Different definitions of a logical system?

Question

From Herre & Schroeder-Heister's "Formal Languages and Systems", on p5

A formal system is based on a formal language $L$, endowing it with a consequence operation $C: 2^L\to 2^L$.

On p6 and p7, a deductive system is a formal system whose consequence operation can be equivalently replaced by a set of inference rules.

On p7

In logic one considers deductive systems over a language L whose expressions, called formulas, are built up from certain basic expressions by means of function symbols, predicate symbols and logical operators. In propositional logic, which for reasons of simplicity is considered in the following, formulas are built up from a subset P of L (the set of ‘propositional letters’ or ‘propositional variables’) and a finite set S of symbols of the alphabet Σ of L (the set of ‘propositional connectives’).

On p261 (also see below), in §1. Logical Systems in XIII. Lindstrom's Theorems in Ebbinghaus' Mathematical Logic

1.1 Definition. A logical system $\mathcal{L}$ consists of a function $L$ and a binary relation $\models_\mathcal{L}$. $L$ associates with every symbol set $S$ a set $L(S)$, the set of $S$-sentences of $\mathcal{L}$. (...)

Questions:

Is it correct that in Herre's article,

• a logical system is implicitly defined to be a deductive system which is a formal system, and therefore a logical system is purely formal? (If yes, can I call such a logical system "a formal logical system"?)
• a logical system is a system for a specific $S$?

Is it correct that in Ebbinghaus' book,

• a logical system $\mathcal{L}$ has both a formal part (function $L$) and a semantic part ($\models_\mathcal{L}$)?

• a logical system is a family of subsystems indexed by $S$?

Is it correct that the definitions of a logical system in the two sources are independent of each other? Is either definition not created based on the other?

Given a logical system (defined in the sense of Ebbinghaus' book), is it correct that its sybsystems do not necessarily correspond to formal logical systems (defined in the sense of Herre's article)? In other words, is a logical system not necessarily able to be formalized?

When does a formal logical system correspond to a subsystem in a logical system? Does that happen if and only if a formal logical system is a formal first-order logical system, and a logical system is the first order logical system?

There might be other definitions of a logical system. Which definition of a logical system is more popular in popular mathematical logic books? (For example, does Enderton's book ever define a logical system?)

Thanks.

See also On the definition of a logical system

Answer 1

This question is a bit all over the place, and certain parts of it (e.g. whether logical systems are "purely formal," whatever that means) seem ill-posed to me; I'll answer the part of it which I think is answerable, namely the relationship between formal/deductive systems in the sense of HSH and logical systems in the sense of EFT.

The relevant definitions are indeed totally independent of each other, but they're not unrelated. Suppose $\mathcal{L}=(L, \models_\mathcal{L})$ is a logical system. We can assign to each symbol set $S$ a formal system $Sys_\mathcal{L}(S)$ as follows:

• The formal language of $Sys_\mathcal{L}(S)$, which I'll call "$A$" to disambiguate it from $L$, is the free Boolean algebra generated by the elements of $L(S)$. (Keep in mind that part of the definition$^1$ of "logical system" involves closure under the Boolean operations.) Note that $A$ is much larger than $L(S)$, but has the same semantic strength so from the point of view of $\mathcal{L}$ there's no serious issue in conflating them. We need to do this, however, since $\mathcal{L}$ doesn't tell us that $L(S)$ looks anything like a formal language.

• The deduction relation of $Sys_\mathcal{L}(S)$ is just the map $$C(\Gamma)=\{\varphi: Mod(\Gamma)\subseteq Mod(\varphi\},$$ or more snappily $C(\Gamma)=\{\varphi: \Gamma\models_\mathcal{L}\varphi\}$.

There is no reason however for $Sys_\mathcal{L}(S)$ to be a deductive system - the most obvious point is that any deductive system is compact, and $\mathcal{L}$ might not be (consider second-order logic). Since we're often interested in non-compact logical systems (e.g. $\mathcal{L}_{\omega_1,\omega}$, SOL, logics with cardinality quantifiers, ...) this limits the relevance of HSH-style systems to the theory of logical systems in the sense of EFT.

Moreover, even when we break a logical system into its "layers" (one for each symbol set $S$), each layer is more than just a formal system: it also has semantics. The formal system $Sys_\mathcal{L}(S)$ "forgets" which structures satisfy which sentences, and so just from knowing the map $S\mapsto Sys_\mathcal{L}(S)$ we cannot reconstruct $\mathcal{L}$. So a logical system is much more than just a symbol-set-indexed collection of formal systems.

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$^1$Actually, I might be getting this wrong - you didn't quote the whole definition (which is understandable! it's long) and I don't have EFT at hand. I might mean "regular logic" instead here. If that's the case, then we can adopt an even sillier solution: we consider the free algebra on the set $L(S)$ in the empty signature. This is just the set $L(S)$ itself. It's not a huge issue though, the main point is that we do have a notion of deduction for each fixed $S$ coming from $\mathcal{L}$.