# Is Newton's first law an “If and only if”

On Wikipedia, Newton's first law is stated as:

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

I read this as $$\mathbf{F}_{\textrm{net}}=0\Rightarrow \mathbf{a} = 0,$$ but does it also mean that $$\mathbf{a}=0\Rightarrow \mathbf{F}_{\textrm{net}}=0,$$ or do you need the second law for that? I would argue that you need the second law, as the first law doesn't say anything about what happens if there is a net force. A force could cause acceleration in some situations, but not in others. Wikipedia seems to disagree since the following can be read further down the page:

The first law can be stated mathematically when the mass is a non-zero constant, as, $$\sum \mathbf {F} =0\;\Leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0.$$

So, who is right?

• well. how is the inertial frame and force defined in order to be used to formulate the first law? Are the newtons laws laws or just definitions?i think if you take this formulation of newton laws literally, you will get into real trubles. It offers itself for a lot of circular reasonings and definitions. – Umaxo Nov 14 '19 at 10:51
• If I remember correctly, some ppl view this as a definition of inertial frame (you assume that the meaning of inertial frame is know a priori) – lalala Nov 14 '19 at 20:30

The first law is a statement about the "natural" state of an object. It does not make reference to the specific form of the second law. But keep in mind that the word "force" does appear in the statement (..., unless...). So you question is more about linguistics and mapping English into pure logic than it is about physics. The second law is not needed to imply the first as the former is a specific relation between force and acceleration. If, for example, we had discovered that $$F_{net} = ma^2$$ or some other law then it would still be true that if $$a = 0$$ $$F_{net} = 0$$ and vice versa.

The word "unless" is typically interpreted in logic in the following manner. $$q$$ unless $$p$$ is equivalent to $$\sim p \Rightarrow q$$, if there is NOT a net force then the motion is uniform. If $$\sim p$$ is false $$q$$ can still be true. This is allowed by the principles of logic. So, the motion could be uniform if there is a net force present. However the contra-positive is always true. If $$\sim q$$ is true then $$p$$ must be true. If the motion is non-uniform then there must be a net force acting on the object. This is a weaker dependence than if and only if.

There have been critiques of this suggesting, by counter example, that there should be a biconditional equivalent of this. The real issue is whether the proper English use of unless is unambiguous and always provides a faithful representation of the speaker's intent. Did Newton mean to state the weaker connection or was this the result of a colloquial use of the term unless?

Some relationship between force and inertial state might be required to make your IFF statement hold but it does not need to be linear. And I would state that such a relation is not necessary to define an inertial state.

• Thanks. So you're saying that who is right depends on how you read the law? From a straight logical point of view I'm right, but you can also argue that the intention was an IFF statement. What made me think of this question, was a student who made this experiment where an object didn't move. The student wanted to conclude that $F_{\textrm{net}}=0$. Do the student refer to the first or the second law to make this conclusion? – bgst Nov 14 '19 at 8:17
• @bgst, I don't mean to be indecisive on this point but the logic texts state unless is equal to an if then, not an iff. The real issue is what does this collection of statement mean to us in the physics community. I think that the 1st law alone, as worded, does not imply a bi-conditional relation. The second law would be needed too. – ggcg Nov 14 '19 at 18:05

In my personal opinion, second law is more or less enough, as other two laws can be derived from it.

### First law:

$$\sum\vec{F} = m\frac{\textrm{d}\vec{v}}{\textrm{d}t} = 0 ,\\ \Rightarrow \vec{v}=\textrm{const}$$

### Third law

$$\vec{F}_{A\to B} \neq 0, \vec{v_{_B}}=\textrm{const},\\ \Rightarrow \vec{F}_{A\to B} + \sum_i\vec{F_i} = 0 , \\ \Rightarrow \sum_i\vec{F_i} = - \vec{F}_{A\to B}$$

In words: If body A exerts force on body B, but body B still keeps it's speed constant, then there must exist superposition of other forces which acts like original force, just in opposite direction. Otherwise, second Newton law would be crushed to peaces, because bodies must react to forces applied. Of course this formulation is not exactly the Third law, because superposition of other forces is involved, thus it's not clear from it if B exerts some force on A too, or just some external force(s) to B is applied or both. In any way, this "third law replacement" comes as a direct conclusion from second law.