Galileo's famous argument against the Aristotle's theory of falling bodies goes like this (taken from THEP forum, now defunct):"Let's say heavy objects do fall faster than light ones. Then it seems the heavier weight will fall with the lighter weight acting, as it were, a bit like a parachute. In that case, the two balls will together fall more slowly than the heavy weight would on its own. On the other hand, once the two weights are tied together and held out over the parapet, they have effectively combined their weights, becoming one greater weight... they must therefore fall even faster than the heavy weight would on its own." Contradiction. Hence weight has no effect on falling rates.

Some philosophers are very fond of this argument. Gendler uses it as a prototypical example of how "reasoning about particular entities within the context of an imaginary scenario can lead to rationally justified conclusions". Snooks goes further saying "it is striking that one leaves the falling balls example with something approaching certainty for its outcome". And Brown goes all the way and claims that Aristotle's theory is "self-contradictory", and we gain a priori knowledge here. The argument does give off that flavor of "synthetic a priori" reasoning, as in geometry but without images. But is it a proof or a fallacy? Even Gendler admits that some "obvious" premises are missing, and Atkinson even calls it a "non-sequitur" for similar reasons. But Galileo's logic is not questioned it seems. Shouldn't it be?

Let's replace weight with cross-section. If objects with smaller cross section do fall faster let's strap two together (next to each other, so that the cross-sections add up) and argue as above. Ergo, cross section can not affect the falling rate either. But objects with smaller cross section do fall faster, because of air resistance, and two objects strapped together with added cross sections will fall slower than each one separately, for the same reason. Something is wrong here, but the reasoning leading to a contradiction is essentially identical to the one above.

I don't think presence/absence of air matters, not that Galileo mentions anything about vacuum. What is most suspicious about his argument is exactly that it is so general. If it works the falling rate should not depend on any (additive) characteristic of objects whatsoever, regardless of the missing extra conditions. We could give those objects electric charges and turn on electric field of our liking, and the argument still seems to go through. But leads to a wrong conclusion that the falling rate is independent of the charges.

Question: It is assumed that Galileo's argument should work with something like "free fall in a vacuum" premise properly spelled out. But I don't see where such premise could be used, or how non-vacuum would alter the conclusion, or why Aristotle's theory is self-contradictory rather than just empirically wrong. Is the problem just in unspelled premises or is the reasoning itself logically flawed? Is there a logically correct "a priori" argument?

EDIT: I believe Quentin's answer gives the right reconstruction of what the a priori aspect of Galileo's argument amounts to. I'll rephrase it somewhat. Suppose bodies move under a single "motive cause" that determines the "fastness" of motion (this formalizes fall in a vacuum). The "cause" could be Newtonian force that determines acceleration, or something else that determines velocity as Aristotle thought, etc., as long as "the stronger the cause the faster the motion" holds. Suppose further that the cause is additive, i.e. its values add up when the bodies are strapped together. Then Galileo's argument shows that the proportionality constant ("weight" in the argument) between the cause and the rate can not also be additive (a.k.a. extensive).

The weakest link is the "additivity of cause" premise. It is true of Newtonian forces, but that part is empirical, and not in an intuitive way, unlike geometry. Quentin points out that two narrowly separated bodies falling differently than two touching ones leads to a counterintuitive discontinuity. That is true, but "touching" is not the same as "strapped together". The strapping introduces rigidity that turns two bodies into a single item, it is not a priori clear that the cause should move this item just as it would two bodies that are only touching. Aristotle would probably reject this premise, especially since to him falling is a "natural" motion rather than "forced". So his theory is not exactly self-contradictory or counterintuitive, but Galileo's argument is still valid under broader assumptions than just Newtonian physics.

EDIT 2: Found this paper that analyzes Galileo's argument in detail, and reaches the same conclusion.

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    I think that it is reductive to see only at the "logical" aspects of the argument. First of all, a scientific discovery is something more than "an argument". In addition, we have to take into account the historical and philosophical context; see at least Peter Damerow and Gideon Freudenthal, Exploring the Limits of Preclassical Mechanics : A Study of Conceptual Development in Early Modern Science (2nd ed 2004) Commented Dec 2, 2014 at 9:04
  • I'd agree with @Allegranza that logic as a tool in physics is not nearly as important as its made out to be; there is a certain kind of physics 'logic', normally put as physical intuition that is altogether more important. Commented Dec 6, 2014 at 10:36
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    Physics is about theory, one can always throw enough axioms to validate Galileo's a priori reasoning such as no internal stress were observed for any lopsided non-uniformly weighted object during free fall (thus same for your emphasized strapped case), or deems such reasoning flawed such as a body is as heavy as a mini blackhole which now we know such weight would surely confound the classic simple result massively intensively... Commented Sep 25, 2023 at 7:42
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    I would like someone to focus on the smaller body.
    – Hudjefa
    Commented Sep 25, 2023 at 8:15
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    Your cross section argument doesn't work. At least not as stated. Suppose that objects with a smaller cross section fall faster, then coupling a small and a big object would slow down the small object and adding the 2 cross sections would ALSO result in a slower object so there is no contradiction. You'd need to make the argument that adding a faster object to a slower one would increase it's velocity but adding the two cross sections would decrease it to have a contradiction.
    – haxor789
    Commented Sep 25, 2023 at 10:02

5 Answers 5


Galileo's argument shows that the magnitude which determines the speed of free fall must be intensive, not extensive.

In the case of charges, the relevant magnitude is charge per mass unit. Same in the case of cross sections: it is cross-section per mass unit. These magnitudes are intensive. When attaching two objects, you double the resistance of the air, but you also double inertial mass, so the two objects will not fall slowlier or faster than each separately (contrarily to what you are saying). Galileo's argument remains valid in this case. Idem in the case of charges attracted by a magnet: two charges won't be attracted faster when tied together, because inertial mass adds up as well as charge.

The premiss on which the argument rests is that a continuous change in the arrangement of a situation cannot produce a discontinuous change in the dynamics of these arrangement (change an arrangement a little and the dynamics should change a little). If an extensive quantity is involved in movement then the principle fails: there'd be a discontinuity between the dynamics of two spheres with an infinitely small gap and two connected spheres, which seems absurd.

In conclusion: you might quibble on the formulation, but given this premiss of continuity of change, Galileo's argument is sound.

EDIT: perhaps the argument is even more general and does not depend on continuous change. Consider a composite system S1+S2. The extensive magnitudes of S1 and S2 will add up while the speeds of S1 and S2 won't (you'll take average speed for the composite system). Insofar as nature's behaviour does not depend on how you name and group things, extensive magnitudes cannot be directly relevant for movement.

  • Your argument assumes that air resistance is a function of mass only and shape doesn't affect it. Or you are assuming that the heavy and light balls are exactly the same shape and that drag coefficients don't exist. Galileo's original experiment was with a large (heavy) ball and a small (light) ball. His thought experiment breaks down because the mass is linearly proportional to the volume, but the volume is not linearly proportional to the crossectional area. Also, tying them together will change the drag coefficient because the air path is changed (no air can flow past the joint). Commented Dec 2, 2014 at 15:46
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    He mostly got lucky that the balls were relatively similar and the drop was relatively short so the difference was too small to notice. If we did the same thing today with a pebble size ball and a car sized ball and dropped them from an airplane, there would be a clear difference - and Galileo would be very confused. Commented Dec 2, 2014 at 15:54
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    I am not assuming this at all, I observe that air resistance is a force, and the resultant acceleration is a=F/m (not F alone), that's the point. That makes acceleration proportional to an intensive magnitude (per mass unit) not extensive like mass. Galileo's argument is perfectly correct: acceleration cannot be proportional to extensive quantities, in pain of contradiction. Commented Dec 2, 2014 at 16:00
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    Just did some additional research outside of the OP statement, and it looks like Galileo meant for this example to be in a vacuum. I had assumed the context was natural air (because it was unstated in the OP). So, with air there are problems, but neglecting air his argument is sound. Commented Dec 2, 2014 at 18:07
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    Succintly the argument is: Speed/acceleration cannot be function of a quantity that adds up when objects are combined (instead of getting averaged) because speed/acceleration don't add up when objects are combined. This in ensured in newtonian physics by the inertial mass term m in F=ma (where force will generally be extensive). Commented Dec 3, 2014 at 7:34

Galileo's logic is correct, but an important part of his reasoning is not so explicit, that it need to be.

The main statement he bases on is that weight is additive. He uses this statement when regards compound body, made from two glued bodies, as the same as two bodies, connected by wire.

What we see as body's weight is it's attraction force towards Earth, which is guided by Newtonian law of gravitation. Newtonian law states, that the force of attraction is proportional to the body's mass F = K * m, where F is an attraction force (weight), m is a body's mass and K is some constant.

This proportionality is the physical cause of Galileo's law. The heavier bodies are attracted stronger, but they are more inert by the same factors (this is from another Newtonian laws -- three famous laws).


From the physics point of view, it is quite possible, that gravitation law was different, than Newtonian. I.e. the force of attraction to Earth could not be proportional to body's mass, but be guided by other rules.

For example, it could be guided so as Aristotle though, causing heavier bodies fall faster. In this case the picture would be exactly as Galileo described it: lighter body, tied to heavier one, would act like a parachute.

But these other rules would automatically cause the violation of weight additivity. We couldn't not use weighters to know the objects composition, for example, the amount of gold inside coins. Entire civilization would be different.

Galileo knew weight is additive and he based his brilliant qualitative reasoning basing on this fact. Any nowadays qualitative proof would be more complex.


Take the example of momentum and apply Galileo's logic. Momentum equals mass x velocity. If you double the mass, you double the momentum. A ball with twice the mass will have twice the momentum. If you take two balls, each having the same velocity, and tie them together, the new ball has more momentum than the heavier ball alone. The smaller ball does not act as a "parachute" to momentum.

Now, if you take the same example and say that the velocities are exactly opposite (equal magnitude but pointed in the opposite directions), then the resulting momentum would be in the direction of the heavier ball, but it would be less than the heavier ball alone due to the "parachute" effect of the smaller ball.

In the case of dropping the ball, Galileo is being contradictory in his decription of the smaller ball. One way is to say "this ball is fast and that ball is faster" and the other way is to say "this ball is fast and that ball acts against motion creating a negative force". The former adds the speed of the ball, the latter subtracts it.

Conceptually, he is equating "less" with "negative". This would be a fallacy of equivocation.

*Edit The above is only true when considering a real-life example with air resistance and other modern common sense knowledge. If you neglect air resistance (as Galileo did for in this example), then Galileo's statement is sound. Aristotle's view was so primative that it's hard to think about it in terms of his restricted knowledge. If a mass has a mystical set speed, then it makes sense that it would slow down a larger mass, but then they are the same mass and should be going faster, so the conclusion Galileo make is that this can't be true - which is correctly reasoned.

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    I also think that the flaw is in arguing "if one ball is slower than the other then it drags on it" in the first part of the argument, the second part leads to the correct conclusion (under Aristotle's theory). But I can't quite put my finger on what the logical issue is exactly.
    – Conifold
    Commented Dec 1, 2014 at 20:48
  • Are you looking for a formal fallacy? There may be one here, but the more obvious one seems to be the False Premise informal fallacy. Falling balls do not, in reality, act like a parachute in the way Galileo is claiming, so his conclusions are invalid, informally. Commented Dec 1, 2014 at 21:16
  • I see, but what is wrong with the parachute reasoning exactly since it seems intuitively convincing to many people?
    – Conifold
    Commented Dec 1, 2014 at 23:48
  • I assume that at this point we have left the logical fallacy argument and you are now asking about physics/math. In a vacuum, this whole conversation doesn't make sense, but Aristotle probably was drawing from observations like steel balls falling faster than feathers or paper in regular air. The reason a ball hits first is because it has a higher terminal velocity than the feather. Terminal velocity exists because the earth pulls you down but air exerts a force pushing you up (drag). Drag increases as a function of geometry - more aerodynamic shapes have higher terminal velocities. Commented Dec 2, 2014 at 0:36
  • Mass also contributes to terminal velocity. Higher mass pushes harder on the air and allows a faster terminal velocity. Terminal velocity is the point where gravity and air resistance are equal. Ok, so a parachute slows things down by decreasing the terminal velocity by using a huge shape to created air resistance. This air resistance is a negative force acting against gravity. That is why things get slower – because of the difference in the shape, not the weight. Commented Dec 2, 2014 at 0:37

Most of physics is logically flawed - the example I usually go for is that of the calculus which Newton used for various arguments in his Physics; it was Bishop Berkeley that pointed out the logical in the arguments; not really to kick physics down but to point out there are various kinds of 'reasoning' - which ought to be obvious when reflected on - the literary imagination is different from the logical; and the theological is different from the physical. I'm speculating here; as I do not know the history very well, but I suspect he was defending traditional Christian epistemology from various attacks which originated in a purely materialist doctrine.

Trying to put physics on a logically coherent foundation is a large question and a large project - there are various axiomatisations of Newtonian physics, and quantum field theory - its an ongoing project.

One of the biggest 'logical' holes in Physics is the use of Occams Razors; for example there is no logic that can prove there are only four forces. Perhaps, at the incredibly high energy levels when the universe is compressed to the size of a golf-ball a new powerful field comes into existence - who knows; possibly and most probably we can never know.

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    Yes, but Newton's logical gaps can be filled in, and were by Lagrange and Hamilton. I don't think the same can be done with this particular argument of Galileo's, it is fallacious even "in spirit" as it were.
    – Conifold
    Commented Dec 2, 2014 at 2:31
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    Physics is not logically flawed. This is logic what has gaps. I.e. logic itself, especially Aristotle's, is incomplete and insufficient to reason in physics domain.
    – Dims
    Commented Dec 2, 2014 at 16:06
  • @conifold:can you explain the 'fallaciousness' in spirit here? I'd say that this is essentially an argument 'in spirit' in terms of a limiting argument (take the spheres to a small limit), and also from symmetry (the two balls being identical as are the circumstances) - these are two very important principles in physics (and mathematics); that is in 'spirit'. Commented Dec 6, 2014 at 10:19
  • @Dim: yes, but did Newton 'know' that? After all it was thought after the development of mathematical logic by Boole that mathematics could be put on a logical foundation - and we know how that worked out ie Gödel. To put the calculus on a rigorous foundation took three centuries - ie Robinsons non-standard calculi and Brouwers/Kocks synthetic geometry Commented Dec 6, 2014 at 10:22
  • @Dim: logic that has gaps in it, in my book, is 'logically' flawed; in the precise sense that not every step can be justified; I agree though that physics 'logic' is different to the precise sense of logic established by Aristotle where every step needs justification, or contemporary logic. Commented Dec 6, 2014 at 10:30

It is true that that the gravitational charge could be different from the inertial mass, and there is no logical contradiction in that theory. It would pretty much be identical to the Coulomb force theory. In such theories, different objects would fall at different rates, but the rates would be unrelated to their inertial mass.

The statement that more massive objects fall faster is clearly stronger than that, because it makes the gravitational accelerations dependent on the inertial mass using an increasing function a(m).

Suppose you join two objects of different inertial masses together using a glue-force. The system must move with a common acceleration. When the system is in free fall, the accelerations due to gravity of the two components are different (by our assumption). If there was no glue force, then the distance between the components would increase.

But the glue force is an attractive force between the components which opposes this, by adding an acceleration to the heavier component in the opposite direction of its gravitational acceleration. Hence the common acceleration of the system is lower than that of the heavier component. Hence we reach the contradiction because the system has a higher inertial mass than the components, and yet falls slower.

So, while there are no contradictions with making the gravitation charge independent of the inertial mass, there is a contradiction with the idea of making the gravitational acceleration dependent on the inertial mass using an increasing function a(m).

In a theory where we make the gravitational charge independent from the inertial mass, the gravitational acceleration would be directly proportion to the charge and inversely proportional to the mass.

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