# Can the fundamental laws of Physics be formalized as axioms written in First Order Logic (FOL), or any other logical system for that matter?

Is it possible to state any fundamental law in Physics as an axiom written in First Order Logic (FOL), or any other logical system (and semantic interpretation) for that matter?

EDIT: in a way that other non-fundamental laws can be deduced as theorems from these axioms by applying inference rules.

• Sure. My logical language has one variable, namely $x$, one predicate, namely $P(x)$, and one axiom, namely "for all $x$, $P(x)$". My semantic interpretation is that $x$ represents a physical system and $P(x)$ represents the statement "Newton's Laws apply to $x$". I've just written all of Newtonian mechanics as an axiom-plus-semantic-interpretation in first order logic. Perhaps this clarifies the downvote (which did not come from me). – WillO May 17 '18 at 5:43
• This is an interesting question, that someone thought to oversimplify ad absurdum. I'd like to see a proper answer – TheEnvironmentalist May 17 '18 at 5:46
• @WillO, in your example, how would you go about proving new theorems from that axiom? (check out the edit to the question) – xwb May 17 '18 at 5:51
• Related: physics.stackexchange.com/q/87239/2451 and links therein. – Qmechanic May 17 '18 at 6:21
• @xwb: I would add some constant symbols to my language, semantically interpret those constant symbols as representing some specific physical situations, and then from the axiom "for all $x$, $P(x)$" I would deduce the theorems $P(c_1),P(c_2),P(c_3)...$. This doesn't seem to be getting any more interesting. – WillO May 17 '18 at 15:19

I'm not sure if such an axiomatization is possible, because axiomatizing real numbers is already impossible to do in a first order manner, as the first order axiomatization of the real is the same as for any of the ultrapower extensions of the real (the first-order axioms of $\mathbb{R}$ are the same as the first-order axioms of $^*\mathbb{R}$, for instance. I think they're just the axioms of $\mathbb{Q}$ overall).

First-order axiomatizations have been done for some theories, such as special relativity. There are many axiom systems, but here is one for instance, from Andréka et al.

The list of symbols, beyond the set theory symbols, is the following :

• $\mathsf{B}$ (bodies), $\mathsf{Ob}$ (observers), $\mathsf{Ph}$ (photons) and $\mathsf{F}$ (field) are unary relation symbols, to be used in contexts such as $\mathsf{B}(x)$ to signify $x$ is a body. We can also use those as sets, so that $x \in \mathsf{B} \leftrightarrow \mathsf{B}(x)$. We also have the relation that both observers and photons are bodies : $\mathsf{Ob}, \mathsf{Ph} \subset \mathsf{B}$
• The usual algebraic symbols $+$, $-$, $\times$, $/$ and $<$, with the usual meanings.
• For an $n$-dimensional spacetime, an $(n + 2)$-ary symbol $\mathsf{W}$, signifying that if $\mathsf{W}(x,y,z_1, z_2, ..., z_n)$, then the observer $x$ observes the body $y$ at coordinates $z^\mu$.

A few definitions :

• $^n\mathsf{F}$ is the set of $n$-tuples of $\mathsf{F}$. It has the structure of a vector space, with the usual axioms.
• The slope of a vector is defined by $$\mathsf{slope}(v) = (v_2^2 + v_3^2 + ... + v_n^2) / v_1^2$$if $v_1 \neq 0$, and the symbol $\mathsf{slope}(v) = \infty$ otherwise.
• The set of lines is defined by $$\mathsf{Lines} = \left\{ \{ q + \lambda v | \lambda \in \mathsf{F} \} | q, v \in {}^n\mathsf{F}, v \neq 0 \right\}$$(so that a single line is the set of points in ${}^n\mathsf{F}$ satisfying that condition for every $\lambda$ and a given origin $q$ and direction $v$)
• The slope of a line $\ell \in \mathsf{Lines}$ is defined by $$\forall p, q \in \ell, p \neq q, \mathsf{slope}(\ell) = \mathsf{slope}(p - q)$$
• The trace of a body $b \in \mathsf{B}$ given an observer $m \in \mathsf{Ob}$ is the set of coordinates at which $m$ sees $b$. $$\mathsf{tr}_m(b) = \{ p \in {}^n\mathsf{F} | \mathsf{W}(m, b, p) \}$$
• Inversely, an event at $p$ seen by an observer $m$ is the set of all bodies seen by $m$ at the coordinates $p$. $$\mathsf{ev}_m(p) = \left\{ b \in \mathsf{B} | \mathsf{W}(m, b, p) \right\}$$
• And finally, the speed of a body is the slope of its trace.$$\mathsf{speed}_m (b) = \mathsf{slope}(\mathsf{tr}_m(b))$$

With all these definitions, here's the list of axioms :

1. AxSelf : Every body sees itself as stationary at the origin. $$\mathsf{tr}_m(m) = \{ \langle t, 0, ..., 0 \rangle | t \in \mathsf{F} \}$$or, equivalently, $$\forall m, p, \left( (\mathsf{Ob}(m) \wedge \mathsf{F}(p_1)) \rightarrow [\mathsf{W}(m, m, p) \leftrightarrow (p_2 = 0 \wedge ... \wedge p_n = 0)]) \right)$$
2. AxPh : The trace of a photon always has slope $1$ (the speed of light is constant for all observers)$$\forall m \in \mathsf{Ob}, \left\{ \mathsf{tr}_m(\mathsf{ph}) | \mathsf{ph} \in \mathsf{Ph} \right\} = \left\{ \ell \in \mathsf{Lines} | \mathsf{slope}(\ell) = 1 \right\}$$or, equivalently, $$\forall m, \forall p, q \in {}^n\mathsf{F}, \left( \mathsf{Ob}(m) \right [\exists \mathsf{ph} (\mathsf{Ph}(\mathsf{ph} \wedge \mathsf{W}(m, \mathsf{ph}, p) \wedge \mathsf{W}(m, \mathsf{ph}, q)) \leftrightarrow (p_1 - q_1)^2 = (p_2 - q_2)^2 + ... + (p_n - q_n)^2])$$
3. Observers see the same bodies.$$\{ \mathsf{ev}_m(p) | p \in {}^n\mathsf{F} \} = \{ \mathsf{ev}_k(p) | p \in {}^n\mathsf{F} \}$$or, equivalently, $$\forall m, k, \forall p \in {}^n\mathsf{F}, \exists q \in {}^n\mathsf{F}, ( (\mathsf{Ob}(m) \wedge \mathsf{Ob}(k)) \rightarrow \forall b \in \mathsf{B}, [\mathsf{W}(m, b, p) \leftrightarrow \mathsf{W}(k, b, q)] )$$

Along with the axioms of a field, these form a basic first-order axiomatization of special relativity. A bit more robust axiomatization uses the additional definition of the worldview transformation (going from the coordinates of one observer to another)

$$\mathsf{f}_{mk} = \left\{ \langle p, q \rangle | \mathsf{ev}_m(p) = \mathsf{ev}_k(q) \wedge \mathsf{ev}_k(q) \neq \varnothing \wedge p, q \in {}^n\mathsf{F} \right\}$$

This is basically the Poincaré transformation from observer $m$ to $k$, for which you can show $\mathsf{ev}_m(p) = \mathsf{ev}_k(\mathsf{f}_{mk}(p))$. Then you can add the axiom :

1. AxSym : The effects of time dilation are symmetric.$$\forall m, k, \forall p, q \in {}^n\mathsf{F}, (\mathsf{Ob}(m) \wedge \mathsf{Ob}(k)) \rightarrow \| \mathsf{f}_{mk}(p)_1 - \mathsf{f}_{mk}(q)_1 \| = \| \mathsf{f}_{km}(p)_1 - \mathsf{f}_{km}(q)_1 \|$$

This is not the only axiomatization of special relativity, nor the best (in particular, all observers are straight lines in this). But it's an example of a first-order axiomatization of a physical theory.

I don't think this will give great results, since as I've mentionned, you can't use real numbers here. Everything will be fairly awkward and something as basic as a square root will be beyond our usage. But it is something you can do, at least. I'm not aware of any such other first-order axiomatization for other theories (though I'm aware of an axiomatization of classical mechanics based on Hilbert's axioms, but this is not first order).

• I edited the title question so that other logical systems might be used if FOL falls short. – xwb May 17 '18 at 16:47
• For other logical system, see here : physics.stackexchange.com/questions/388849/… – Slereah May 17 '18 at 16:56