There Is No Difference Between A Proposal, Hypothesis, Model, Theory or Law

Ready for some Logic 101? Something real easy, I promise. We’ll use it in service to show why falsificationism is not that interesting, or useful, and we’ll need it in judging how good models, theories and the like are.

If you have a valid sound argument that proves a certain proposition is false, then your riches have doubled, for you also have in hand an argument that proves that the contrary of that proposition is true. Prove one, get one free!

By contrary I of course mean the logical contrary.

For instance, P = “It is raining”. And your argument is, “I’m looking out the window, and it’s not raining (and, implicitly, my observation is sound, and, implicitly, the meanings of all the words, grammar, and punctuation I’m using)”, then P is false. Which makes P’ true (as it is sometimes written), i.e. the logical contrary of P is true. In this case, that contrary is easy to speak: “It is not raining.” (Sometimes people write not P’ but Pc or other similar things.)

Easy promised, easy delivered.

Want even better news? This simple bit o’ logic applies to all scientific hypotheses, models, theories, laws and whatnot.

Now I claim, and with the blessing will prove to you, that all these labels speak of the same thing. Logically, there is no difference between a hypothesis, model, theory, or law. There may be, and are, many practical separations which are used as bookkeeping, tools to divide labor, and trust given to different labels, as it were. But logically speaking, there is no difference between any of them.

All of them take the form of a list of propositions speaking about some conclusion, just like the argument about the rain. That list of propositions, explicit and implicit, is the argument. In science, the conclusion, itself too a proposition, says something about Reality.

We did the “law” of gravity before. Quoting the argument portion:

“F = GMm/r^2,

where F is the force, G is a constant, M and m the masses of two bodies, and r the distance between them.”

As an argument, this is incomplete, as the linked article proves. It speaks of a part of Reality (the masses, for instance), but it doesn’t say how to apply it to real objects. For instance, the earth is here and the moon there, so what happens to both at some future point? We can’t tell just from this. That only means we take all those extra propositions as implicit when we speak of “laws” like this. They are there, but not written down. Just as the meanings of the symbols are there, even though not written down.

A hypothesis is the same thing. If all theses things hold, the hypothesis says, the “all these things” forming an argument, then this-and-such will happen in Reality, plus or minus. That “plus or minus” is optional. Think statistical or quantum mechanical hypotheses.

Again, the same is true for theories and models.

Of course, we speak of different parts of the arguments, those propositions in the middle, as it were, in all sorts of ways, giving them individual names, like with mathematical theorems that are components of most scientific theories. Or like how we speak of the force part of the “law” of gravity, as if it exists by itself (it does not). Yet the only way the “law” itself is accorded any weight is because of the tacit premises that made predictions possible.

With all that, here is what all theories, models, hypotheses, propositions, laws look like:

Premises -> Bit of Reality.

Which is to say, an argument. The Bit of Reality may be certain, or only probable, or even false. Some or all of the premises may be mere assumptions. Whatever.

Like with unconditional probability, which is impossible to write down, I invite skeptical readers to show how any of these, models etc., do not fit into the scheme I outline. Can’t be done.

For ease, I’ll call all entities models. They are all models because they are not Reality, but abstractions of it. See the original gravity post for details about the differences between “laws” (i.e. models) and causal powers.

Expanding the prior formula, all models are arguments of the form:

M = P_1 & P_2 & … & P_m -> Bit of Reality,

where each of these individual propositions P_i are often themselves compound propositions, containing logical ors, ands, ifs, and so forth, including observations. The tacit and implicit premises are all there, which again they are usually not written. Here they are all there. Which means even the simplest model is huge, rich in premises (word definitions are there!).

That makes the little “m” quite large, at least for any model of non-trivial importance. Because if there’s any math in the model, we have all the propositions supporting that math in the list, too. And so on.

Here’s the falsification part.

Suppose a model said Y was impossible. Y could not happen. It’s not conceivable that Y could occur, according to our model. M says, “Y is out of the question. So there.”

Yet, one day, Lo!, Y is observed!

This is just like the rain argument. We have proved our model is false. Which—we are now back at the beginning—means we have also proved its contrary is true! M’ is true, given the observation Y is observed. (That is a separate argument!) That contrary of M is this:

M’ = (P_1 & P_2 & … & P_m)’.

That only means that at least one of these propositions inside M is false; not that all are (though that could be true). And since each P_i may itself be compound, if P_i is the one false proposition, it could only be some element of P_i that is the offending part.

Now M itself has been (in our example) falsified. It must therefore be abandoned. Tossed out. Thrown away. Spoken of with disdain. And, worst of all, unfunded.

But that’s not what happens. When a model is falsified, especially a beloved or well funded model, it is tweaked. Some minor change in one of the P_i, is made, or more usually some new P_(m+1) is added. The result is not M, but something else which is called M. Since so much of the original M remains intact, it’s natural to call the new model M, too. Even though M has been falsified.

In practice, it’s not only small tweaks, but large ones, too. Surgery to the original M can be gruesome, like some aging celebrity whose face is pulled into the shape of a demonic cat. But still the new thing will be called M. Because there is something in that original M that retains a powerful grasp on the minds of those who love it.

Technically—by which I mean logicallyany change in M probative about Reality results in a new model. Even each new (probative) data point added results in a new model, logically speaking, though most people will go on using the same old name. That’s to make bookkeeping easier, and because the alternative is not known.

I keep saying “probative” because we could add any number of propositions in M that have nothing to do with the Reality spoken of, even necessary truths (which is like multiplying a simple equation by 1). But none of these change what M says about Reality.

Which means that to judge M we can only look at those propositions which have something to say about Reality.

That is enough for now. We’ve done a lot but haven’t yet put our finger on the real problem. Which is how some people judge the model’s goodness in reference to itself, and not in its performance. That will have to wait.

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Categories: Philosophy, Statistics

19 replies »

  1. Slightly related:
    Gravity is not a force. Why not? Because forces work through an exchange of energy. Gravity does not.
    Forces alter the kinetic energy of a particle, thus changing its momentum.
    Gravity alters the momentum of a particle, thus changing its kinetic energy.

    It takes an infinite amount of energy to accelerate a massive object to the speed of light, something which the gravitational attraction of a black hole does every day and twice on Sunday.

  2. It’s not the contrary that is proven, it’s another form of opposition, that is the contradictory. I know, it was a lapsus mentis, it happens all the time

  3. The contrary to white is black. If something is not white, it doesn’t mean it’s black, could be blue, gray or green. The contradictory is non-white. If something isn’t white, then, it’s non-white or not white

  4. “Suppose a model said Y was impossible. Y could not happen. It’s not conceivable that Y could occur, according to our model. M says, “Y is out of the question. So there.”

    Ahh…you mean like ahh…men in the lady’s room? (now that’s real power)
    Some say gravity is not instantaneous.

  5. Right – but convoluted. Consider the easy way: argue that there are no natural laws, only (usually partial) descriptions of natural processes.

  6. It might be helpful to describe the context of the “law” of gravity as discussed here and in the linked article. [It should be noted that the discussion here is confined to the classical context, without relativity (special or general) or quantum mechanics. We could discuss those things, but it would be even longer than this comment is now.]

    The statement:
    “As an argument, this is incomplete, as the linked article proves. It speaks of a part of Reality (the masses, for instance), but it doesn’t say how to apply it to real objects. For instance, the earth is here and the moon there, so what happens to both at some future point? We can’t tell just from this. That only means we take all those extra propositions as implicit when we speak of “laws” like this. They are there, but not written down. Just as the meanings of the symbols are there, even though not written down.”

    ignores the context in which “models” in physics are applied and evaluated against observations or experiments.

    It is true that “F = GMm/r^2” does not explicitly write down the definitions of the terms used, but those definitions of terms are nevertheless “there” and “written down” in that they are assumed in the statement of the relationship and are part of the general context of classical mechanics. In particular, Newton, who is credited with first writing down the above equation, previously had written down the definition of a force, namely the so-called Newton’s second law, F = ma. The second law might be viewed as a “law” but also can be viewed as simply a definition of the concept of force (measured in kg-m/second^2) which says that if an object with mass m (strictly speaking the mass in this equation is what is called inertial mass, defined as the proportionality constant for an object between a force acting on it and its acceleration) and the acceleration of the object, which is defined as the second time derivative of its position, a purely kinematic quantity. So, the second law says 1) measure the mass of an object, 2) measure the acceleration of an object then 3) the force acting on that object, whatever its origin, is equal to ma.

    [It is useful to note that these statements, in the form of equations, are vector statements for each spatial dimension so that when “acceleration” or “force” is mentioned it should be proceeded by “net” i.e. the mass times the net acceleration of an object equals the net force on an object. The “net” would apply to several different types of forces, or it could just apply to the different vector components of a force from a single specific interaction. In other words, a force of 1 newton (1 kg-m/s^2) in the x-direction and 1 newton in the y-direction would result in an acceleration of a 1 kg object of 1 m/s^2 in the x-direction and 1 m/s^2 in the y-direction, for a net acceleration of sqrt(2) m/s^2 in a direction along a line 45 degrees from the x-axis. The vectorial statement of the “law” of gravitation would be something like
    F_vector = m_1*m_2*r_hat1,2/(r_1,2)^2
    saying the force between object 1 with mass m_1 and object 2 with mass m_2 is equal to the product of the two masses times the vector pointing from one to the other divided by the square of the distance between the two, i.e. the force acts along the displacement vector between the two. The direction of r_hat should be written so that the force of m_1 ON m_2 is directed such that m_2 feels a force in the direction of object 1. Similarly, by Newton’s third law, the displacement is written so that the force ON m_1 exerted BY m_2 is equal and opposite to the force ON m_2. But here we can neglect to carry all that vector stuff along and assume it is implicitly understood.]

    Now back to the “law” of gravity. Understanding the definition of a force in classical mechanics, the “law” of gravity actually says an object of mass m in the vicinity of an object of mass M should have its motion described by

    ma = MmG/r^2 or a = MG/r^2, where a here means the acceleration of the object of mass m

    and where all the vector stuff is suppressed but understood such that m is accelerated TOWARDS M. The “law” also says the mass M in the vicinity of an object with mass m should have its motion described by (using capital A for the acceleration of capital M)

    MA = MmG/r^2 or A = mG/r^2

    where again the vector stuff is also suppressed but such that M is accelerated TOWARD m.

    So what the “law” of gravity actually says is that the acceleration of an object of mass m in the vicinity of another mass M should be given by the result of the above equations and we can numerically or analytically solve the differential equations (thanks in part to Newton, but also others) and integrate the acceleration twice for the position as a function of time in the future, using the measured initial conditions of the system.
    If m and M are similar in magnitude the integration is a little complicated (but do-able) since both m and M are accelerated and move about the center of mass. Fortunately, there are a lot of situations in which M is much much greater than m and the acceleration of M due to m is very, very small and can essentially be neglected, such as the solar system, where the sun has about 3e5 the mass of the earth, or about 5e6 the mass of Mercury. The acceleration the sun feels from Mercury works out to about 10 nanometers/s^2 and from earth about 20 nanometers/s^2 so for most calculations it is a very good approximation to assume the sun does not move due to the accelerations of earth or Mercury.

    Within that level of approximation, orbits of the planets (or any satellite around a much larger body) can be analytically or numerically integrated to predict things like: the orbits should be elliptical, the orbits should sweep out equal areas in equal times (as observed by Kepler, but not predicted quantitatively until Newton about 60 years after Kepler’s death) and various relationships between orbital radii and periods, etc.

    So the statement that F = MmG/r^2 does or does not say something about “reality” in some probabilistic formulation misunderstands 1) the definition of the terms in the equation (which are not independent propositions, but in fact definitions of the terms) and 2) that “reality” is not the object of the endeavor but instead simply a quantitative prediction of the future position as a function of time, given initial conditions, within the accuracy of any approximations made. The latter is rather gloriously achieved by the simple force law.

    One could make the same statement that the Coulomb’s force law,
    F = k* q1*q2/r^2 does not describe the positions of the charges at future times, but that would also be due to lack of understanding the definitions of the terms. One could just as easily argue that the expression for Coulomb’s law does not describe motions because the value of k is not explicitly defined, but that would be patently specious. Or one might state that the expression for the dissipative force of air resistance (i.e. drag) on a falling body F = (1/2)rho*v*2*C_D*A does not explicitly describe the position as a function of time, but that is only due to not knowing the details of the application of the force to a moving body. It would be like saying the recipe for a sauce does not describe the complete preparation of a dish.

    There are other assumptions in the “law” of gravity that are worth considering, namely that the classical “law” assumes that inertial and gravitational mass are the same. By that is meant that the mass in F = ma, i.e. the proportionality between any force and the resulting acceleration, is exactly the same mass that acts as a source for the gravitational force in mMG/r^2. This equivalence is called just that, the so-called weak equivalence principle. Galileo tested it at the Tower of Pisa by dropping different mass objects (and also using pendulums) and could conclude the two masses, inertial and gravitational, are equivalent to about 1 part in 100. Tests have gotten progressively more accurate so that now the equivalence has been determined to be within about 1 part in 10^(-15).

    In summary, mapping the “law” of gravity to some kind of probability statement about “reality” is a bit uninformed at best. Instead, the “law” simply gives a method to predict the future positions of one body under the gravitational influence of another body and it does remarkably well in doing so within the uncertainties of measurements of positions and masses of the bodies concerned. But like all “models” or “laws” in physics it is never “right” only “not wrong” so far. In fact, it was observed that the orbit of Mercury had a peculiar feature that its perihelion precessed (changed over time slightly) and the prediction of that precession based on the Newtonian “law” of gravitation as discussed here did not predict that movement within the uncertainty bounds of the measurements. This is one of the well-known cases in which Einstein’s formulation of gravity predicted the correct value (within measurement uncertainties) and the Newtonian theory does not. This is how the “laws” of physics work, they don’t make probability statements about “reality” they predict outcomes of measurements, in this case measurement of the future positions of massive bodies under the influence of gravity. When they predict outcomes that fall outside measurement uncertainties they are eventually replaced with a more accurate model. The Newtonian model still works in the limit that the general relativistic corrections become small, and for most purposes calculating satellite, planetary or asteroid trajectories using the Newtonian “law” predicts results well within measurement uncertainties.

  7. ”We’ve done a lot but haven’t yet put our finger on the real problem.”

    The Real Problem is a slippery devil. Just when you think you’ve gotten to the bottom of the mess and pinned the devil down he pops up again somewhere else, grinning. You can nail him six ways from Sunday and get his pointy butt deported but then he sneaks back across the border and in short order is mayor of your town.

    Briggs has chased down the Real Problem, tackled him, got him in a chokehold, put a knee on his neck, handcuffed, zip-tied, hog-tied, leg-ironed, straight-jacketed and convicted that slippery devil then got him locked away in Sing Sing on death row waiting for a date with Ol’ Sparky. But the devil starts a prison riot, escapes, is pardoned by the governor, and in short order is running for president. Hell’s bells!

    None of this deters Briggs. He just picks up his bag of nails, hammer, zip-ties, straight-jackets, et cetera, and sets off again like Javert hot on the trail of Jean Valjean. Like Holmes after Moriarty. Like Snydley Whiplash chasing Sweet Polly Purebred. Never mind that last one. At any rate, it’s always instructive and a jolly good show.

  8. Seeing the names Snydley Whiplash and Polly Purebred after 50 years is icing on the cake of a great article! But, wasn’t Snydley chasing Penelope Pitstop? Or, maybe he jumped cartoons – transUnderdod

  9. If the sun disappeared in an instant would it take eight minutes for its
    absence to be felt on earth?

  10. Good luck explaining these nuances to your College Physics 1301 freshmen when they are in the hypothesis/theory/law discussion of Chapter 1.

  11. Newton’s second law, F = ma

    Which says that the acceleration [a] on an object subjected to a given impressed force [F] is determined by its mass ]m]. We can test this by dropping a dollar bill from the Tower of Pisa at the same time as we drop a crumpled up dollar bill. They will not experience the same acceleration, demonstrating that mass is not the only factor in play. Like all scientific laws, F=ma is ceteris paribus, “everything else being equal.” So, we exclude light bodies, restrict to motion of uncharged bodies, in a vacuum, et cet. par.

    Newton failed to solve the Moon’s orbit because it is a three-body problem. The Sun is also highly influential. There is no analytical solution, only a numerical approximation. The law is ‘good enough’ fapp [for all practical purposes] when you trade off precision for term, say high precision/short term vs. low precision/long term. The model is ultimately chaotic.

  12. Pk,
    Snidely was always after Nell, Dudley Do-right’s GF. Polly Purebred was Underdog’s squeeze. Her nemesis was Simon Barsinister.

  13. Science is now the state religion. Unlike previously established religions it doesn’t have to have any connection to phenomenal reality or the human condition, e.g. String Theory.

  14. Two anecdotes: I started my career as an experimental physicist in controlled thermonuclear research. “I gave you an MHD model, special price, just for you.” There was an “altnerative design” (not tokamak) device at MIT where I worked post grad. The model predicted (by a very 70s stoned hippie) that a population of hot electrons at each end of the machine would stabilize the “plasma”. After a year, we finally turned on device without the “electron anchors”, and then we turned on the electron anchors and, Ooops! Plasma — HE GONE! At our next staff meeting, I raised my hand and said “So what are we going to do for the next three years? The electrons destabilize the plasma; they don’t stabilize it.” Chief Scientist put on his politics hat, “we’ll tweak this, yadda yadda”, but Mr Big Mouth here didn’t endear himself to Boss Man. Anyway, we ended up doing lots of useless face time at the lab, but it wasn’t all bad.

    Second anedote: worked as equity analyst for a big bank in London. To “help” me to structure a equity research report, my boss said “You gotta put your earnings and valuation model at the beginning. That’s all investors want to see”. My reply: “Why should I start with the most bullshit section? “

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