Science and the principle of noncontradiction

Discussions on the philosophical foundations, assumptions, and implications of science, including the natural sciences.

Science and the principle of noncontradiction

Postby TLK on January 9th, 2017, 7:09 pm 

There are certain things that need to be assumed to do science; e.g., it is assumed that events will unfold in the same patterns in the future as they appear to have unflolded in the past and unfold in the present (that formulation probably needs some clarification). Mathematics is essential to science - quantification of observations and quantification of patterns inferred from those observations is central to science. Different "realms" of math operate by different postulates and science is about "finding" the right math to fit the observed patterns of quantified data. But what about logic? Of course, mathematics has a logic (or logics) to it, but there are other formal logics. Should science be constrained by any formal logic other than logic found in mathematics?

A basic principle adhered to in most formal logics is the principle of noncontradiction - two contradictory statements cannot both be true (or a statement cannot be both true and false). Should the principle of noncontradiction be assumed in science? How would science operate if it allowed contradictory statements to be considered true? Are there examples found in science as it is practiced (by most scientists) now where contradictory statements are both considered to be true (or where a statement is said to be both true and false)?

Here is a little background on the question:

http://www.math.wichita.edu/history/topics/logic.html

One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is the study of formal reasoning based upon statements or propositions. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics. At the most fundamental level, mathematics is a language and it is a language of choice and must be communicated with great precision. (Wheeler, 1995) The idea of logic was a major achievement of Aristotle. In his effort to produce correct laws of mathematical reasoning, Aristotle was able to codify and systemize these laws into a separate field of study. The basic principles of logic center on the law of contradiction, which states that a statement cannot be both true and false, and the law of the excluded middle, which stresses that a statement must be either true or false. The key to his reasoning was that Aristotle used mathematical examples taken from contemporary texts of the time to illustrate his principles. Even though the science of logic was derived from mathematics, logic eventually came to be considered as a study independent of mathematics yet applicable to all reasoning. (Kline, 1972)


I only asked about the principle of noncontradiction (law of contradiction or law of noncontradiction), but the same question applies to the law of the excluded middle.
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Re: Science and the principle of noncontradiction

Postby someguy1 on January 16th, 2017, 5:11 pm 

TLK » January 9th, 2017, 5:09 pm wrote:I only asked about the principle of noncontradiction (law of contradiction or law of noncontradiction), but the same question applies to the law of the excluded middle.


LEM may not last another century. Intuitionism is making a comeback through, of all things, computer science. After all there are sets of natural numbers such that neither the set nor its complement are computable. In math, modern intuitionism is expressed via Homotopy type theory (HoTT), which lends itself to proof by computer.

Aristotelian logic is under attack not by anti-rationalists, but by science itself. If everything meaningful is computable (a common belief these days), you cannot have the law of the excluded middle.

https://en.wikipedia.org/wiki/Homotopy_type_theory

https://en.wikipedia.org/wiki/Intuition ... ype_theory

Indeed even in classical math we have the same problem. 20th century independence proofs show that in any axiomatic system of sufficient power, there are propositions that can neither be proven nor disproven. Just because you can't prove something true, does not necessarily mean it's false.
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Re: Science and the principle of noncontradiction

Postby TLK on January 17th, 2017, 3:09 pm 

someguy1 » Mon Jan 16, 2017 2:11 pm wrote:
TLK » January 9th, 2017, 5:09 pm wrote:I only asked about the principle of noncontradiction (law of contradiction or law of noncontradiction), but the same question applies to the law of the excluded middle.


LEM may not last another century. Intuitionism is making a comeback through, of all things, computer science. After all there are sets of natural numbers such that neither the set nor its complement are computable. In math, modern intuitionism is expressed via Homotopy type theory (HoTT), which lends itself to proof by computer.

Aristotelian logic is under attack not by anti-rationalists, but by science itself. If everything meaningful is computable (a common belief these days), you cannot have the law of the excluded middle.

https://en.wikipedia.org/wiki/Homotopy_type_theory

https://en.wikipedia.org/wiki/Intuition ... ype_theory

Indeed even in classical math we have the same problem. 20th century independence proofs show that in any axiomatic system of sufficient power, there are propositions that can neither be proven nor disproven. Just because you can't prove something true, does not necessarily mean it's false.


If I understand intuitionistic logic correctly, I can see in empirical sciences where it is possible that there are situations where truth values of propositions may be unknown and such a logic may be more useful than a bivalent logic. I'm not sure I of that, though. Can you give some examples in empirical sciences where intuitionistic logic is widely used and accepted? Or examples in empirical sciences where intuitionistic logic could be used effectively.

Again if I understand it correctly, accepting the use of an intuitionistic logic would strengthen the requirement to retain the law of noncontradiction even more than in classic logics. I am still very curious if anyone can give an example of an empirical science where contradictory statements are both accepted as true (or a single statement is said to be both true and false).
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Re: Science and the principle of noncontradiction

Postby TLK on January 17th, 2017, 4:15 pm 

If anyone is interested in these questions and has a few minutes to read what appears to be an informed argument, I think this paper is worth a read:

http://www.scielo.br/scielo.php?script= ... 1000100013

I'm not sure I agree with his conclusions and I don't have enough background to judge the accuracy of some of his claims, but he does say some provocative things; e.g.:
ABSTRACT

Any inconsistent theory whose underlying logic is classical encompasses all the sentences of its own language. As it denies everything it asserts, it is useless for explaining or predicting anything. Nevertheless, paraconsistent logic has shown that it is possible to live with contradictions and still avoid the collapse of the theory. The main point of this paper is to show that even if it is formally possible to isolate the contradictions and to live with them, this cohabitation is neither desired by working scientists not desirable for the progress of science. Several cases from the recent history of physics and cosmology are analyzed.

Keywords: Consistency. Cosmology. Contradiction. Logic. Physics. Relativity.


INTUITIONISTIC LOGIC

Intuitionistic logic is weaker than classical logic. Everything that can be deduced in an intuitionistic system can also be deduced classically. Of course, the interpretation is not the same in both cases (it is constructive in the first case, structural in the second).

The system of classical logic is obtained from intuitionistic logic by adding some new axiom, like Excluded middle (φ∨¬φ) or Double negation (¬¬φ→ φ).

Intuitionists were afraid of inconsistency. In fact, they were much more afraid of contradiction than classical mathematicians were. From that point of view, intuitionism is more strongly opposed to paraconsistent logic than classical logic is.

A great advantage of intuitionistic logic was supposed to be that it was safer than its classical counterpart. But Gödel proved in 1932 that intuitionistic logic is not safer than classical logic. Any classical formula can be translated into an intuitionistic one in such a way that any classical contradiction would immediately produce an intuitionistic contradiction.


Newton da Costa and other logicians have shown us that it is possible to build consistent systems of paraconsistent logic, systems that allow us to isolate the found contradictions and so preserve the inconsistent theory from collapse or "explosion". Paraconsistent logics are weaker than classical logic, they allow for fewer inferences to be drawn. It is this weakness that makes it possible to contain the deleterious effects of contradictions. But here lurks a problem. Many domains of science -from physics to economics- are extensions of underlying mathematical theories, and it is not clear that all the power of the underlying mathematics can be preserved under a paraconsistent reconstruction.
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Re: Science and the principle of noncontradiction

Postby TLK on January 17th, 2017, 4:17 pm 

Sorry - instead of revising the original post when I edited it, the forum put up the edited version as a new post
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Re: Science and the principle of noncontradiction

Postby someguy1 on January 17th, 2017, 5:43 pm 

TLK » January 17th, 2017, 1:09 pm wrote:Can you give some examples in empirical sciences where intuitionistic logic is widely used and accepted? Or examples in empirical sciences where intuitionistic logic could be used effectively.



Empirical science is a long way from the kind of philosophical considerations I mentioned. My only point was to indicate that intuitionism and the rejection of LEM are making a comeback these days through the influence of computer science and contemporary work on mathematical foundations.

I don't understand intuitionism or constructive mathematics in the sense that I don't understand any system of math in which the intermediate value theorem is false. But a lot of smart contemporary thinkers are going in this direction. I'm not qualified to address developments in empirical science that relate to these issues. I did link to the Wiki articles on Homotopy type theory and Intuitionist type theory. I don't think this kind of thinking has filtered to experimental science yet (if that's what you mean by empirical science). But we live in the age of computers, and computational thinking is going to become more and more important going forward. And as I understand it, you can't hold both LEM and the belief that everything important is computable.
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Re: Science and the principle of noncontradiction

Postby someguy1 on January 18th, 2017, 1:26 am 

Since you are interested in empirical science, I wanted to add an observation about the distinction between falsification and refinement.

To answer your original question, "Should the principle of noncontradiction be assumed in science?" I would say that this is pretty standard. If I am trying to prove X, and professor so-and-so proves not-X, then my research project is pretty much toast unless I can find an error in professor so-and-so's work. In that sense, the law of contradiction is operative in standard science.

But there's another sense in which the notions of true and false are too blunt to describe what's really going on. Suppose Newton proposes his theory of gravity, which we'll called X. Then professor Einstein comes along and falsifies X. But Newtonian gravity is not wrong; it's simply been refined and extended.

In other words if I say, "Newtonian gravity is true," and "Things fall up!" those statements are both technically false. But the second statement is false in a very different way than Newtonian gravity is false.

Point being that there is some subtlety involved. When professor so-and-so shows that theory X is false, it might be that X is flat out wrong. Or it might be that X is still correct, but only under restricted circumstances; and the "not-X" theory reduces to X under these circumstances. That's the case with Newtonian and Einsteinian gravity.

In that sense, the law of contradiction isn't absolute in science. Newtonian gravity is false in an absolute sense. But then again since scientific theories are merely models that fit experiments and are never the ultimate truth of reality, every scientific theory is false. In science, truth and falsity are nuanced and relative.

Isaac Asimov wrote a nice essay about this, The Relativity of Wrong. http://chem.tufts.edu/answersinscience/ ... fwrong.htm

From Asimov's essay:
My answer to him [someone who'd written Asimov a letter] was, "John, when people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together."

The basic trouble, you see, is that people think that "right" and "wrong" are absolute; that everything that isn't perfectly and completely right is totally and equally wrong.

However, I don't think that's so. It seems to me that right and wrong are fuzzy concepts, and I will devote this essay to an explanation of why I think so.


I hope my post has at least the virtue of being relevant to your question. My excursion into neo-intuitionism was off the mark.
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Re: Science and the principle of noncontradiction

Postby Scott Mayers on January 18th, 2017, 3:16 am 

Most science is based on assuming the three main laws of logic. (1) Law of Identity, (2) Law of Non-contradiction, and (3) Law of Excluded Middle.

Without delving into the specific depths of all the different interpretations, these three are actually just forms of one that demands "consistency". A "law" is about those things that are consistent or that have patterns or rules that are consistent. So most of science has defaulted to these traditionally.

I think that a partial divorce and confusion came about with quantum mechanics though because much of the interpretation of things like superposition HAVE to abandon these laws or prove inconsistent with other areas of science.

I've already argued before on this site that a non-consistent logic must be assumed prior since they can be closed over all logic and reality. But it would have to be such that it diverges into places (worlds) that have consistency versus those without consistency. I doubt any world could be completely ever closed consistently without becoming 'dead'. I personally believe that 'contradiction' acts as a function of force in its own right. If reality is inconsistent as a whole, reality competes to try to be consistent deriving our kind of world but motivated perpetually because it cannot possibly meet that goal.
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Re: Science and the principle of noncontradiction

Postby TLK on January 18th, 2017, 12:40 pm 

someguy1 » Tue Jan 17, 2017 2:43 pm wrote:
TLK » January 17th, 2017, 1:09 pm wrote:Can you give some examples in empirical sciences where intuitionistic logic is widely used and accepted? Or examples in empirical sciences where intuitionistic logic could be used effectively.



Empirical science is a long way from the kind of philosophical considerations I mentioned. My only point was to indicate that intuitionism and the rejection of LEM are making a comeback these days through the influence of computer science and contemporary work on mathematical foundations.

I don't understand intuitionism or constructive mathematics in the sense that I don't understand any system of math in which the intermediate value theorem is false. But a lot of smart contemporary thinkers are going in this direction. I'm not qualified to address developments in empirical science that relate to these issues. I did link to the Wiki articles on Homotopy type theory and Intuitionist type theory. I don't think this kind of thinking has filtered to experimental science yet (if that's what you mean by empirical science). But we live in the age of computers, and computational thinking is going to become more and more important going forward. And as I understand it, you can't hold both LEM and the belief that everything important is computable.


I hadn't realized that intuitionistic logic had an important role to play in computation. I appreciate you sharing that. I must admit that I don't understand what you mean by "belief that everything important is computable", though.
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Re: Science and the principle of noncontradiction

Postby TLK on January 18th, 2017, 1:21 pm 

someguy1 » Tue Jan 17, 2017 10:26 pm wrote:Since you are interested in empirical science, I wanted to add an observation about the distinction between falsification and refinement.

To answer your original question, "Should the principle of noncontradiction be assumed in science?" I would say that this is pretty standard. If I am trying to prove X, and professor so-and-so proves not-X, then my research project is pretty much toast unless I can find an error in professor so-and-so's work. In that sense, the law of contradiction is operative in standard science.

But there's another sense in which the notions of true and false are too blunt to describe what's really going on. Suppose Newton proposes his theory of gravity, which we'll called X. Then professor Einstein comes along and falsifies X. But Newtonian gravity is not wrong; it's simply been refined and extended.

In other words if I say, "Newtonian gravity is true," and "Things fall up!" those statements are both technically false. But the second statement is false in a very different way than Newtonian gravity is false.

Point being that there is some subtlety involved. When professor so-and-so shows that theory X is false, it might be that X is flat out wrong. Or it might be that X is still correct, but only under restricted circumstances; and the "not-X" theory reduces to X under these circumstances. That's the case with Newtonian and Einsteinian gravity.

In that sense, the law of contradiction isn't absolute in science. Newtonian gravity is false in an absolute sense. But then again since scientific theories are merely models that fit experiments and are never the ultimate truth of reality, every scientific theory is false. In science, truth and falsity are nuanced and relative.

Isaac Asimov wrote a nice essay about this, The Relativity of Wrong. http://chem.tufts.edu/answersinscience/ ... fwrong.htm

From Asimov's essay:
My answer to him [someone who'd written Asimov a letter] was, "John, when people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together."

The basic trouble, you see, is that people think that "right" and "wrong" are absolute; that everything that isn't perfectly and completely right is totally and equally wrong.

However, I don't think that's so. It seems to me that right and wrong are fuzzy concepts, and I will devote this essay to an explanation of why I think so.


I hope my post has at least the virtue of being relevant to your question. My excursion into neo-intuitionism was off the mark.



The important notion of falsification in science is definitely one of the things I was thinking about when I asked the question about the role of the principle of noncontradiction in science. I think your distinction between falsification and refinement is an important one - although in a very strict sense a refinement could be seen as being based on falsification of at least some crucial aspect of a theory opening up an avenue to a theory that doesn't have the same flaw(s). Falsification is a test of a theory and the theory may not pass. That does not imply that the "falsified" theory doesn't provide a fairly good approximation of how some patterns of events in nature unfold. The history of science is full of examples of theories that are good approximations giving way to theories that are better approximations - a process of refinement.

This gets into another question I have. "Findings" in science are always tentative - open to revision with new information. In that sense, in science can we really comfortably use a logic that "works" based on assessing the truth or falsity of statements? Can we express scientific notions in true or false statements and yet still maintain the inherent contingency of scientific statements?


- As I said in my earlier post, I found the information about intuitionistic logic to be really interesting so I'm glad you posted it.
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Re: Science and the principle of noncontradiction

Postby TLK on January 18th, 2017, 1:56 pm 

Scott Mayers » Wed Jan 18, 2017 12:16 am wrote:Most science is based on assuming the three main laws of logic. (1) Law of Identity, (2) Law of Non-contradiction, and (3) Law of Excluded Middle.

Without delving into the specific depths of all the different interpretations, these three are actually just forms of one that demands "consistency". A "law" is about those things that are consistent or that have patterns or rules that are consistent. So most of science has defaulted to these traditionally.

I think that a partial divorce and confusion came about with quantum mechanics though because much of the interpretation of things like superposition HAVE to abandon these laws or prove inconsistent with other areas of science.


It does seem like assuming consistency is crucial to at least some aspects of science. The assumption that events will unfold in the future in patterns that have been observed in the past is an example of something very basic about science that is about consistency.

As long as a scientist isn't making any ontological claims about the nature of things like waves and particles, I'm not sure that we have to abandon the idea of consistency with respect to QM. The Schrodinger equation can be used to predict the probabilities of quantum level events and do so very consistently. The Schrodinger equation has no necessary ontological implications, though. I can see the problem of consistency in the notion of superposition in the Copenhagen interpretation (ontology) of QM but I'm not sure that the many-worlds interpretation (ontology) of QM has a problem with consistency with respect to the concept of superposition. Could you explain what you mean somewhat more? I'm not sure I understand.


I've already argued before on this site that a non-consistent logic must be assumed prior since they can be closed over all logic and reality. But it would have to be such that it diverges into places (worlds) that have consistency versus those without consistency. I doubt any world could be completely ever closed consistently without becoming 'dead'. I personally believe that 'contradiction' acts as a function of force in its own right. If reality is inconsistent as a whole, reality competes to try to be consistent deriving our kind of world but motivated perpetually because it cannot possibly meet that goal.


I guess I'd have to see the arguments you have presented previously because, unfortunately, I am not understanding what you are saying here.
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