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Byron Jennings | TRIUMF | Canada

View Blog | Read Bio

The Siren Call of Logical Positivism

For every problem, there is a simple solution: neat, plausible and wrong.

The philosophers such as Rudolf Carnap (1891 – 1970) and the Vienna Circle considered logical positivism the received view of the scientific method.  In the early to mid twentieth century, it dominated the philosophy of science discussions but is now widely viewed as seriously flawed—or as A. J. Ayer (1910 – 1989), a former advocate, put it: “I suppose the most important [defect]…was that nearly all of it was false.” Pity. But it was good while it lasted. So, what is logical positivism? It is sometimes defined by the statement: Only verifiable statements have meaning—note verifiable not falsifiable. The doctrine included opposition to all metaphysics, especially ontology and synthetic a priori propositions. Metaphysics is rejected not as wrong but as having no meaning.

Logical positivism is very nice idea: we work only with observations and what can be deduced directly from them. No need for theories, models or metaphysics. I can hear the cheering now, especially from my experimental colleagues. It was partially in response to the revolutions in physics in the early twentieth century. Quantum mechanics and relativity completely upended the metaphysics and philosophy built around classical mechanics, so the logical positivist wanted to eliminate the metaphysics to prevent this from happening again; a very laudable goal.

So what went wrong? As Ayer noted, almost everything. First, metaphysics tends to be like accents—something only the other person has. The very claim that metaphysics is not needed is itself a metaphysical claim.  Second, observations are not simple. As demonstrated by optical illusions, what we see is not necessarily what is there.  The perceptual apparatus does a lot of processing before the results are presented to the conscious mind. The model of the universe presented to the conscious mind probably has more uncontrolled assumptions than any accepted scientific model. But that is what the logical positivists took as the gospel truth. In addition there is Thomas Kuhn’s (1922 – 1996) claim that observations are model dependent. While that claim is disputable, it is clear that the interpretation of observations depend on the model, the paradigm or if you prefer the metaphysics; something beyond the observations themselves.

Third as Sir Karl Popper (1902 – 1994) argued, in general, scientific models cannot be verified only falsified (and one can argue that even that is impossible, see the first post in this series).  Thus, Only verifiable statements have meaning would exclude all of science from having meaning. Indeed, it would exclude even the statement itself since the statement Only verifiable statements have meaning cannot be verified.

Logical positivism: neat, plausible and wrong. Well can anything be salvaged? Perhaps a little. Consider the statement: In science, only models that can be empirically tested are worth discussing. Not to be overly broad, I restrict the statement to science. The criteria in mathematics are rather different and I do not wish to make a general statement about knowledge, at least not here. Second, I have replaced statement with model since by the Duhem-Quine thesis individual statements cannot be tested since one can make almost any statement true by varying the supporting assumptions. In the end it is global models that are tested. Science is observationally based, so the adjective empirical. I use tested to avoid complaints about the validity of verification or falsification. Tested is neutral in that regard. Finally, meaningful has been replaced by worth discussing. To see why consider the composition of the sun. In the late nineteenth century, it was regarded as something that would never be known. At that point the statement “The sun is composed mainly of hydrogen” would have been considered meaningless by the logical positivists and certainly, at that time, discussion of the issue would have been futile. But with the discovery of spectroscopic lines, models for the composition of the sun became very testable and the composition of sun is now considered well understood. It went from not worth discussing to well understood but the composition of the sun did not change. I would consider the statement “The sun is composed mainly of hydrogen” to be meaningful even before it could be tested; meaningful but not worth discussing.

My restatement above does, however, eliminate a lot of nonsense; like the omphalos hypothesis, the flying spaghetti monster, and a lot of metaphysics, from discussion. But its implications are more wide ranging. During my chequered career as a scientist, I have seen many pointless discussions of things that could not be tested: d-state of the deuteron, off-shell properties, nuclear spectroscopic factors and various other technical quantities that appear in the equations used by physicists. There was much heat but little light. It is important to keep track of what aspects of the models we produce are constrained by observation and which are not. Follow the logical positivists, not the yellow brick road, and keep careful track of what can actually be determined by measurements. What is behind the curtain is only interesting if the curtain can be pulled aside.

To conclude: Don’t waste your time discussing what can’t be empirically tested. That is all that’s left of logical positivism once the chaff has been blown away. And good advice it is—except for mathematicians. Either that or I have been lured to the rocks by the siren call of logical positivism and have another statement that is neat, plausible and wrong!

Additional posts in this series will appear most Friday afternoons at 3:30 pm Vancouver time. To receive a reminder follow me on Twitter: @musquod



  • Kea

    You know, I really do enjoy your little essays, but I always find myself frustrated by them. Please don’t confuse logical positivism with empiricism!

    No one would argue that the early 20th century style of positivism is outdated, but to my mind the central problem in Theory today is a false anti-positivism, which completely fails to understand the core issue. The case in point is Mach. Mach was of course an anti-atomist, and is much laughed at today for persisting with this view. Now most physicists today have little or no education in philosophy or history of science, and so fail to realise where Mach’s anti-atomism comes from. It is an important lesson. The problem was the entrenched materialist ontology of the time, where one could not speak of atoms without invoking discrete objects in an a priori background, aether if you will. Mach always understood that the basic components of reality could not come down to such an ugly ontology, and he was right, as any decent modern theorist would tell you. Measurement principles are more fundamenatal than an emergent spacetime, and although today this is no longer logical positivism as such, I think logical positivism is a helpful anchor from the past, in the destructive sea of a materialism that is long past its use-by date.

  • It seems that LHC-experiments have falcified the basic models of theoretical physics (Higgs, a superstring and etc.). Now remains nothing except the logic analysis of the Standard Model.

  • Glad you enjoy them. The problem with logical positivism is that they try to go directly from observation to meaning with the necessary intermediate step of a model.

  • Kea

    You mean, WITHOUT the intermediate step? But if they truly did that, their theories and philosophies would be pure empiricism without any semantic depth, which they are not, at least not in the case of Mach! It is a question of what we permit to take on Meaning. You might argue that all true theories contain unobservable elements, and I would agree with that. But one can demand that the principles of the theory rely heavily on empirical fact, and that unobservable consequences are only derived features of the formalism. That seems very reasonable to me.

  • Torbjörn Larsson, OM

    Besides being a philosophical description, it sounds like an attempt to axiomatization. But that can’t be done in physics. (Say, quantization resists it, I am told.)

    “The very claim that metaphysics is not needed is itself a metaphysical claim.”

    It is is an observation. You know, what philosophy can’t handle, since you can’t falsify any philosophy. Thus philosophy has no bearing whatsoever on empirical matters.

    @ gunn:

    “Higgs … Standard Model”. This is inconsistent, since Higgs _is_ a part of the Standard Model.

    As it happens, LHC has not falsified anything yet. But have rather promising hints of a ~ 125 GeV Higgs.

  • Torbjörn Larsson:

    I mean Cтандартную model Sheldon Glashow’s — SM without Higgs (http://www.pteponline.com/index_files/books_files/quznetsov2011.pdf).

    Similar hints since LEP was observed not less than three times – and what?

  • It should be said that Carnap and other logical positivists recognized the basic element of the Duhem-Quine thesis that one does not test individual statements but whole sets of principles (funnily enough Carnap attributed the basic observation that we test theories against sets of assumptions to Duhem and Poincaré). The logical positivists and the Vienna circle were unified more in their attempt to apply the successful methods of what they saw as best practice in science (and mathematics), their particular positions were at least somewhat diverse.

    For example Carnap (who I’ve been reading a little lately), actually takes something like the approach you suggest. He takes it that even empirical questions are only meaningful and decidable in a “language” (which is a complete system of description and inference and may include positing things like physical laws). He takes it as standard scientific practice to recognize these languages/assumption and work within them or switch to other ones where it is useful. So all non-empirical stuff just becomes a convention of the language assumed and therefore tautology, observations are not tautologies. So he does end up with something like the only empirical statements have meaning, but the inferences from observations are neither fish nor fowl (neither tautology nor simply empirically given). He would reject metaphysical statements for having no meaning, but more because say exist only has a meaning in a system and is only provable in a system (the species [genus?] Kangaroo “exist” in biology given biology’s classification scheme, standards of evidence and observations made is a perfectly sensible statement for Carnap), metaphysical statements become nonsense by being misapplied (the species Kangaroo “exist” by some non-biological sense of exists). Carnap advocated for a neutrality about languages and deciding which ones to use on pragmatic grounds. I think this is messy and if it does not end up as one of the post-positivist positions (Kuhn, Poppers or Quine’s) certainly threatens to.

  • Ron Maimon

    This is a nonsense critique of straw-man positivism. The position of the logical positivists was coherent, but the formalization by Carnap had a few minor flaws, which were not particularly significant or important. The problem was academic politics— the positivists simply resolved the age-old mysteries of philosophy, and resolved them for good, by showing, not that they are deep mysteries, but gibberish and wankery. This creates a lot of hostility.

    You have stated the principle of positivism more or less as Mach stated it, but it needs more careful statement. Physicists starting with Mach, but not ending with Mach, developed positivism in the 20th century. Essentially all physicists continue to be positivists until today, without understanding that they are adopting Mach’s philosophy, because, in Einstein’s words “They have imbibed it in their mother’s milk”.

    The proper statement of positivism is that the sensory experience is the fundamental object which is invariant to theorizing away, you can’t theorize away a direct observation, so that it forms the foundation of knowledge. The other conceptions are important only to the extent that they have an impact, direct or indirect, on sensory experience. So you need to give a path from the object you are hypothesizing directly to a sensory experience to be sure you know what you are talking about.

    This principle is extremey important, and it is clarified by some examples from physics. The first notion which is clarified is the “electric field”. What is a “field”? The answer is that you can measure the field by placing charges at a position in space, and seeing how they accelerate. This procedure defines what the content of the field is in a postivist way, and allows you to be sure that the field is a sensible thing to introduce into a formalism.

    Another example is a “gauge”. The gauge of electromagnetism, unlike the fields, does not have any impact on observations— any gauge choice gives the same answer as any other. Does that mean that the postivists reject a choice of gauge? This is explicity false! Their experience was from GR and the choice of coordinate system there, Einstein’s positivism, and this experience made it clear that you CAN talk about a gauge, but you consider the question “which gauge is real” to be meaningless. That’s not the same as saying “you don’t need to choose a gauge”, or “gauge’s can’t be chosen”, or “gauges don’t exist”. It’s the statement that “gauge choice is immaterial, and any one choice is as good as any other, and you can translate freely between the gauges. No argument depends on choice of gauge in a crucial way, although some arguments might be simpler or clearer in one gauge rather than another.”

    This is the positivist insight that Carnap et al were formalizing in various degrees of formality in the 1940s and 1950s. It is not the same as saying that ONLY sense experience is admissible in a model of nature (as Mach sometimes overstated a little), because, as you say (and as Riechenbacher and Carnap understood very well), you need all sorts of apparatus to make sense of the primitive bit-values in the sense-data. It is next to impossible to disentangle sense-data from non-sense data, because you need to consider the data in the processing stages of the conscious mind, which, as you say, involves a lot of uncontrolled approximations.

    But there is a principle here, a simple one: within our given framework, we can talk about sense data. Within the enlargements of the framework to new concepts, there are questions which make claims about future or present sense-data, and questions which do not make any such claims. The answer to questions which do not make any claims about sense-data are A FREE CHOICE, they are up to you, just like a gauge in electromagnetism. You can translate freely from any answer to any other, and the question “which answer is correct” is nonsense, it is metaphysical, and has no answer because it has no meaning.

    This is the fundamental insight of the positivists. It can’t die, because it is true. It is used ALL THE TIME in physics, and it forms the core confusion that non-physicists have when discussing physics. If you don’t accept that a question which has no impact on future sense-data is arbitrary and meaningless, you can’t make the framework shifts required for the theoretical advances in physics.

    Here is a list of meaningless questions in physics (that are trivial once you understand logical positivism, but hopeless otherwise):

    1. Are gauge ghosts actual particles?
    2. Does the electron have a position in the ground state of Hydrogen, if you aren’t measuring where it is?
    3. Are confined particles, like quarks, actual particles?
    4. Do objects cross the black-hole event horizon or get smeared on the surface, never falling through?
    5. Are Green-Schwarz superstrings the same as RNS superstrings?

    I could go on all day. All these superficially sensible questions are obviously nonsensical in logical postivism, and to simply study physics, you can’t avoid imbibing the entire philosophy right at the outset, and definitely when you study quantum mechanics.

    The result is a complete disconnect between physics and philosophy, due to the deranged reaction against positivism in philosophy departments. The positivism is not “wrong”, it is RIGHT! It is the only right answer. There is no other answer, there is no POINT in looking for another answer, the question has been resolved, and it has been resolved FOR GOOD, and EVERYONE IN PHYSICS KNOWS IT, and gets PISSED OFF AT THE IDIOTS AT THE PHILOSOPHY DEPARTMENT FOR SCREWING UP THIS OBVIOUS TRIPE WITH POLITICAL OBFUSCATION.

    The problem here is that the discourse in philosophy is political, and permanently broken. These people are basically nitwits, and should not exist. The exceptions are those that do mathematical stuff, or study philosophy of political movements and things like this, but usually not. The problems stem from the subservience of philosophy to the hierarchical order of society, so that philosophers are genteel, high-class, well-spoken, mild-mannered, bourgeoise types. Science is done by proles, and logical positivism is the revenge of the proles.

    Thankfully, the internet exists, and so it can no longer be suppressed. Now, unlike in the 1950s, it can bury old philosophy for good.

  • “… you can’t theorize away a direct observation….”

    Optical illusions. Mirages. Uri-Geller-type closeup-magician (illusionist) techniques. Human neurological limitations, that can make rapid intermittent signals seem continuous, or a continuous beam of varying frequency (at the edge of perception range, e.g. violet/ultraviolet) seem intermittent. Pareidolia/apophenia, seeing patterns in what is merely random data, like animals in clouds or faces in mounds on Mars. Misunderstood observations, the effect of fool’s gold being consistently reported as gold, or vice versa. For how long has the Sun been “observed” circling the Earth, rising and setting just as does the Moon?

    And then there is the brain-in-the-box (or Mi-Go cylinder) setup, something like a Star Trek holodeck, Matrix simulation, or 3D sensurround videogame: perhaps all your sensory data are merely fakes being fed to you. (This idea goes back to Plato’s Cave.)

    I suppose the judges of the Salem Witch Trials could have made your statement, “you can’t theorize away a direct observation”, about Spectral Evidence, their witnesses claiming to see the spirits of the accused coming into court and tormenting them, invisibly to everyone else. That was solid enough evidence to get the accused hanged; perhaps the judges shared your logical positivism.

    So add dreams, hallucinations, suggestibility, and sheer imagination to the list, cf. N-rays.

    “The positivism is not ‘wrong’, it is RIGHT! It is the only right answer. There is no other answer….”

    You’re overlooking the distinction-and-difference between “verifiable” and “falsifiable” statements/hypotheses. Today we consider only falsifiable hypotheses to be “scientific” — it must be possible to test them and prove them wrong if they are, otherwise they’re not about science. Nothing is accepted as “true” except to the extent that it hasn’t been falsified yet.

    On this point, logical positivism’s “verifiability” is indeed the wrong answer.

  • Ron Maimon

    These are not what is meant by “direct observations” in positivism, these are all (false) inferences regarding higher order structure. Also, are you my personal minder? What are you doing commenting on a not-particularly widely appreciated debate in physics philosophy?

    A “direct observation”: I see the following jpg-data in my eyes. I hear the following wmv data in my ears. There is NO EXTRA STRUCTURE imposed over this, like “I see a face”, just the direct sensory data. You can’t theorize this away means this is irreducible data. This is what you build yout theory out of, whether that theory is “this is gold” or “this is fool’s gold”. “I am hallucinating” or “this is reproducible”.

    It can’t be lying to you, because, in positivism, it is what everything else is made of, it is the foundation of what you are after explaining. The notions of “fool’s gold”, “mirage”, “illusion”, “psychosis” are just more abstract categories which are used to make predictions about future sense impression. In the case of fool’s gold, what a gold-broker would say to you if you tried to sell him some, in the case of hallucinations, about some consistency properties of the impression.

    There is no further claim about reality than the statement that it involves a coherent meshing of impressions among folks who communicate, and there is no a-priori notions used to make sense of sensory experience, just what you are given in the sense data.

    It is important in physics, because when you have a scientific revolution, like relativity, quantum mechanics, etc, you end up changing all the high-level interpretation while the sense data is unaltered, so that it’s like finding out that a bunch of stuff you thought was gold is all fool’s gold. It is also important because putting together a notion of reality in quantum mechanics is impossible without positivism, and even then, it is weird and unreal. It is an essential part of modern physics, and it is rejected in philosophy, for stupid reasons.

  • With a mirage, you actually see the jpg of the oasis in your eyes; unfortunately there is not a real oasis where it appears to be (though there may possibly be one much further away), due to the heated layers of air bouncing light/images across the desert.

    Likewise, optical illusions can construct a jpg for your eyes in the absence of a physical object corresponding to what you think you’re seeing: a piece close up, a piece farther away, etc., and they all line up in your sight to create an object — but not for the people standing to either side of you. “Look,” you say, “there’s an ____!” And they look, but can’t see it. Confusion ensues.

    As for jpgs of pareidolia, surely you’ve seen pictures of clouds that “looked like” something or other; did they or didn’t they, to you? Did you see the face in The Face on Mars?

    Have you ever seen iron pyrite (“fool’s gold”), or pictures of it?

    I won’t bother to ask whether you’ve seen (“directly observed”) sunrises and sunsets, or the sun inbetween.

  • Ron Maimon

    You DON’T UNDERSTAND ANYTHING! The jpg of the oasis IS the basic data you start with. There is no interpretation on this, NADA, NOTHING. No “This is an oasis”, no “this is an illusion due to light curvature”, nothing but a jpg, a list of pixel intensities. That’s it, no extra structure, just the basic data. The interpretations, even the most primitive ones like “hey, that toast looks like Jesus” come from doing INFERENCE on the basic data, using formal logic (usually in your brain, sometimes on a computer), and being open to competely changing the inference based on new sensory impressions. When you are talking about illusions, you are talking about a sudden change of inference due to more careful observation, i.e. more data, with no change in the basic data at all.

    There is no a-priori requirement in positivism that the sensory impressions reported by different observers have to come together to make a coherent single picture of reality. This just is an observation that is true in the classical world, and one that arguably fails in quantum mechanics.

    The interpretation “There is an oasis there” is an INFERENCE, a THEORY. It is open to getting replaced with any other theory that explains the same sense data, including “I was hallucinating” or “my senses were misfiring”. All of these are plausible hypotheses, and they are deduced from the basic sense-data, which you can’t argue with, because this and Occam’s razor is pretty much all you have.

    That’s POSITIVISM.

    The basic data comes with NO INTERPRETATION AT ALL, not even the basic scanning of a human brain. It’s just pure data.

  • Yes, the problem is that the jpg of a mirage looks just like the jpg of an actual oasis. The “basic data” you get to work with is the same either way. But one set of “basic data” will lead you to a real oasis with real life-saving water, and the other will lead you into empty desert with dry sand where you will die of thirst. If you don’t think that matters, you have a very low life expectancy.

  • Ron Maimon

    Exactly— these things you talk about, the frameworks of interpretation, are all making predictions about FUTURE DATA, and you learn to make these predictions by synthesizing them from previous sense data, and making predictions about future sense data, and seeing which predictions are correct and which predictions are not correct, using new data.

    The framework for interpreting the sense data is subject to evolutionary change, and to some a-priori constraint, from Occam’s razor. if you like, you choose the simplest framwork among those that maximize your life expectancy. But since this is not Darwinian evolution, it is better to say, you choose the framework consistent with sense data so as to maximize the correct predictions of future observations.

    In positivism that’s ALL you are maximizing— the correct predictions for future observations. You make no a-priori assumptions that do not help in predicting future impressions, and you don’t give any credence to any structure that is not a sense-impression, or a framework for making predictions about sense-impressions.

    That’s how physics philosophy works, and it is essential to doing physics, because in quantum mechanics, the sense-data reported by different observers is not consistent with a traditional classical reality underneath, but with something else. Some take it to be a many-worlds reality, others with a purely positivistic unreality. So long as the predictions are unaltered, which point of view you take between those that predict the same impressions is a free choice, the two positions are not distinct, they are “equivalent in positivism”. The equivalence means that it is senseless to ask which framework is correct, when the sense-data predicted is the same. The question “which framework among these two predicting the same impressions” is correct is a Carnap non-question, it’s nonsense. Carnap called it “metaphysics”, but it includes questions like “is many-worlds correct, or Copenhangen correct?” which are not traditionally considered metaphysics. That’s the main upshot of positivism— identifying together as equivalent any two philosophical positions which are superficially different in their mental structure, but which predict the exact same impressions in all cases. A clear example from physics is the choice of gauge in electromagnetism— any two gauges make the exact same predictions in every case, so it is senseless to ask which is correct. Rather, you can choose one, and freely translate to any other, without thinking you are somehow changing anything at all, even though the mathematical description superficially changes between the gauges. Ditto for change of basis in quantum mechanics, change of picture in string theory, introducing ghosts, doing a different path integral, all things physicsts swap around all the time, but philosophers stop and say “hold it! Which one is right in reality?” A stupid question, because positivism shows that not only has no answer, it is not
    even a real question.

    This point of view regarding how knowledge is built up was accepted in philosophy from the 1940s to the 1960s, but went out of fasion in the 1970s, as people made up certain “problems” with it. The only problem is that is is correct, and cutting, in that it shows that most of philosophy is empty prattle, as it has no significance in predicting future sense-impressions, or organizing past ones.

  • More to the point: of the two approaches (“verifiability” or “falsifiability”) we could take toward the hypothesis “that is an oasis”, the better survival strategy is “falsifiability”.

    If we set out thinking “We’re going to prove that truly is an oasis”, then nothing but reaching it will satisfy us — even if it keeps receding and receding and receding… which seems like a dismal survival prospect.

    If we set out suspecting “That may not be an oasis, we must be alert for evidence it isn’t, and turn back if we discover it’s a mirage” — we’re less likely to be drawn further and further on, because the first disappearance will probably falsify the image for us.

    This scenario exemplifies why scientists should be trying to falsify their own hypotheses, and why it should set off quiet alarms in your mind when you hear someone say he’s trying to prove his own hypothesis “true”.

  • Ron Maimon

    That’s Popperian nonsense. There is no difference between verifiability and falsifiability, they are related by the operation of “logical not”. If the oasis recedes as you approach it, it is a mirage, and whether this is a falsification of “this is water” or a verification of “this is is not water” is trivially logically identical. The two statements are negations of each other!

    The distinction between “falsifiability” and “verifiability” is nonsense in formal logic, because they are obviously equivalent, since “not” is a primitive operation. So why is it that Popper was able to go on with this nonsense for so long?

    It’s because you need scientific induction to make frameworks. When doing inference that involve scientific induction, one direction of induction is usually easy (falsified!) and one direction of induction is hard (verified!) This asymmetry is due to the fact that theories usually make precise predictions in one direction only, for example “water will get closer to me as I walk towards it”, which are easy to check, because they are precise. Conversely, usually the other direction “a mysterious new phenomenon that looks like water” which usually doesn’t come with precise predictions, and which cannot be checked.

    But the reason is simply the asymmetry in induction produced when there is a precise theory. To falsify a precise theory is easy, because it predicts a ton of precise sense-data, while to verify the same theory involves an induction step, because at some point you need to accept the theory, which makes an infinite number of predictions, using only finitely many observations.

    The problem of induction in positivism is solved using a precise formulation of Occam’s razor which uses Kolmogorov complexity (ciomputer program length). The criterion of simplicity is made precise using the length of the shortest program which makes the predictions, and the statement of induction is that you continuously update the program predicting the sense-data to the Kolmogorov simplest one which is in accordance with the data so far.

    This principle includes both “Falsification” and “verification” and relates them by logical not. There is no difference.

    Popper made up this “falsification” nonsense so as to steal “verification” from the positivists. The positivists were usually politically incompatible with Popper, as a lot of them were socialists, and he was a traditional liberal. “Falsification” is prattle which is propped up because it sounds nice to an ignorant student, it is a primitive form of verification, and if you add induction (which you need to do anyway), it is completely equivalent to verification, except it allows Popper to steal credit from the positivists for this idea.

  • “There is no difference between verifiability and falsifiability….”

    As much as to say “there is no difference between the requirement to prove someone innocent [as in the Roman and Napoleonic Codes] and the requirement to prove someone guilty [as in the Anglo-Saxon and UK/US laws]”? There is no difference between the prosecutor and the defense attorney?

  • Ron Maimon

    While the negation of “the defendent is guilty” is “the defendent is not guilty”, the negation of “prove a defendent guilty” is “not prove a defendent guilty”. It isn’t “prove a defendent innocent”.

    And indeed, the job of the defence attorney in a modern state is to show that the prosecuter “did NOT prove the defendent guilty”. The job is not to prove that the defendent is NOT guilty. If you don’t know formal logic, you sometimes make ridiculous mistakes like this.

    These higher structures are very complicated anyway, they are human, involving abstract ideas of “fairness” and “justice” and “authority”, in a situation of extremely imperfect knowledge. They are many steps removed from direct sensory experience, or the reading of instruments. When you can acquire as much data as you like, through future careful controlled experiments, and when you are in a regime where all competing theories are required to be simple, even the difference between “prove A” and “prove not A” disappears.

  • “… the fact that there is a legal difference between ‘proving someone innocent’ and ‘proving someone guilty’ in terms of burden of proof does NOT mean that the same gap exists in formal analysis of scientific knowledge….”

    I think you missed the point. If there were “no difference between verifiability and falsifiability”, then that would apply to the question(s) of guilt/innocence as well.

    Yet the [Roman/Napoleonic] legal systems that placed the presumption on guilt and the burden of proof on innocence had different outcomes than the [Anglo-Saxon/UK-US] legal systems that placed the presumption on innocence and the burden of proof on guilt.

    That difference in outcomes is practical (affecting the real lives of real people), not just “legal”, and if your “formal analysis” can’t account for that, then it’s flawed as shown by this failure.

  • Ron Maimon

    I did not miss any point, the question of proving guilt and proving innocence are not related by “logical not”. They are treated asymmetrically because there is an asymmetry of power between the state and the individual.

    When two individuals are both suing each other, the burden of proof on the two sides is the same, and it is preponderance of the evidence. There is no requirement that one side have overwhelming evidence, unlike the case of a criminal trial.

    These asymmetries are entirely due to political factors, who has power, who has authority, which side is in a position to intimidate or coerce witnesses, and so on and so on. These factors were not present (and should never be present, although our world is imperfect) when you are deciding whether Brans-Dicke gravity is better than General Relativity, except to the extent that the scientists involved are also human, and also politicians.

    When you have scientific data and hypotheses, you prune the hypotheses using Occam’s razor, and you distinguish between them when you have explained ALL the data using one of the ideas, and ruled out the other explanation using the same data. If the data is ambiguous, you get more. This is a process which does not require political meddling, and this is the process which positivism places at the core of the theory of knowledge. This process also appears in a courtroom, as the testimony of expert witnesses, like fingerprint analysts, or DNA experts, who claim to be reporting the conclusion of such a process.

    When you are talking about a judge or jury, where they are making and testing hypotheses regarding some activity they aren’t sure about, these people have to build up much more complicated narrative structures of people, their relations, their exchange of money, and relate these to the law, to authority, to credibility, admissibility of evidence, and so on, and these structures are purely political. These political structures include asymmetric burden of proof, because of asymmetries of power, and they are a headache when you are evaluating scientific theories.

    Scientific theories are not immune from this sort of political nonsense, as Aristotelianism and Phlogiston show. Indeed, from a human point of view, because of the power-structure of established doctrine in science, you also do need to bend over backwards to listen to claims of crazy observations that conflict with established dogma, and keep testing the dogma, because there is always the possibility that the dogma is all wrong, and the conflicting observations were politically suppressed. But this is ultimately a game with credibility and politics, and in science, because you are dealing with nature, you can always resolve these disputes unambiguously by just repeating the experiments and observations until you know exactly what was going on, by making and testing the hypotheses repeatedly. The politics gets in the way, sure, but the burden of proof for a theory is to explain EVERYTHING, not just some things better, and the theory is only definitively accepted (in an idealization) once ALL the data is consistent with the theory, and the mistake which led to all inconsistent data is identified.

    It’s always the same theory of knowledge, but in the sophisticated areas, the knowledge is talking about very sophisticated abstractions, where you’ll never get a chance to do such a careful inquiry. In that case, you make a meta-theory about probabilities of different scenerios, and try to distinguish between their plausibility. Within physics, you see the pure mechanism of knowledge production, because the extraneous political complications are stripped away.

  • “… the question of proving guilt and proving innocence are not related by ‘logical not’. They are treated asymmetrically because there is an asymmetry of power between the state and the individual.”

    And apparently in your view, under the UK-US “presumption of innocence”, the asymmetry of power is all on the individual’s side, just as under the Roman/Napoleonic Code it was on the state’s side. Well, that’s a very benevolent view of the UK-US legal system!

  • Ron Maimon

    The assumption is that the state can gang up politically on an individual, and create charges out of thin air by selectively choosing witnesses, bribing or coercing informants to lie, create social consensus through media releases, and cherry-pick evidence that suggests guilt (like states tend to do all the time). The requirement of “beyond reasonable doubt” is to protect the individual from the massive power of the state. It was really designed to protect wealthy individuals from the various charges that come when the powers in government decide to put them away, because of the political resentment of others who are not so wealthy. But it works to protect dissidents and so on, more or less, from capricious charges also. It doesn’t work so well when private power is the one fabricating charges against an individual— sometimes private corporations can get an individual hounded or put away using similarly oppressive methods, as in the case of Aaron Schwarz. The issue is how you produce a free society, where the power is distributed broadly, and this has nothing to do with logical positivism.

    Look, this type of philosophy is valid and important, but very far removed from questions of semantics and knowledge which positivism is addressing. The positivism is just a foundation, it tells you how to acquire knowledge with confidence, it tells you nothing about how these complicated things are supposed to work.

  • “What are you doing commenting on a not-particularly widely appreciated debate in physics philosophy?”

    Hardly my first comment on falsifiability etc.; see for instance this from Aug.10, 1997, over 17 years ago.

    Or on examples of the history of mistaken science and medicine, over a year earlier….

  • “The distinction between ‘falsifiability’ and ‘verifiability’ is nonsense in formal logic, because they are obviously equivalent, since ‘not’ is a primitive operation.”

    Earth and not-Earth (the entire rest of the Universe) are obviously equivalent, as are you and not-you (the entire rest of the Universe), since “not” is a primitive operation. Oh, really?

    My goodness, has your study of formal logic not included Venn diagrams? A tiny circle marked “A”, within a universe-enclosing circle marked “¬A”? One is the exclusion of the other, but they are not thereby “equivalent”.

    Type I and Type II errors — incorrect rejection of a true null hypothesis or failure to reject a false null hypothesis, respectively — are likewise opposites, but not equivalents.

    Type I, a false positive, false alarm, may lead you to cry wolf when there is no wolf, react needlessly. But Type II, a false negative, failed alarm, may lead you not to cry wolf when there is a wolf, fail to react when needed — and be devoured. One outcome is worse than the other.

    A false negative on a disease test, vs. a false positive, these do not have equivalent effects on your health. Even though “not” is a primitive operation.

  • Ron Maimon

    Stop talking nonsense, and please learn formal logic. The operation of “not” is not the same thing as the operation of set-complement, it is a PRIMITIVE LOGICAL operation, not realized on sets, as the complement of a set in the universe is not itself a set. If you want to talk about logical operations, you should talk about classes. In this case, the class of Earth and the class of not-Earth are completely equivalent objects in set theories with classes, anything you can state about one, you can state about the other, and there is no difference between the statements despite your superficial untrained intuition.

    The operation of LOGICAL not (not set theoretic complement) relates verifiability and falsifiability in all cases trivially. The concept of falsifiability is exploiting an intuition that is false, namely that one direction of verification is always easy, and doesn’t require deep induction, while the other direction is always hard.

    This is in practice often the case, where the theory makes many precise predictions, but it is sometimes not, as for example, the big bang only makes a finite number of predictions, essentially the CMB fluctuations. Further, the theory that electrons are responsible for Faraday’s law has exceptions, sometimes the deposition is half the predicted amount. To account for this, you introduce doubly-charged ions, you don’t throw away the Faraday theory. This is not a fudge making Faraday theory untestable, it is a development that teaches you something new, that there are doubly charged ions.

    Theories in sociology are statistical and vague, although still knowledge (for example, that inflationary spending reduces unemployment, or how to control hyperinflation). These theories are full of exceptions, yet you muddle through using verification/falsification as best you can, and the main tool is Occam’s razor, and the ability to produce new hypotheses.

    The distinction between verifiability and falsifiability is never useful in scientific contexts. The important thing is that there are predictions for sense impressions, and that the framework is simple. These two constraints together obviate any need to distinguish between falsifying and verifying, which is good, because formal logic can’t distinguish these.

  • Ron Maimon

    Ok, but you should realize that the Popperian notions are not incompatible with positivism, but form a primitive approximation to the idea which is attempting to steal credit from the positivists, who came up with a better formulation first. This dispute is mostly over extremely minute details of what constitutes a “verification”, but I insist on the positivist formulation, because it is the one that is correct when you formalize logic.

    I should add that formalizing logic does not preserve all untrained intuitions. In particular, verifying that “all swans are white” is IDENTICAL to verifying that “all nonwhite things are not swans” formally. The intuitive distinction between them comes from the number of things that are swans as opposed to the number of non-swans, and the Baysian confidence you gain from observing something nonwhite and concluding it isn’t a swan is relatively small compared to the observation of a white swan.

    But this unformal intuition is simply being exploited by Popper to make a folk-philosophy which is less precise and less rigorous than positivism. But since it is mostly just a primitive version of the same idea, it doesn’t lead to infighting, and positivist like Hawking have even (mistakenly) attributed the positivism to Popper, despite his claim to be an anti-positivist! The reason is that the distinction between falsification and verification is essentially meaningless in the presence of Occam’s razor, and the dispute, when both positions are sufficiently precise, is nonexistent, as they are the same exact position (although Popper is emphasizing an intuitive but formally meaningless distinction, just to do academic politics using people’s untrained intuition).

  • “…. the complement of a set in the universe is not itself a set.”

    [Sigh] The complement of the set of all even numbers, in the universe of integers, is the set of all odd numbers.

    Wikipedia: “Absolute complement”, in “Complement (set theory)”

  • “… verifying that ‘all swans are white’ is IDENTICAL to verifying that ‘all nonwhite things are not swans’ formally.”

    Count: “all NONwhite things are NOT swans” involves a double negative, making its logical identity with the affirmative statement entirely unsurprising. “To recross is not to cross,” as G. Spencer-Brown wrote in Laws of Form; that is, the second annuls the first. What is inside the subset (or class, swans) is inside the superset (or class, white things), therefore what is outside the superset is outside the subset.

    But this is a far cry from claiming that the set (or class) of swans/Earth/you and the set (or class) of NON-swans/Earth/you are “equivalent”. You cannot verifiably assert of NON-swans that they are all white or all NON-white or all any other category of color (except perhaps “all colors” — which you could have said without bringing swans’ whiteness into it).

    That is, our NON NOT double negative ain’t told us nothin’ about items outside the subset (non-swans) yet still inside the superset (white) — of which there might be none or many, for all that formal logic based on “all swans are white” could say.

  • Ron Maimon

    The equivalence between “all nonwhite things are nonswans” and “all swans are white” is exactly the type of thing that Popper claims is asymmetric with respect to verification. It just isn’t. It isn’t quite a double negative exactly, that’s nothing at all in usual non-intuitionistic logic. This is the contrapositive, which is nontrivial.

    To show you a contrapositive without two “not” statements: saying “no swans are black” is the same as saying “all black things are not swans”. The word “not” is used exactly once in both places, but the two statements are identical in content.

    In formal logic, classes and negation behave the same way, so you can speak about not-you and you, so that “I am not tall” is the same thing as saying “all tall things are non-me”, and the statement can be understood as a formal statement about classes, and the evidence for the statement is the same as the evidence for the statement “I am not tall”.

    The rules for evidence are that you make a hypothesis, and test it using the predictions for sense-impressions. The tests are the same whether you are talking about the formal class, or it’s complement, the predictions don’t depend on this detail.

    The intuition Popper has is not necessary. All you need is Occam’s razor and predictions, and you can start culling theories and making better knowledge.

    Also, this dispute is very minor, as I am sure you see. The positivists would not complain about Popper, except to the extent that he was claiming credit for their work, and this is a political issue, not an issue of real philosophical dispute.

  • Ron Maimon

    The notion of complement is INSIDE A GIVEN SET. There is no complement relative to the UNIVERSE, because the UNIVERSE IS NOT A SET. This is why people talk about proper classes.

    Sets were ORIGINALLY designed to be collections where all the laws of logic had a counterpart in operations on a set. That was Boole and Frege’s original project. This idea DIDN’T WORK, because of Cantor’s and Russells’ paradoxes. So now sets mean something else, a tower of ordinally constructed objects without a notion of complement, while the word “classes” has taken over for objects which are constructed in correspondence with the laws of logic.

    You can sigh all you want, you don’t know logic or sets beyond the vague intuitions of the late 19th century, and pretending to understand formal mathematics doesn’t work.

  • “The notion of complement is INSIDE A GIVEN SET. … So now sets mean something else, a tower of ordinally constructed objects without a notion of complement ….” [boldface added]

    The above splendid self-contradiction makes it pointless to even argue with you, or again refer you to that linked section on “absolute complement” in set theory.

    “There is no complement relative to the UNIVERSE….”

    Try entering the command “define absolute-complement” in your browser’s address bar.

  • Ron Maimon

    The CONCEPT of absolute complement exists, it is just INCONSISTENT and NONSENSICAL and REJECTED by set theorists since around 1900. The concept is not included in any modern set theory (you can enter “Naive set theory” in your search bar to read about this).

    To see why the concept of absolute set complement is formally inconsistent— consider the empty set, take it’s complement in the universe, and consider the ordinals which are inside this complement, find the limit of these ordinals, and add one. This ordinal is not in the complement (since it is bigger than all the ordinals in the compement), but it should be. Contradiction.

    That’s just one of Cantor’s paradoxes. Another is “take the set of all subsets of the complement of the empty set”. This is larger in cardinality than the complement, but the complement supposedly includes all nonempty sets. This is another Cantor paradox.

    Knowing about these paradoxes, Cantor explicitly rejected considering the universe as a set, or taking complements of set in the universe. This advice was not appreciated by Frege and Boole, who wanted the laws of sets to mirror the laws of logic. Then Russell forced Frege to take Cantor’s objections seriously by stating the simplest formulation of this paradox, which demolished Frege’s first attempt at axiomatizing set theory.

    Russell’s paradox comes from observing that the complement of any normal set, like the empty set, must contain itself (since this complement contains all nonempty sets, including pathologically large sets like itself). Russell then divided sets into two types “pathological” sets which contain themselves, and normal sets, that don’t. He then asked “what is the set of all normal sets?”, i.e. “What is the set of all sets that don’t contain themselves?” This question is obviously paradoxical, as this set can neither contain itself or not contain itself, as a moment’s reflection shows. From this point on, everyone got it, including Frege, and the first good axiomatizations appeared a decade later, first with Russell’s theory of types, then with Zermelo’s axiomatic set theory.

    Axiomatic set theory goes along the lines suggested by Cantor’s intuition, and it is entirely without the notion of absolute complement. The sets in any modern set theory are built up from the empty set by the processes of unions, separation, replacement, and powerset, along with trivial things like pairing. The relevant axioms are “Zermelo Fraenkel set theory”, and these do not include the idea of universal complement. The axiom of foundation guarantees that all sets are built from the empty set by a quasi-finite process (meaning that set membership has no loops, if you look at a member of a member of a member of …. a given set, you terminate on the empty set after a finite number of steps). A good no-nonsense modern introduction to axiomatic set theory is in Paul Cohen’s “Set Theory and the Continuum Hypothesis”.

    To make logical operations into a “set theory” of the naive kind, Godel and Bernays reaxiomatized the laws of logic and sets so that they talk about “classes” and “sets” both. Classes are not sets, they are logical collections of sets, which are usually too big to also be sets. All sets are classes, but not all classes are sets. You can then talk about “The class of all sets that don’t contain themselves”, and this is not a paradox, because this class is not a set. This class is also the entire universe, because in modern set theory, no set contains itself.

    The formalization of set theory went in tandem with the development of formal logic. You need to understand these things to comment on them, not only because it is interesting, but because you are currently commenting with your head up your ass (although in good faith).

  • “The CONCEPT of absolute complement exists, it is just INCONSISTENT and NONSENSICAL and REJECTED by set theorists since around 1900. The concept is not included in any modern set theory…. Cantor explicitly rejected… taking complements of set in the universe.”

    Yet I only brought up the term “absolute complement” after *you* did exactly that:

    “…. the complement of a set in the universe….”

    If you’re going to speak in terms presuming the existence of such a complement, why should I not reply in such terms? Had you brought up a Euclidean framework, and I replied in those terms, would have you then have switched to non-Euclidean geometry and mocked me for talking about “NONSENSICAL and REJECTED concepts” — when you yourself had initiated the topic?

  • Ron Maimon

    The absolute complement DOES EXIST for classes, you simpering nitwit. I used it for classes, not sets. I always used the concept correctly.

  • Yet what you yourself wrote at the start of this “complement” subthread was
    > “…. the complement of a set in the universe….” [boldface added]
    > “…. the complement of a class in the universe….”

  • But the equivalence you claimed before was between “the class of Earth and the class of not-Earth”, and therefore here should be between “the class of swans and the class of not-swans” — yet you are making it between “the class of swans and the class of non-white things”, reversing the order of subclass/superclass inclusions (as you would have to for the negations, showing that the negations are indeed NOT equivalent to the affirmatives: A⊂B ∴ ¬B⊂¬A).

    Had Earth and not-Earth been entirely equivalent, as you claimed, one could sensibly state A⊂B ∴ ¬A⊂¬B — but one can’t.

  • “If you want to talk about logical operations….”

    Actually, I’d like you to address the part after Venn diagrams, pointing out the non-equivalent effects/outcomes of opposites:

    Type I and Type II errors — incorrect rejection of a true null hypothesis or failure to reject a false null hypothesis, respectively — are likewise opposites, but not equivalents.Type I, a false positive, false alarm, may lead you to cry wolf when there is no wolf, react needlessly. But Type II, a false negative, failed alarm, may lead you not to cry wolf when there is a wolf, fail to react when needed — and be devoured. One outcome is worse than the other.A false negative on a disease test, vs. a false positive, these do not have equivalent effects on your health. Even though “not” is a primitive operation.

  • … and that’s the basic problem, isn’t it? You’ve fundamentally misunderstood contraposition.

    Quoth Wiki: “The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of P → Q is thus ¬ Q → ¬ P .”

    The contrapositive of the statement has the same truth value, but this is altogether different from claiming that P and ¬ P (e.g. Earth and not-Earth, or swans and not-swans) are entirely equivalent “objects”, or that Q and ¬ Q are entirely equivalent “objects”, let alone P and ¬ Q….

  • Ron Maimon

    Read it again, you insincere git. The complement of a set in the universe is a CLASS, because every set is also a CLASS. Just not vice versa.

  • Ron Maimon

    Of course not, little buddy. The contrapositives I gave were correct. P and Q are PROPOSITIONS, i.e. sentences, not things. So P cannot equal “Earth”, because “Earth” is not a proposition. It can equal “The Earth is blue”, or “rain is wet”.

    Once you have CLASSES, then you can have propositions as objects, each proposition is the class of sets that satisfy it. When you have SETS and mathematical objects, and you assume “all mathematical objects are sets”. Then you can speak about “Earth” as a set, a VERY UNNATURAL set, the set of “Earth” considered as a mathematical object.

    But now you can’t speak about “not Earth” as a set, because “not Earth” is not a set. The complement of a set in the universe is always a class.

    So the proper symmetric way to treat “Earth” and “not Earth” as set-like things is to treat them both as proper classes. But it is still unnatural. The calculus of logic applies most naturally to propositions, to sentences. Class language just turns sentences with variables into nouns, the collection of objects for which the sentences becomes true.

    Please stop arguing for rhetorical points, it is obvious that you don’t know anything at all. Your insistence on runing your mouth without bothering to learn the most basic material is absolutely galling, and shows your disrespect for intellectual discourse.

  • Ron Maimon

    I know it’s difficult for you, but the contrapositive of the proposition “All swans are white” is “all nonwhite things are not swans”. It doesn’t look like the formal thing immediately, because you don’t understand formal logic.

    In formal logic “all swans are white” reads

    X is a swan -> X is white

    The contrapositive is

    not X white -> not X is a swan

    This is the most rudimentary formal logic. Please learn it first, and not just take ten seconds on Wikipedia to prove you never read a book.

  • No, Ron, the problem is that you have not been using the concept of “contrapositive” truth-equivalence to refer to statements but to classes, e.g.: “In this case, the class of Earth and the class of not-Earth are completely equivalent objects in set theories with classes, anything you can state about one, you can state about the other….”

  • “P and Q are PROPOSITIONS, i.e. sentences, not things. So P cannot equal “Earth”, because “Earth” is not a proposition. It can equal “The Earth is blue”, or “rain is wet”.”

    Then why did you think you could sensibly state: “In this case, the class of Earth and the class of not-Earth are completely equivalent objects in set theories with classes, anything you can state about one, you can state about the other….” ?

    If the Earth is blue, is not-Earth also blue? If the Earth has such-and-so diameter, mass, age, chemical composition, human population, etc., can the same truthfully be stated about not-Earth as well?

    Because, if this is how you’ve interpreted contraposition, then (in the words of Inigo Montoya) I do not think it means what you think it means.

  • “The complement of a set in the universe is a CLASS….”

    Ohhhhh, so there exists such a complement!
    Yet you’d said:

    “The notion of complement is INSIDE A GIVEN SET. There is no complement relative to the UNIVERSE…”


    “The CONCEPT of absolute complement exists, it is just INCONSISTENT and NONSENSICAL and REJECTED…. Cantor explicitly rejected… taking complements of set in the universe.”

    But now you’re saying (again) there exists, not just a complement of a class in the universe, but also a complement of a set in the universe? Are you sure this time?

    Given the contextual universe of “integers” (or, say, “cardinal numbers”), will you accept “even numbers” as a set? How about “odd numbers”? Are those two not “absolute complements” (of each other)?

  • Ron Maimon

    Get AWAY, you pestering locust. You are not competent to read.

    In PURE SET THEORY WITHOUT CLASSES, e.g. ZFC, there is no complement relative to the universe.

    In AN EQUIVALENT SET THEORY WITH CLASSES, e.g Godel-Bernays, the complement relative to the universe exists and is a CLASS, not a SET.

    Since ALL SETS ARE ALSO CLASSES, the complement of a set is its complement considered as a class.

    I never said anything inconsistent, I just assumed I was talking to someone who understood the first thing about set theory, not an incompetent twit.

    In a LIMITED UNIVERSE, you can talk about complements, but then it doesn’t correspond to Popper’s ideas, because these universes are too limited.

    Please stop wasting people’s time with nonsense. Your incompetence is galling,

  • Ron Maimon

    Since you insisted on conflating propositions with objects. I took the charitable interpretation where THIS IS ALLOWED. You make a mathematical model of the Earth and everything in it, and consider the Earth as a mathematical object.

    If the surrounding mathematical universe is set theoretic, then, the EARTH in the model, is a set, and it’s complement is a CLASS.

    Then any statement you make about the mathematical object “EARTH” is equivalent to a statement about the class “NOT EARTH”. And these are both classes, but only one is a set. This is the only way to talk about statements and their negations in a way so as to pretend to be talking about objects.

    If you talk about logical predicates, as you do 90% of the time, there is nothing to discuss. You don’t need to bring up classes, you just talk about the logical predicates.

    This stuff is extremely rudimentary, and it is appaling that you would discuss it without learning it, and further, pretend you have some knowledge in the subject.

  • Ron Maimon

    “Earth is blue” is equivalent to “non-blue is non-Earth”. The statement about classes is that the set/class consisting of the single mathematical object “EARTH” in your mathematical model is included in the class of “blue things”.

    It is also the statement that the absolute complement of the class of “blue things”, i.e. the class of nonblue things, is included inside the class “non-Earth”, the absolute complement of the class “Earth” (every set is also a class).

    That’s the complete and proper translation of all statements by someone who knows rudimentary logic. Notice, any statement about Earth, you can translate into a statement about the class “non-Earth”. Notice that complements require talking about proper classes. Notice that everything I said is both precise and correct.

    This is assuming a model where the Earth is a mathematical object, so that you can talk about it as if it were a mathematical object. This is not the usual way people speak, but it is an acceptable way to speak, because it’s not like “the Earth can be considered as a mathematical object” is a statement with any bearing on the senses.

    I USED THE WORDS CORRECTLY IN EVERY CASE, you simpering twit. The discussion here is simply an unwitting display of your own lamentable mathematical ignorance.

  • “Then any statement you make about the mathematical object ‘EARTH’ is equivalent to a statement about the class ‘NOT EARTH’.”

    … blue, such-and-so diameter, mass, age, chemical composition, human population, etc….

    According to your sentence quoted above, any statement “(1) Earth is X.” is equivalent to a statement “(2a) Not-Earth is X.” (Notice that Earth and not-Earth are the subjects [antecedents] of the sentences.) But in fact the truth-values of these two statements have no predictable relationship to each other. (It all depends on X.)

    Meanwhile the contrapositive statement, “(2b) Not-X is not-Earth.”, has the same truth-value as (1). The contrapositive flips what’s at the beginning and end of the statement, arrow (→), sub/super relationship (⊂).

    You seem to keep missing that part; it is no longer a statement with anything regarding the planet (or its negation) at the start. The Earth revolves; not-Earth does what? — there is no equivalent statement of what not-Earth does, because not-Earth is not the subject of a statement.

  • “‘Earth is blue’ is equivalent to ‘non-blue is non-Earth’.”

    Those statements have equal truth-values. But what you claimed before was:

    “In this case, the class of Earth and the class of not-Earth are completely equivalent objects…, anything you can state about one, you can state about the other, and there is no difference between the statements….”

    If so, then above you should be able to state “Earth is blue, not-Earth is blue” — because “anything you can state about one, you can state about the other, and there is no difference between the statements”.

    That’s what gives me the impression you don’t understand what you type.

  • Ron Maimon

    You are focusing on superficial linguistic accidental nonsense!

    “The Earth is blue” is logically

    “Earth class is inside blue class”

    and to the contrapositive

    “The not blue class is included in not-Earth”
    (not-Earth is object)

    and also

    “not-Earth includes not blue”
    (identical sentence where not-Earth is subject)

    Where you put the subject or the object is an immaterial detail of English grammar, depending on what form of verb you prefer to use, and has nothing to do with the formal sentence or the content of the sentence in a formal interpretation. There is no difference at all in the formal content of the last two sentences, and this is something one has to get used to when discussing formal sentences.

    The statement “not Earth contains not blue” is most definitely a statement stating a property of “not Earth”, just as “not blue is included in not earth”. It is also COMPLETELY EQUIVALENTLY a statement about the Earth.


    “The circumference of the Earth is 40,000 miles”

    is contrapositive equivalent to the statement

    “Not Earth includes all objects whose diameter is not 40,000 miles”

    Now it’s a statement about a property of the class “not Earth”.

    These formal things are not obvious from informal language, but they are not deep. At least you sound sincere again, not making rhetorical points.

  • “You are focusing on superficial linguistic accidental nonsense! … Where you put the subject or the object is an immaterial detail of English grammar….”

    At least you read my text as carelessly as you read and write your own. Take note this time:

    “Notice that Earth and not-Earth are the subjects [antecedents] of the sentences. … The contrapositive flips what’s at the beginning and end of the statement, arrow (→), sub/super relationship (⊂).” (or as quoth Wiki, “The contrapositive of the statement has its antecedent and consequent inverted and flipped….”)

    These are not “immaterial details of English grammar”, and it makes no difference whether you use active or passive voice to put one item or the other first in a sentence. “Antecedent→Consequent” and “Consequent←Antecedent” are identical statements because “what’s at the beginning and end of the arrow” remained identical.

    When you claim that “the class of Earth and the class of not-Earth are completely equivalent objects…, anything you can state about one, you can state about the other, and there is no difference between the statements….” — that leaves Earth and not-Earth as (equivalent) antecedents in otherwise identical statements: Earth→blue, not-Earth→blue, no matter how you phrase it grammatically. Your claim was senseless, and you have spent 8 full days yammering and throwing insults around in a fruitless attempt to cover up that mess on the carpet.

  • Ron Maimon

    The flipping of implication arrows in the logic description is a flip in the containership properties of the classes, which is completely insignificant, since either “contains” or “is contained in” are equally good properties of a class.

    “The Earth is blue” is a property of Earth.
    “The Earth is not a gas giant” is also a property of Earth.
    “The Earth in 1955 is a special case of Earth” is also a property of Earth.
    “The Earth did not exist in 10,000,000,000 BC” is also a property of Earth

    The contrapositive statements in class language is:

    “Not-Earth is contained in the class not-blue”
    “Not-Earth contains the class gas-giants”
    “Not (Earth-in-1955) includes Not-Earth and is not the full universe”
    “Not (Earth-in-10,000,000,000 BC) is the full universe”

    Every statement about Earth considered as a class can be converted trivially to the statement about the class not-Earth, as I explicitly did for you in the four examples. You can do it yourself now.

    The question of which way the containership relation goes is an idiotic red-herring you made up. All these statements are equally good properties of Earth (or not-Earth), as you can see explicitly in the examples, and the containership properties go every which way, depending on the sense of the statement.

    You can continue to advertise your twerpy ignorance, but it’s really not worth it. It’s better to read an elementary logic book.

  • “The flipping of implication arrows in the logic description is a flip in the containership properties of the classes, which is completely insignificant, since either ‘contains’ or ‘is contained in’ are equally good properties of a class.”

    “Equally good properties” for a class to have? Oh, yes, along with being regularly dusted, waxed, and polished for passing viewers to admire. They’re all good, Ron.

    What they’re not is equivalent. The difference between them, the effect of a “flip” (A→B vs. B→A), is not “insignificant”.

    Is it true that all Mafiosi are Sicilians (M→S)? Or is it true that all Sicilians are Mafiosi (S→M)?

    Sicilians may possibly contain Mafiosi or be contained in, say, “Europeans”; Mafiosi may possibly contain Sicilians or be contained in “gangsters” — so both classes could equally well both contain and be contained in other classes; but it is not insignificant which contains the other.

    Cause→effect. By your reasoning, flipping that statement to say “Effect→cause” would be equivalent, equally good, because that flip was “completely insignificant”.

  • “Every statement about Earth considered as a class can be converted trivially to the statement about the class not-Earth…”

    As contrapositives, inverted (that is, with the attribute also negated) and flipped (reversing positions on either end of the arrow). Yes, but these contrapositive statements about the one are not the same statements as about the other — which was the claim you made:

    “In this case, the class of Earth and the class of not-Earth are completely equivalent objects in set theories with classes, anything you can state about one, you can state about the other….”

    By that reasoning, all the “converting” you should need to do is substitute “not-Earth” for “Earth”, or “Earth” for “not-Earth”, in any of those sentences, and change nothing else: if Earth→blue, then not-Earth→blue; “anything you can state about one, you can state about the other”.

  • Ron Maimon

    As by now YOU VERY WELL KNOW, you insincere potted plant, “contained in” and “contains” are both properties of a class, one property is not distinguished as special over the other, so they are equally good— meaning neither kind of property is preferred over the other, and you can’t say “these containerships are properties of Earth” and “these other kinds of containership are properties of not-Earth”. Both containerships are equally well thought of as properties of either one, equivalently, because of the freedom in formal logic to negate.

    Your examples are mentally retarded. “All mafiosi are sicilians” is reversed to “all non-sicilians are not mafiosi” as I have explained many times, and all these statements can be interpreted as properties of “mafiosi”, “sicilians”, “non-mafiosi”, and “non-sicilians” equally well. There is no distiniguished interpretation, nor any preference for a statement over its contrapositive.

    The contrapositive just complements all classes and flips one kind of containership to the other, and neither is distinguished. So there is no significance to Popper’s idiotic false intuitive distinction between “verification” and “falsification”, or for that matter, to any other intuitive distinction which hinges on an informal intuition that there is some sort of asymmetry between negated statements. The formal language simply doesn’t allow you to treat these asymmetrically, when you speak precisely, despite Popper’s brain-damage.

    I explained how to do the translation from a property of a class to the property of the complement precisely above, so that you can see that it is easily done. What I was saying is precisely correct, and your response is MENTALLY DEFECTIVE RHETORIC. Do you honestly think that any of your examples are confusing? Obviously not. You just want to sound like you aren’t as ignorant as you have unfortunately revealed yourself to be. That doesn’t work with formal mathematics, fortunately. You are just busted.

  • Ron Maimon

    No, you do the conversion by doing a proper logical operation, by doing a contrapositive, not by your mentally defective substitution drivel.

  • You appear to have forgotten that it was you who wrote the ‘drivel’ that set the terms of the substitution: “… the class of Earth and the class of not-Earth are completely equivalent objects… anything you can state about one, you can state about the other, and there is no difference between the statements….” — which makes no allowance for inversion or flipping (i.e. for contraposition), but only for substitution of “Earth” and “non-Earth” for each other in otherwise identical statements.

    It was to that senseless claim of yours that I replied, and you are now on your ninth day of ducking, dodging, and diving to avoid admitting error, and even at this point apparently trying to pull a switcheroo and pretend *I’m* the one who made *your* senseless claim.

  • “Both containerships are equally well thought of as properties of either one, equivalently, because of the freedom in formal logic to negate.”

    “Equally well thought of”? Oh, my. Held in equally high esteem by their social peers? Invited to parties at Downton Abbey equally often? Members of equally fine exclusive clubs in London? *sniff* As if anyone cared! Here’s how you used “equivalent” the first time:

    “the class of Earth and the class of not-Earth are completely equivalent objects… anything you can state about one, you can state about the other, and there is no difference between the statements….”

    “No difference” excludes negating-attributes and flipping-arrows to differentiate them, Ron.

    The effect of that statement is “If you can state about Earth, ‘Earth→blue‘, then you can state about not-Earth, ‘not-Earth→blue‘, no difference between the statements.” Yeah, that’s a senseless claim, but it’s the claim you made.

    The terms of your claim did not allow differences like changing “blue” to “not-blue” or flipping antecedent and consequent, let alone both, to get “not-blue→not-Earth“.

    That’s not my doing, Ron. It’s your very own.

  • Ron Maimon

    The claim I made is accurate. Anything you can state about “Earth” you can state about “not Earth”, using trivial and obvious inversion and flipping, which I showed you by example. Your retarded interpretation of this statement shows that you must have suffered a blow to the head. I never made the claim that you do a simple substitution, nor could I even conceive in my wildest fantasy that ANYONE would think this, considering how stupid it is. Guess I was assuming I wasn’t talking to the victim of blunt head trauma.

  • Ron Maimon

    “Equally well” means that all statements are the same. You don’t consider one kind of statement special and the contrapositive different. none are special.

    Since you don’t understand, I see that, on the other hand, you must be a different kind of special.

    “No difference” does not exclude negating attributes and flipping arrows. It means the two statements, after the proper flipping of arrows, have exactly the same content.. It takes a really special person to misunderstand this obvious point.

  • “… all statements are the same.”

    The true ones and the false ones?

    The sensible and the senseless?

    Well, that explains the flurry of comments you’ve posted here!

    “It means the two statements… have exactly the same content.”

    And the problem is that you’ve claimed “the same content” can be expressed toward both the original class and its complement, toward Earth and not-Earth — that, as both classes are “completely equivalent objects… anything you can state about one, you can state about the other….” — thus if Earth→blue then not-Earth→blue, etc.

    To use the contrapositive, not-blue→not-Earth, is to state “the same content”not about not-Earth,* the apparent topic of the statement, but about Earth, the same topic as the original statement, due to the double-negative.

    * The contrapositive makes no claims about the colors of the rest of the universe, which may in fact contain many blues, blue stars, blue nebulae, etc., or be entirely blue for all that the single datum “Earth→blue” tells you in strict logic! It is in fact a statement about [whether or not a tested object is falsified as] the Earth. (Verification is not an option here.)

    That you think the contrapositive states attributes of the complement is exactly what shows that you do not understand contraposition.

  • “I never made the claim that you do a simple substitution….”

    Well, let’s take (2+2) on the one hand, and (3+1) on the other.

    “… anything you can state about one, you can state about the other, and there is no difference between the statements….”

    Yeah, they both equal 4, are half of 8, are positive integers, cardinal numbers, etc. — those are indeed “completely equivalent objects”, your claim in the third paragraph above is actually true about them, and simple substitution works. Had one of them been preceded by a minus (negative) sign, none of that would be true. So negation makes a difference.

    Yet you did not realize this. Hmm.

  • Ron Maimon

    You pathetic sap, you are simply crazy. there is no “double negative” in the contrapositive. Double negative is something else altogether, where two nots cancel each other out, because they are applied to the same thing. In a contrapositive, the two negations are applied to separate things, and don’t cancel each other out. It’s not a simple example of a double negative.


    “The Earth is not red” is a proposition. The contrapositive is “Not Earth contains class red”. Notice that there s only one negative in each statement. Nothing to cancel out.

    “The Earth is blue” contraposes into “Not Earth contains class not-blue”, and it is not a double negative either, despite your pathetically ignorant untrained intuition, because the two “nots” are acting on separate things.

    An actual example of a double negative is “The complement of the class not-Earth is blue”. This is a double negative equivalent to “The Earth is blue”. Double negatives are trivial in formal logic, because you can get rid of all pairs of consecutive nots. Contrapositive is less trivial, precisely because it is not a double negative. That’s the reason they gave it a separate name, you see.

    The contrapositive sentence converts a property of Earth into a property of Not-Earth. The Earth is in this class tells you that not earth contains this other class. It’s not the normal way you say “The Earth is blue”, but it’s completely logically equivalent. This is why it helps to study formal logic, because you can’t figure this stuff out just by blabbing ignorantly like you are trying to do. You have to use an organ called “the brain”, which organ, unfortunately, you are missing.

    The contrapositive of Earth properties give for each one a property of not-Earth. You can’t distinguish them formally from properties of Earth, you can’t say “these here are properties and these other statements are not properties” (or, rather, you can if you like being ARBITRARY, but hey, so far you don’t care about even seem to care about being correct, so maybe I shouldn’t presume). These statements contain the exact same content, except speaking about the complement. Moron

    There isn’t anything more to say about this, I explained this in detail several times, with detailed examples, and you continue to splatter diarrhea onto this page. Have you no shame, asswipe?

  • Ron Maimon

    I didn’t say “Earth” and “not Earth” were equal, you pap smear, I said that any property of one can be equivalently stated as a property of the other, and this is a simple not-even-counterintuitive fact from formal logic, just a fact that you are obviously far too dim to appreciate. Must be tough with so little light up there.

  • “There isn’t anything more to say about this….”

    Oh, that’s such a pity, Ron. So then you won’t have a chance to change the deep impression I’m sure you’ll leave in every future reader’s mind by your debatediscussion tactic of emulating a monkey throwing his droppings from a tree. Too bad. Of course, in the absence of having any sensible argument or other mode of persuasion, it’s unlikely that tactic would have changed anyway….

  • Ron Maimon

    Trust me, the tactic worked. I look like a monkey, you’re covered in poop, and I’m objectively right on the mathematical logic. In a debate between someone who knows something technical and someone who uses nice political language to disguise knowing nothing, the best way to remove authority from the idiot is to insult.

  • “… I’m objectively right on the mathematical logic.”

    Alas, no. Consider two fill-in-the-blank statements:

    (1) _____ is blue. (X→blue, i.e. X⊂blue.)

    (2) _____ contains not-blue. (X←¬blue, i.e. X⊃¬blue.)

    You may or may not have noticed that these are two different statements; they are not logically equivalent — although they may be “equally well thought of”, they will not have equal truth-table values. For instance, fill in the blank with a solid-blue object, and statement (1) will be true but statement (2) will be false. A solid-red object will reverse the situation.

    Now, according to your original claim — “… the class of Earth and the class of not-Earth are completely equivalent objects… anything you can state about one, you can state about the other….” — if we can state (1) about Earth, we can state (1) about not-Earth as well:

    (1) Earth is blue. [and this is true]
    (1) not-Earth is blue. [but this is false]

    Also, according to your original claim, if we can state (2) about not-Earth, we can state (2) about Earth as well:

    (2) not-Earth contains not-blue. [and this is true]
    (2) Earth contains not-blue. [but this is false in the sense of not-blue→Earth]

    Yet you think you were “objectively right“?

    Not until now, after repeated corrections to your distorted efforts, have I gotten you to even state a well-formulated contrapositive! Yet you refuse to admit the error in your original claim… let alone retract it.

  • Ron Maimon

    Hey, wow, you’re stupid again. The correct things that go in your “fill in the blanks” are X and THE COMPLEMENT OF X, not X and X.

    X is blue

    class (not X) contains class (not blue)

    These are the statements with identical truth tables, you challenged moron. I have explained this about a dozen times.

    not-Earth contains not-blue

    is equivalent to

    Earth is blue


    Earth is CONTAINED IN class blue.

    not the stupid thing you wrote.

    Again, the same thing a dozen times. You see, you insincere lying wretch, it doesn’t help to claim the same stupid thing 20 times when you are objectively wrong, and I call you an idiot to your face.

    Your claim regarding “solid blue” show you don’t understand the class attributes. Saying “Earth contains blue” does NOT mean that “Earth consists of oceans and land”, and has nothing to do with the solidness of color of the Earth, you idiot. What “Earth contains blue” REALLY means in this context is the nonsensical claim that the collection of the one thing “Earth” contains the collection of ALL BLUE THINGS, including every blue gas giant in the Andromeda galaxy. The collection “Earth” does not contain “the sky”, even though the sky is part of the Earth. It doesn’t contain “Krishna” either, even though all Hindus currently happen to live on the Earth. The formal statement about classes is very different from the idiotic meaning your mathematically untrained little brain gives to the informal statement.

    Now, you have been so stupid for so long, it is clear you don’t actually sincerely believe your own crap. Rather, you are trying to do propaganda. That fortunately doesn’t work when you have the mathematical sophistication of a zombie.

  • “The correct things that go in your ‘fill in the blanks’ are X and THE COMPLEMENT OF X, not X and X.”

    But Ron, you stated firmly that the two were “completely equivalent objects” and that “… anything you can state about one, you can state about the other, and there is no difference between the statements….”

    So, X or the complement of X, whichever one you use to fill in the blank, the statement will be equally true, according to your own original claim.

  • Ron Maimon

    Yes, I see that you are too stupid to understand what I type, so I explained it in excruciating detail below.

  • Ron Maimon

    Not exactly. That’s according to YOUR IDIOTIC MISUNDERSTANDING of my original claim. Nobody could possibly take that obtuse literal interpretation, because nobody else could be that stupid. It’s the interpretation of an autistic savant.

    “Anything you can state about one, you can state about the other” is correct, and “there is no difference between the statements” is also correct— the statements are contrapositives of each other. Only a lying twerp purposefully doing propaganda could pretend to get confused in this way.

  • “Anything you can state about one, you can state about the other” is correct….

    No, poor Ron: anything you can state about one, you can state as a contrapositive involving the other (and the complement of the asserted attribute, and with the implication-arrow flipped so antecedent becomes consequent and vice versa).

    This is not the same thing at all.

    The first suggests that the statement “_____ is blue” can have its blank filled in by either “Earth” or “not-Earth”, with equal truth, because the two are, as you said, “completely equivalent objects”. (Well, yes, if we were discussing an entirely blue universe….)

    The second suggests that “Earth is blue” ‘s counterpart would involve ‘not-Earth’, ‘not-blue’, and a reversed relationship such as “not-blue is not-Earth” — which (if “Earth is blue” is a statement-about-Earth) looks on its face like a statement-about-not-blue… and as a contrapositive reduces back to a statement about Earth. (In that sense resembling a double-negative.)

    … and “there is no difference between the statements” is also correct— the statements are contrapositives of each other.

    You’re contradicting yourself. The contrapositive statements “Earth is blue” and “What is not-blue is not-Earth” are indeed equally true… BUT they are NOT each making the same assertion (blueness) about “one and the other” (Earth vs. not-Earth)! They are both only making that same assertion with regard to Earth, and the second statement even hints at an opposite assertion with regard to not-Earth (though what if the universe is entirely blue?!).

  • “… below” ?

  • “Double negative is something else altogether, where two nots cancel each other out, because they are applied to the same thing.”

    I’ll give you a very simple counter-example from algebra, -2x=-y, where the two negatives are not applied to the same thing, they are on opposite sides of the equation, and for that reason can be canceled out to 2x=y. Likewise, -x>-y ∴ x<y … and note the necessary reversal of the “arrow”! Just like contraposition in logic.

    (Even in plain English, the examples of double negative given at yourdictionary com include several where the two nots are not applied to the same thing, e.g.: “Nobody with any sense isn’t going.” “It ain’t right to not paint the house.”)

  • Scott

    If -2X is equal to -y, in what way are the two negatives not applied to the same thing? Not negative y is equivalent to not negative 2X which is the same as saying 2x=y is = to -2X = -y; but as Ron pointed out, only after the appropriate flipping of arrows would that be true; only after this flipping can it be said that the content of the two statements is equivalent. That’s how I understand Ron’s description, but maybe I misunderstood?

  • A “double negative” in the self-cancelling sense would be -(-2x)=y, or for an English-language example, “We couldn’t not go.”