How to turn into conditional statements the equations of physics? Looking for basic examples ( at the high school physics, college physics level).

Question

My question is related to philosophy, but I do not ask for a philosophical answer. I would be interested in a technical answer from a logician's / mathematician's point of view

In basic philosophy of science texts, the name of Karl Popper is often associated the claim that the basic form of a law of physics is

For all x ( if P(x) , then Q(x)).

Remark.- According to Popper, the fact that laws of physics are conditional statements is what makes them interesting, what gives them a cognitive value. That's what makes them risky statements, statements that can be tested.

What I am looking for is some examples of mathematical physics laws that could actually be shown to have this conditional form. The problem ( for me) is that laws of physics are equations, I do not really know how to turn an equation into a conditional statement.

In basic texts, the examples given are oversimplistic, like :

For all x ( x is a raven --> x is black).

Remark.- I have no knowledge of advanced physics.

Answer 1

If $O$ is an object that started at rest and has since been falling without resistance in a uniform gravitational field, if $g$ is the gravitational acceleration in that field, if $t$ is the time that $O$ has been falling, and if $s$ is the distance that $O$ has fallen, then $s = \frac12 gt^2.$

What might be missing from an oversimplified view of Popper's claim is that $x$ might have many components (in the case above, we can identify at least $O,$ $g,$ $t,$ and $s$) and that $P(x)$ might have a large number of clauses.

To put it another way, if you write down any equation of physics, an equation $Q(x)$ tells us nothing about the physical world until someone has said what each of the symbols in the equation means. The assignment of meaning to the symbols is $P(x),$ and it's what makes $Q(x)$ part of a physical law rather than just a random, meaningless equation.

Answer 2

For all objects $O_v$ moving with velocity $v>0$: if there is no force applied to $O_v$, then $O_v$ maintains its velocity $v$.

For all objects $O_{m,v}$ in the universe with rest mass $m \geq 0$ and velocity $v>0$: if the rest mass $m$ of $O_{m,v}$ is strictly positive, then the velocity $v$ of $O_{m,v}$ is smaller than the speed of light.

For all thermodynamic systems $T_{E}$ with internal energy $E$: if $T_{E}$ is a closed system, then $E$ must stay constant for all time.