Is second order logic logic?

I have uploaded next to no philosophy yet on this webpage and I think that this is a shame. So I have decided this holiday to gift you with my Philosophic Logic module tasks (if I deem them of a high enough quality).

I am going to start with a short essay outlining a few arguments to take first order logic as the correct logic.

Why we should take first order logic to be logic

The best argument for taking first order logic is that it is the tool kit for analysing just propositions. Any statement that makes a claim about the world is a proposition, and if you really want to get to the bottom of things, you need a tool kit capable of analysing propositions. Luckily, this isn’t hard to come by now days after the work of Gödel, Russell, Hilbert, Frege, etc. Those people, plus many more, have defined a sound and complete set of rules, with respect to a very agreeable semantic model, that you can use in a wholly uncontroversial way. But what about second order logic? Anything valid in first order logic is trivially valid in second order logic meaning that you aren’t going to get contradictory results by using it, but it’s level of expressibility is far greater than that of first order logic allowing formula such as $\exists F \forall x (Fx)​$ be expressed. Is greater expressibility in a formal system a bad thing? I will put forward a few good arguments that try to demonstrate why it might not be a good idea.

Ontological Commitment Argument

To begin, I shall consider the different ways of interpreting a formula such as $\exists x (Fx)$. The first way is to use some kind of model theory and maybe find the value of $V_\mathscr{M} (\exists x (Fx))$. But this is a meaningless endeavour unless we know what proposition we were trying to formalise. Taking F as a metalinguistic variable that should be substituted with the denotation of $F$, you might be tempted by the sentence “There is an $x$ such that $x$ is $F$.” But what is it, exactly, that an $x$ is? I don’t know either. Maybe we could try “There is a thing, $x$, such that the thing we have named $x$ is $F$.” But this seems unsatisfactory as well. We have introduced far too much. The best way, in my option, is to express it as “Something is $F$.” We have made no claims over and above what our initial formula said. But what about the second order sentence $\exists F \exists x (Fx)$. There seems to be no way of translating this into English without creating some kind of ontological claim. The best I could do is “There is a thing, $F$, such that something is $F$.” Although this doesn’t seem like a very controversial claim, it presupposes the existence of abstract objects. You might have no issues with accepting the existence of such things, but surely a truly universal logic shouldn’t presuppose anything and you will certainly have trouble convincing a person who does not share your views on abstract objects using a logic that may rely on them. This argument may seem unintuitive to anyone that has worked with an axiomatic proof system before. In Ted Sider’s $QL$, we are given the axiom $\forall \alpha (\phi \to \psi) \to (\phi \to \forall \alpha \psi)$. “But James! Surely $\phi$ and $\psi$ are the very abstract objects you warned me about!” I hear you cry. I understand the concern. I would seem too tempting to see that, given that $\phi$ and $\psi$ can be replaced with any predicate, you could see this as an open sentence and bind it (as we have done with terms) to make $\forall \phi \forall \psi (\forall \alpha (\phi \to \psi) \to (\phi \to \forall \alpha \psi))$. But this isn’t a valid way of thinking. Whenever $\phi$, $\psi$, etc are involved, we are dealing with proof schema. These are not $QL$-wffs as they make no semantic claims, only syntactic claims and shouldn’t be confused.

Completness Argument

One of the more strikingly worrying things about second order logic is the fact that there is no proof of completeness meaning that $\neg (\Gamma \vDash \phi \to \Gamma \vdash \phi)$. But does this matter? So what if logic didn’t turn out the way we wanted it to? Its just classic philosophy! Bound to make you feel like Hilbert! Surely it is more important that it is sound and consistent, meaning that any deduction we can make in the logic is true in the real world? That way we can use this logic as a tool for analysis without worrying that our inferences are wrong. When considering a deductive logic, a statement should only be true if it is a logical consequence of its premises. This is important because you need truth to be demonstrable and clearly linked to its premises. How can we assign something the status of being ‘logically true’ if the conclusion hasn’t been reached through deduction or inference and how can we analyse it in a systematic and formal way to confirm the validity of an argument or proposition? As a logician, I don’t want to be reduced to eternal skepticism when I don’t have to be! So I shall simply reject second order logic and accept first order logic as logic.