People would like to think that since the number of concepts involved needs to be kept small, that the specific statement of all laws of physics, is, at any given moment, expressible in some many-sorted first-order logic.
(More directly, this is related to "Henkin semantics" on higher order logics, where if you accept limitations on what quantifiers consider, you can safely use second-order language in a way that is no more risky than first-order language. I would suggest that the way science tests and defines things requires its quantifiers to have Henkin semantics.
But I think that to get consensus a scientific principle must be even more stringent about its referents -- it needs to be prescriptively clear about the 'sorts' of things to which any given claim applies, or such a claim will not be demonstrable (much less falsifiable). This reduces it to a sub-recursive type theory and makes it a many-sorted first-order system.)
The need for higher orders of logic arises largely in mathematics and psychological arenas, where things or people can refer to themselves. But then a lot of those things are second-order by nature, and some of them should be irreducibly second-order.
If those irreducibly second-order ideas are somehow real, then physics will not be able to encompass them. Physics and biology are unlikely to be able to explain why we imagine space should be infinitely divisible, for example. Notions like infinity, the continuity of space, and other deep mathematical anchor points do not seem to arise from their survival value. They are not supported by actual physics. They do little but waste our time thinking about them. And they seem to arise far too late to be evolved.
As Terrence McKenna points out, many of us live in a world of linguistic structure more than material implications (especially those who often visit a psychedelic state.) And very little of that linguistic structure is truly first order. Grammarians have tried to establish an adequate set of parts-of-speech, which is basically a many-sorted first-order classification of words as functions or propositions, for a very long time, and lots of usages still escape our modern Natural Language Processors.
Philosophy's ongoing obsession with finding a materially plausible meaning for 'meaning' is a perfect example. We can come close by encoding meaning as use and action (a la Wittgenstein's games), or by encoding meaning as a matrix of relations between abstract referents with deep commonly-reoccurring psychological underpinnings (a la Lacan's master signifiers), but neither of those is truly reducible to statable rules.
At the same time, these linguistic constructs also do not seem accidental. They have made for easier development of engineering and legal structures in many places. This implies some force of simplicity or cogency of thought that acts as a force on our intellectual development.
You can presume that the drive toward simplicity and elegance itself survives by wasting the energy of those who are not driven to simplify. But that does not explain how the simplifications themselves come to exist. Thinking creates ideas bigger than reality all the time, and we use those to genuine effect.
And as we use them, we enter an order of complexity in our reasoning that is beyond what can be encapsulated by the kind of rules we prefer in science. Science hopes the world is describable in first-order statements, perhaps with a second-order sorting. But our mathematics, as a linguistic structure, is not. So a statable theory of the world is not capable of predicting our mathematics and other pure linguistic objects, or their future development.