# Is Herbrand semantics a kind of term formalism?

Michael Genesereth and Eric Kao describe Herbrand semantics as follows:

Herbrand semantics is an alternative semantics for First Order Logic based on truth assignments for ground sentences rather than interpretations for object, function, and relation constants. A model is simply a truth assignment for the ground atoms in our language. (Equivalently, it is an arbitrary subset of the ground atoms in our language.) In Herbrand semantics, there is no external universe and no interpretation function for constants. In effect, all ground terms are treated as opaque - they "represent" themselves.

This is in contrast to Tarskian semantics which "is based on the notion of interpretations of constants".

Alan Weir distinguishes between two forms of formalism. On the one hand there is term formalism:

The term formalist views the expressions of mathematics, arithmetic for example, as meaningful, the singular terms as referring, but as referring to symbols such as themselves, rather than numbers, construed as entities distinct from symbols.

On the other there is game formalism:

The game formalist sticks with the view that mathematical utterances have no meaning; or at any rate the terms occurring therein do not pick out objects and properties and the utterances cannot be used to state facts. Rather mathematics is a calculus in which ‘empty’ symbol strings are transformed according to fixed rules.

This makes me think that Jacques Herbrand's semantics is best used by a term formalist (unless the formalist requires an uncountable domain). I also suspect the game formalist doesn't need a semantics at all since mathematics is merely a calculus. However, I am not sure I am on the right track.

I am primarily interested in Herbrand semantics for countable domains in logic not necessarily mathematics and hence the question: Is Herbrand semantics a kind of term formalism?

Genesereth, M. and Kao, E. Herbrand Semantics. Stanford. Retrieved on October 5, 2019 at http://logic.stanford.edu/herbrand/herbrand.html

Weir, Alan, "Formalism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2019/entries/formalism-mathematics/.

• Herbrand semantics is a technical device, just as Tarskian semantics, it is not bound to any kind of philosophical perspective. Of course, it is natural for a realist to prefer Tarskian semantics, and for a term formalist to prefer Herbrand's, but both can employ either if they wish. Game formalists may prefer the stance advocated most vocally by Rapaport, semantics is syntax (and many AI researchers do after the Chinese room). To them, even Tarskian semantics maps one formal theory unto another (set theory), i.e. syntax unto more syntax. Oct 6, 2019 at 9:53
• You could say that there are two ways to treat technicalese, literal or deflationary. Herbrand semantics taken literally gives us term formalism, just as Tarskian gives us set-theoretic platonism, Heyting's gives us intuitionism, etc. But deflation by paraphrase allows us to treat any technical creatures as non-literal surface fictions and substitute whatever metaphysics we want as the "depth semantics". It is the same with quantum mechanical formalism, etc. Oct 6, 2019 at 10:13

I would agree that the philosophical stance which most directly advocates for Herbrand semantics as the "correct" semantics is term formalism. However, as conifold commented Herbrand semantics is not bound to any particular philosophical framework. (Indeed, one of the strengths of mathematics is its foundation-independence.)

However, there is an interesting tension here between the formalist stance in general and the "logical wildness" of Herbrand semantics. Herbrand entailment is extremely complicated: in particular, the set of sentences which are Herbrand entailed by even as weak (in the usual sense) a theory as Robinson arithmetic is extremely complicated (this can be precisiated and proved by computability theory). As such, Herbrand semantics is hard to accept for someone who adopts what I'd call the "strong formalist thesis" that only mathematics which is reducible to formal manipulations of symbols is meaningful - and this is something which seems to be strongly suggested, if not outright demanded, by game formalism. It's hard to escape the conclusion, then, that Herbrand semantics (or more accurately, the stance that Herbrand semantics is the "right" notion of semantics) has a nontrivial element of realism to it.

Of course, we can "tame" Herbrand semantics by a canonical trick: if T is any theory, then letting T' be the same theory in the larger language gotten by adding infinitely many new constant symbols we have that Tarski entailment and Herbrand entailment coincide for T', and so in particular every sentence in the original language of T which is Herbrand entailed by T' is Tarski entailed by T. But it's difficult to see this as truly satisfying, since (i) it forces us to adopt an infinite language and (ii) it's hard to motivate without already granting some value to the Tarskian approach.

There's a kind of irony here. Essentially - and phrased from a realist perspective for succinctness - the complexity of entailment in Herbrand semantics comes from the power required to quantify over exactly the "(nicely) definable" objects. That is, unbounded quantification becomes problematic because the domain of discourse is forced to be small! When we work with Tarski semantics, the range of "admitted models" is so large that everything not "clearly forbidden" is actually permitted, which winds up leading to computational simplicity. More generally, I'd say that one of the themes emerging from modern logic is:

The logical notions associated to "naive realism" are extremely well-behaved from a formalist perspective, despite the tension between formalism and realism as philosophical stances.

And this is in my opinion a Really Cool Thing (which, funnily enough, both formalists and realists can argue is evidence for their position!).