Mass additivity and a priori entailment


The principle of mass additivity states that the mass of a composite object is the sum of the masses of its elementary components. Mass additivity is true in Newtonian mechanics but false in special relativity. Physicists have explained why mass additivity is true in Newtonian mechanics by reducing it to Newton’s microphysical laws. This reductive explanation does not fit well with deducibility theories of reductive explanation such as the modern Nagelian theory of reduction, and the a priori entailment theory of reduction that is prominent in the philosophy of mind. Nonetheless, I argue that a reconstruction of the explanation that incorporates distinctively philosophical concepts in fact fits both theories. I discuss the implications of this result for both theories and for the reductive explanation of consciousness.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    In special relativity the mass \(\hbox {m}_{\mathrm{c}}\) of a composite composed of \(N\) particles each with energy \(E_i\) and momentum \(p_i\) is \(m_c =\left[ {\left( {\mathop \sum \nolimits _{i=1}^N \frac{E_i }{c^{2}}} \right) ^{2}-\left( {\mathop \sum \nolimits _{i=1}^N \frac{p_i }{c}} \right) ^{2}} \right] ^{1/2}\). See Okun (1989, p. 632; 2000, p. 1271).

  2. 2.

    The name comes from Dizadji-Bahmani et. al. (2010, Sect. 3.1) who contrast this view with Schaffner’s (1993, ch. 9) own so-called generalized reduction-replacement model. The latter is stronger as it involves specific claims about bridge principles that the former stays neutral on. See also Schaffner (2012).

  3. 3.

    For AET see Chalmers and Jackson (2001), Gertler (2002), and Chalmers (2012). For the use of AET to argue for the irreducibility of consciousness see Chalmers (1996).

  4. 4.

    “As regards active specialists they answer in perfect unison insofar as their scientific work is concerned: the mass is independent of velocity, it is not additive [...] there is no disagreement among researchers on the definition of mass. [...] According to modern terminology, both terms, ‘relativistic mass’ and ‘rest mass’, are obsolete” (Okun 2000, p. 1270). Compare Field (1973, p. 469) and Feyerabend (1962, pp. 80–1).

  5. 5.

    Examples are Kibble and Berkshire (2004, pp. 11–12), and Lindsay (1961, pp. 19–21). Feather (1966) is an important attempt to generalise these explanations, see Sect. 4 below.

  6. 6.

    E.g. let two one-dimensional particles have unit mass, and be located two meters apart at \(-1\) and \(+1\) respectively so that the origin is at 0. In that case (15) puts the composite at 0. But move the origin so the coordinates of the particles are now 99 and 101 and (15) puts the composite at 200\(\sqrt{2}\).

  7. 7.

    See Lewis (1983, pp. 370–373; 1984). Naturalness is not an external constraint on reference but is imposed by our conventions: Schwarz (2014). A formal account of the semantics of theoretical terms governed by such conventions can be given in terms of the so-called Unique Best Deserver semantic theory, see: Elliott et al. (2013, Sect. 4.1).

  8. 8.

    The choice of subscripts comes from Dizadji-Bahmani et al. (2010).

  9. 9.

    One might object that ‘composite’ is not contained in the reducing microphysical theory, but is clearly in the statement of mass additivity. But here one could treat unrestricted composition (or one’s preferred composition principle) as a logical or a priori truth. Alternatively one could follow Dizadji-Bahmani et. al. (2010, p. 404) and treat the correct composition principle as an entity association law that is part of the reducing theory, as opposed to a property association law, which can’t be.

  10. 10.

    Even this is not clear given the Force Additivity Problem. Thus Nagelians require my solution to the Force Additivity Problem, or something similar, to work.

  11. 11.

    See Kim (1999) and Chalmers (2012, pp. 304–305). In what follows ‘consciousness’ means phenomenal consciousness: Chalmers (1996, p. 11).

  12. 12.

    And indexical truths (p. 408)—a complication I ignore for present purposes.

  13. 13.

    See also Kim (1992, p. 127), McLaughlin (1997, p. 38), and Chalmers (2012, pp. 291–292)

  14. 14.

    To make the analogy even more vivid, let M\(_{\sim A}\) instead say ‘something is massive’ and let P stand for a simple microphysical description of some electromagnetic radiation composed entirely of zero-mass photons, given in the language of special relativity. In that case (d’) is shown to be false in Gabovich and Gabovich (2007).

  15. 15.

    Gabovich and Gabovich (2007), Okun (2009).

  16. 16.

    Prima facie to deduce \(F_1 =-F_c \) just infer that C exerts \(F_1 \) on particle 1 in virtue of C’s components exerting \(F_1 \)on particle 1. Then apply the third law to C and particle 1 to deduce the force on C: \(-F_1 \). But the third law cannot obviously be applied to more than two particles: K&B (2004, p. 7) and the applicability of the third law to composites is something we should deduce, not assume. So let’s try another option.


  1. Butterfield, J. (2011). Emergence, reduction and supervenience: A varied landscape. Foundations of Physics, 41, 920–959.

    Article  Google Scholar 

  2. Chalmers, D. J. (1995). Facing up to the problem of consciousness. Journal of Consciousness Studies, 2, 200–219.

    Google Scholar 

  3. Chalmers, D. J. (1996). The conscious mind. New York: Oxford University Press.

    Google Scholar 

  4. Chalmers, D. J. (2002). Does conceivability entail possibility? In T. Gendler & J. Hawthorne (Eds.), Conceivability and possibility. Oxford: Oxford University Press.

  5. Chalmers, D. J. (2012). Constructing the world. Oxford: Oxford University Press.

    Google Scholar 

  6. Chalmers, D. J., & Jackson, F. (2001). Conceptual analysis and reductive explanation. The Philosophical Review, 110(3), 315–361.

    Article  Google Scholar 

  7. Diaz-Leon, E. (2011). Reductive explanation, concepts, and a priori entailment. Philosophical Studies, 155, 99–116.

    Article  Google Scholar 

  8. Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2010). Who’s afraid of Nagelian reduction? Erkenntnis, LXXII(3), 303–322.

    Google Scholar 

  9. Einstein, A. (1948). Letter to Lincoln Barnett, 19 June 1948. partially reproduced in Lev B. Okun. “The concept of mass”, Physics today. 1989 June.

  10. Elliott, E., McQueen, K., & Weber, C. (2013). Epistemic two-dimensionalism and arguments from epistemic misclassification. Australasian Journal of Philosophy, 91(2), 375–389.

    Article  Google Scholar 

  11. Feather, N. (1966). The additivity of mass in Newtonian mechanics. American Journal of Physics, 34(6), 511.

    Article  Google Scholar 

  12. Feyerabend, P. (1962). Explanation, reduction, and empiricism. In H. Feigl & G. Maxwell (Eds.), Scientific explanation, space, and time. Minnesota studies in the philosophy of science III. Minneapolis: Minnesota UP.

    Google Scholar 

  13. Field, H. (1973). Theory change and the indeterminacy of reference. The Journal of Philosophy, 70(14), 462–481.

    Article  Google Scholar 

  14. Gabovich, A. M., & Gabovich, N. A. (2007). How to explain the non-zero mass of electromagnetic radiation consisting of zero-mass photons. European Journal of Physics, 28(4), 649–655.

    Article  Google Scholar 

  15. Gertler, B. (2002). Explanatory reduction, conceptual analysis, and conceivability arguments about the mind. Nous, 36, 22–49.

    Article  Google Scholar 

  16. Jammer, M. (2000). Concepts of mass in contemporary physics and philosophy. Princeton: Princeton University Press.

    Google Scholar 

  17. Kibble, T. W. B., & Berkshire, F. H. (2004). Classical mechanics (5th ed.). London: Imperial College Press.

    Book  Google Scholar 

  18. Kim, J. (1992). Downward causation in emergentism and non-reductive physicalism. In A. Beckermann, H. Flohr, & J. Kim (Eds.), Emergence or reduction?—Essays on the prospects of nonreductive physicalism. Berlin, New York: de Gruyter.

  19. Kim, J. (1999). Making sense of emergence. Philosophical Studies, 95, 3–36.

    Article  Google Scholar 

  20. Lange, M. (2002). An introduction to the philosophy of physics: Locality, fields, energy, and mass. London: Wiley-Blackwell.

    Google Scholar 

  21. Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377.

  22. Lewis, D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62, 221–236.

    Article  Google Scholar 

  23. Lindsay, R. B. (1961). Physical mechanics (3rd ed.). Princeton: Van Nostrand.

    Google Scholar 

  24. Marras, A. (2005). Consciousness and reduction. British Journal for the Philosophy of Science, 56(2), 335–361.

    Article  Google Scholar 

  25. McLaughlin, B. (1997). Emergence and supervenience. Intellectica, 2(25), 25–43.

    Google Scholar 

  26. Nagel, E. (1949). The meaning of reduction in the natural sciences. In R. C. Stauffer (Ed.), Science and confirmation. Wisconsin: Madison.

    Google Scholar 

  27. Nagel, E. (1961). The structure of science: Problems in the logic of scientific explanation. New York: Harcourt, Brace and World.

    Google Scholar 

  28. Nagel, E. (1979). Issues in the logic of reductive explanations. In M. K. Munitz (Ed.), Teleology revisited and other essays in the philosophy and history of science. New York: Columbia University.

    Google Scholar 

  29. Nickles, T. (1973). Two concepts of intertheoretic reduction. Journal of Philosophy, LXX(7), 920–959.

    Google Scholar 

  30. Okun, L. B. (1989). The concept of mass (mass, energy, relativity). Soviet Physics Uspekhi, 32, 629–638.

    Article  Google Scholar 

  31. Okun, L. B. (2000). Reply to the letter ‘what is mass?’ by R I Khrapko. Physics-Uspekhi, 43(12), 1270–1275.

    Article  Google Scholar 

  32. Okun, L. B. (2009). Mass versus relativistic and rest masses. American Journal of Physics, 77, 430–431.

    Article  Google Scholar 

  33. Pettit, P. (1994). Microphysicalism without contingent micro-macro laws. Analysis, 54(4), 253–257.

    Article  Google Scholar 

  34. Pockman, L. T. (1951). Newtonian mechanics and the equivalence of gravitational and inertial mass. American Journal of Physics, 19(5), 305–312.

    Article  Google Scholar 

  35. Primas, H. (1998). Emergence in the exact sciences. Acta Polytechnica Scandinavic, 91, 83–98.

    Google Scholar 

  36. Schaffner, K. F. (1993). Discovery and explanation in biology and medicine. Chicago and London: Chicago University Press.

    Google Scholar 

  37. Schaffner, K. F. (2012). Ernest Nagel and reduction. Journal of Philosophy, 109(8–9), 534–565.

    Google Scholar 

  38. Schwarz, W. (2014). Against magnetism. Australasian Journal of Philosophy, 92(1), 17–36.

    Article  Google Scholar 

Download references


I would like to thank David Chalmers, Dan Marshall, Tim Maudlin, Daniel Nolan, Jesse Robertson, Raul Saucedo, Craig Savage, Wolfgang Schwarz, Jonathan Simon, Michael Simpson, Jonathan Tapsell, and two anonymous referees, for helpful feedback. This publication was made possible in part through the support of a Grant from Templeton World Charity Foundation. The opinions expressed in this publication are those of the author.

Author information



Corresponding author

Correspondence to Kelvin J. McQueen.

Appendix: A solution to the force additivity problem

Appendix: A solution to the force additivity problem

K&B successfully deduce (6) and therefore also: \(F_1 =-(m_2 +m_3 )a_c\). But to deduce (7) we must first deduce: \(F_1 =-F_c\). Here I provide a defensible suggestion.Footnote 16

Recall (5): \(m_1 a_1 =-m_2 a_2 -m_3 a_3\). Substitution using the second law gives: \(F_1 =-\left( {F_2 +F_3} \right) \). So we must deduce: \(F_2 +F_3 =F_c\) or “the additivity of force” as that will give: \(F_1 =-F_c\). I do this in four steps. Step one deduces the claim that C exerts some force on particle 1. Step two deduces that this force must be \(F_1 =-\left( {F_2 +F_3} \right) \). Step three deduces the claim that particle 1 exerts some force on C. Step four deduces that this force must be equal but opposite to the force that C exerts on 1 i.e. \(F_2 +F_3\).

Step one: necessarily, removing C from the microphysical explanans removes particle 2 or particle 3 or both. This removes force \(F_2\) or \(F_3\) or both. But this reduces the force on particle 1, and so given 1’s constant mass and \(F_i =m_i a_i\), 1’s acceleration reduces. So C induces an acceleration on 1 so C exerts some force on 1.

Step two: deduce the value of \(F_{1c}\) i.e. the force that C exerts on 1. We prove by reductio that \(F_{1c} =F_1 =-\left( {F_2 +F_3} \right) \). We know that particles 2 and 3 exert \(F_1 =-\left( {F_2 +F_3}\right) \) which (by stipulation) is the total force on particle 1 and we know that \(F_{1c} \ne 0\) (from step one). Now assume that \(F_{1c} \ne -\left( {F_2 +F_3} \right) \). But if \(F_{1c} \ne -\left( {F_2 +F_3}\right) \) then the total force on 1 is not equal to \(-\left( {F_2 +F_3}\right) \), hence a contradiction. So C exerts the force that its components exert, that is: \(F_{1c}=F_1 =-\left( {F_2 +F_3}\right) \).

Step three: to deduce the claim that particle 1 exerts a force back on C we appeal to considerations similar to those used in step one. If we remove 1 we affect C’s acceleration. Since removing (or even changing the state of) particle 1 affects C in this way there is a force on C due to 1. The remaining question is what the value of this force is.

Step four: to deduce the value of \(F_{c1}\) we appeal to considerations similar to those used in step two. We prove by reductio that \(F_{c1} =-F_1 =F_2 +F_3\). We know (by stipulation) that \(F_2\) is the total force on particle 2 and that \(F_3\) is the total force on particle 3, and we know (from step three) that \(F_{c1} \ne 0\). Now assume that \(F_{c1} \ne F_2 +F_3\). Since there can be no changes in C without changes in particles 2 and 3 then \(F_{c1}\) must to some extent apply to particles 2 and 3. But then \(F_2\) is not the total force on 2 or \(F_3\) is not the total force on 3 (or both), hence a contradiction. So C experiences the sum of the forces that its components experience, that is: \(F_{c1} =F_2 +F_3\).

Since C is only interacting with particle 1, \(F_{c1}\) is the total force on C. So \(F_{c1} =F_c =F_2 +F_3\) such that \(F_1 =-F_c\). We now return to \(F_1 =-m_2 a_2 -m_3 a_3\) and substitute: \(F_c=m_1 a_1+m_2 a_2\) Assuming \(a_1 =a_2\) we then deduce (7) and solve the Force Additivity Problem. (Note that in deducing force additivity we have also deduced that the third law scales up to composites.) Thus, mass additivity is deducible from Newtonian microphysics when components have identical accelerations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

McQueen, K.J. Mass additivity and a priori entailment. Synthese 192, 1373–1392 (2015).

Download citation


  • Reduction
  • Explanation
  • Mass additivity
  • Nagel
  • Newtonian mechanics
  • A priori entailment
  • Scrutability
  • Consciousness