= Solution
There are two different issues: whether a law follows logically from the other laws, and whether an experiment can test it without assuming anything else. <Newton's laws of motion> have empirical content when combined with rules for identifying <inertial frames>, measuring <mass> and specifying <forces>. An experiment tests predictions of that combination. Calling every observed <acceleration> an otherwise unspecified <force> would remove the empirical constraint rather than verify <Newton's second law>. The distinction is central to the <empirical testing of Newton's laws of motion>.
<Newton's first law> contributes the claim that a common family of <inertial frames> exists: bodies with zero net <external force> have uniform straight <motion> in those frames. It is not enough to choose coordinates which straighten the <trajectory> of just one body; an admissible frame must accommodate many independent bodies and interactions. Once that frame and the ordinary constant-positive-<mass> form of <Newton's second law> are assumed,
$$
\mathbf F= m\frac{d\mathbf v}{dt}=0
\quad\Longrightarrow\quad
\mathbf v(t)=\mathbf v_0.
$$
This conditional deduction does not establish the existence of the required frame, or independently identify the physical circumstances in which the net <force> vanishes. Defining an <inertial frame> and then counting a straight <trajectory> as proof of the definition would not be a test.
Nevertheless, the <law of inertia> has testable implications in a network of assumptions. Consider low-friction motion along an air track: determine <positions> at measured <times>, reduce the known residual influences, and ask whether the remaining <acceleration> approaches zero in the same laboratory frame for different bodies. Rotating the frame changes the description through <Coriolis forces> and <centrifugal forces>; this dependence must be accounted for. No finite experiment establishes that every unmeasured influence is absent, but varying the apparatus can discriminate an interaction from a frame effect. A failed prediction is evidence against some part of the package, not automatically the isolated verbal statement of <Newton's first law>.
<Newton's second law> is more than a definition if the <force law> and <mass> assignments constrain predictions before the new <acceleration> is measured. For instance, independently characterize a spring's reproducible extension response, specify the same interaction in several configurations, and compare the resulting <accelerations> for different bodies. If its strength and every <mass> are freely refitted to each outcome, agreement is empty. If they are fixed by an independently checked model and calibration, the law predicts held-out <trajectories>. Such a test remains conditional on the spring model, <initial conditions> and frame identification; this is an instance of the <Duhem problem>, not a reason to deny all empirical content.
<Newton's third law> supplies a different constraint. For two bodies exchanging <forces>, <Newton's second law> gives
$$
\frac{d}{dt}(\mathbf p_1+\mathbf p_2)
=\mathbf F_{12}+\mathbf F_{21}
$$
when there is no net <external force>. Equal-and-opposite interaction <forces> imply <conservation of momentum> for the pair. The two second-law equations alone do not require that cancellation: assigning $\mathbf F_{12}=\mathbf f$ and $\mathbf F_{21}=-2\mathbf f$ is algebraically consistent with both second-law equations while violating the third law. This is a logical countermodel, not a proposed isolated physical interaction. Measuring paired <impulses> or checking the pair's total <momentum> therefore tests an additional constraint, conditional on the assumed absence of momentum exchange with anything else.
The system boundary matters. In <electromagnetism>, matter can exchange <momentum> with the <electromagnetic field>. Instantaneous mechanical <forces> between two particles need not satisfy the simple pairwise statement, although total matter-plus-field <momentum> is conserved under the appropriate closed-system conditions. Likewise, <Noether theorem> can derive <conservation of momentum> from a translationally invariant <action>; that uses an extra symmetry assumption and does not derive the third law from the second alone.
Thus the first law is not an additional differential consequence once the full second-law framework is fixed, and the third law is not a consequence of that framework. All three can contribute to empirical tests, but none is tested in total isolation from frame, interaction and measurement assumptions. This preserves the difference between logical dependence and experimental testability.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Axioms, or Laws of Motion: Laws I–III and the discussion following Law III]{ref}
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