Three-valued logic deals with frictional paradoxes

Our new paper “Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes” by Michael Nosonovsky and Alexander D. Breki is published in Entropy 2019, 21(6), 620;


Paradoxes of dry friction were discovered by Painlevé in 1895 and caused a controversy on whether the Coulomb-Amontons laws of dry friction are compatible with the Newtonian mechanics of the rigid bodies. Various resolutions of the paradoxes have been suggested including the abandonment of the model of rigid bodies and modifications of the law of friction. For compliant (elastic) bodies, the Painlevé paradoxes may correspond to the friction-induced instabilities. Here we investigate another possibility to resolve the paradoxes: the introduction of the three-value logic. We interpret the three states of a frictional system as either rest-motion-paradox or as rest-stable motion-unstable motion depending on whether a rigid or compliant system is investigated. We further relate the ternary logic approach with the entropic stability criteria for a frictional system and with the study of ultraslow sliding friction (intermediate between the rest and motion or between stick and slip).

Painlevé paradox

An example showing two sliders of equal mass m connected by a rigid link with a constant length l at an angle φ with the sliding surface. The upper slider is frictionless, while the lower slider is frictional with the COF μ. An external positive force P is applied to the upper slider. The normal force R acts at the first slider, so that T = R/sinφ is the tension/compression force in the link, and F=μ|R|sign(V) is the friction force on the second slider (while V is the sliding velocity).

Two solutions exist when V<0 and μ tan⁡φ>2, one solution exists when V<0 and μ tan⁡φ<2, one solution when V>0 and μ tan⁡φ<2, and no solution when V>0 and μ tan⁡φ>2.

Historical implications

Throughout the history of pre-modern mechanics, the state of rest and the state of motion were considered two opposite states of a mechanical system, rather than rest being a special case of motion. This is because in Aristotle’s physics, no motion by inertia was possible, and motion always implied the presence of a moving force or an effective cause of motion.

Several paradoxes have emerged accompanying the opposition of rest vs. motion, including the classical Zeno’s arrow paradox, formulated by Aristotle as “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.”

Furthermore, it was a matter of discussion, whether rest can be obtained as a combination of two uniform motions, until the so-called Tusi couple was discovered in the 13th century. The Tusi couple is an imaginary device used for the Copernican astronomical model. The Tusi couple consists of two spheres with a smaller sphere rolling inside a larger sphere having twice the same diameter. A point on a smaller sphere performs an oscillatory motion. At an extreme point of its trajectory, the point changes its direction for an opposite one, so that the instantaneous velocity is zero. While the mechanism is similar to the crank-slider linkage, which converts rotation into the reciprocating motion, and it was known from ancient times, the Tusi couple demonstrated that continuous rotation can produce motion with instantaneous zero velocity, which was not obvious until the concept of instantaneous velocity was suggested.

It took significant efforts, until the motion by inertia (without cause) was discovered by Galileo in the early 1600s (in fact, Giuseppe Moletti had already established that objects of different weight fall with the same acceleration). This required the realization that friction is what prevents moving objects from continuous motion by inertia. Therefore, friction and inertia were in a complimentary relationship: without identifying friction, inertia could not be recognized.

H. A. Wiltsche pointed out that pre-Galilean Aristotelian mechanics studied natural occurrences as opposed to the study of phenomena (“the invariant forms that allegedly underline natural occurrences”) introduced by Galileo. The latter systematically excluded causal accidents as impediments, and friction largely fell as a victim in the search of refined and purified phenomena. Even today, despite almost universal occurrence of friction, it is studied by materials scientists and engineers much more often than by physicists.


In the classical Newtonian and Lagrangian mechanics of particles, a state of the system with N degrees of freedom is characterized by a set of coordinates and velocities. We have suggested an extension of this description with a logical variable, which may attain three values: true (the system at rest), false (the system in motion, which is assumed stable), and undefined. The undefined state can be interpreted as a paradoxical situation (when no solution or non-unique solution exists), as an instability, or as an intermediate state between rest and motion in a certain sense (e.g., the ultraslow sliding). The overall state of the system is given by the conjunction of the degrees of freedom. Thus, the system is at rest if all velocities are defined and equal to zero, the system is moving if at least one velocity is defined and non-zero, and the system is undefined otherwise.

Certainly, the actual behavior of a mechanical system does not depend on the logical apparatus used. The paradoxes are related to shortcomings of the models, which describe mechanical systems with friction. The main advantage of the proposed three-valued logical description is that it formally addresses and often resolves frictional paradoxes in mechanical models (when no solution exists or a solution is non-unique) in a natural way, and allows to clearly distinguish between stable motion, unstable solutions, and rest as qualitatively different states of a mechanical system. The proposed method can be applied to the analysis of unstable motion and to the analysis of situations, when intermediate states between rest and motion exist, such as the ultraslow motion.

Related blog entries

Paper about ultraslow sliding friction of “Jo blocks” in Appl. Phys. Lett. (December 10, 2018)

Who was Painlevé and why his paradoxes are so important for the study of friction (October 12, 2016)

The Missing Link between Nasir al-Din al-Tusi and Nicolaus Copernicus? (July 9, 2018)