Contradiction. See this paper for a discussion of logical pluralism: https://www.cambridge.org/core/journals/think/article/abs/guide-to-logical-pluralism-for-nonlogicians/EDFDFA1C9EB65DB71848DABD6B12D877

Pragmatically, in understanding our world, classical logic is a good first assumption. However, sometimes it breaks down. So the remaining candidates for "One True Logic" are non-classical logics which can encompass multiple other logics, including classical logic, as pragmatic local approximations.

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If you'll permit a simpler explanation than Kristian Berry's ...

A contradiction, the issue therein, revolves around negation. If I negate the statement "God exists", I mean "God exists" is false; this negation looks like "God does not exist"

Let's suppose contradictions are true, an example of which is "God exists" and "God does not exist" ( G & ~G).

Notice the logical operator negation ("~" before G); if contradictions are true, it does absolutely nothing with respect to the truth value of the proposition "God exists" (G). Negation is functionally meaningless/inert.

Now suppose we claim that no, negation does what it's supposed to do (switch the truth value from true to false and false to true). G is true (one conjunct) and ~G is true (the other conjunct). If so G is false because the negation of G (~G) is true. Taking T = true and F = false, if x is the unknown truth value of a proposition and ~x = T then, x = F (negation) , but if I say x = T also, the conclusion is inevitable, x = T = F. The same applies to G & ~G. The truth value of G is T. If the truth value of ~G is T then, the truth value of G is F (negation). If the truth value of G is represented as g then, g = T = F i.e. T = F. The distinction true vs. false is lost and the entire notion of truth collapses.

Logics used to handle contradictions are described as contradiction tolerant which means, in me book, contradictions are permitted as conclusions but they're not allowed as premises (re ex falso quodlibet). That answers your question I believe.

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    I appreciate the effort to turn Kristian Berry's answer into something more understandable for non-logicians such as you and I. I am not sure you were successful, and I tried my hand at it as well.
    – Dcleve
    Jan 3, 2023 at 19:17
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    Apologies ... my plan backfired. I'll edit it, not now, but later; check back in about 7 - 8 hours.
    – Hudjefa
    Jan 4, 2023 at 3:51
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"«Something that leads to a contradiction can not be true», is this principle always true?"

Yes, this statement is always true. The proof of its truthness is ofcourse in data (observation).

Its never observed that something both exist and not exist at the same time, in the same place, in the same sense.

Once you get to the atoms, the indivisible parts, the parts that cannot be simplified / broken down any more then the world is binary, things exist or don't exist, thats what our observations tell us.

This is the basis of Aristotelian Logic. Things exist or don't exist, never both, never neither.

The indivisible parts may be "at same time" meaning a thing can exist at some time and don't exist at other times, for example if you destroy a car then the car don't exist any more; at same place, ofcourse you are where you are right now, not also in some other place; in same sense, a person is dead physically but his memory is alive say.

Nobody goes against this fundamental thing. People will argue about what sense one mean that a thing exist (or don't exist) but once the sense, the atoms, are agreed upon everybody agree on a binary / either-or.

As noted above, the truth of all this is in data. As always data should be kept at top and all theories should be derived from it.

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    Atif -- you will find, if you study modern physics, that virtual particles (which don not clearly exist or not exist) are a central aspect of how our physics works. Your Classical Logic categories don not actually constrain our world, only your thinking.
    – Dcleve
    Jan 3, 2023 at 18:59
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    What you call modern physics has nothing to do with physics aka mechanics, the how; and is just abstract maths. Has nothing to do with real world. How do I know? Unlike many who just read headlines and swallow ready made conclusions I actually looked at data of the experiments. The maths proposed dont fit with data. How? The so-called physicists make placeholder terms they dont even have a definition for to fill holes in their equations. They invented concept of Dark Energy for example. A term without meaning. Chat with me in private because its off-topic here. There is a long list.
    – Atif
    Jan 3, 2023 at 19:07
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    Atif -- your rejecting what physicists say about physics, because it does not fit your Aristotelian presuppositions, is not something I expect to learn anything useful from. For a discussion of dogmatism, see this answer: philosophy.stackexchange.com/questions/95735/…
    – Dcleve
    Jan 3, 2023 at 19:13
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    You want to throw logic out of window, its your choice. It dont make a thing what its not. You look too impressed with big names to use your own head. Like putting a whig on your head dont make you a judge unless you sit in a court listen to people's cases no matter your degrees, a person is not a physicist if he dont do experiments and just sit with a pen and paper have abstract thoughts, do thought experiments, make equations that dont fit even with observations, then go make terms like Zero Point Energy, Dark Energy, Orbitons, Uncertainty Principles, Particle-Wave Duality etc.
    – Atif
    Jan 3, 2023 at 19:21
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    The concept of virtual particles is a made-up concept that has nothing to do with real world. A particle if exist even for a very short period of time do exist during that as much as you and I exist. But the so-called physicists call them virtual because they want them to do only some things during their existence, not all as it would really, because otherwise their equations are proved wrong. Likewise, data shows more gravity in universe than any theory they have suggests, so instead of correcting the equations they introduce made-up terms like Dark Energy.
    – Atif
    Jan 3, 2023 at 19:26

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