What's a Logic?

Question

I'm taking a logic course, and so far we've seen

1. Logical Symbols
2. Logical Axioms
3. Rules of Inference
4. Non-Logical Symbols
5. Non-Logical Axioms

Now, I'm confused about definitions.

• Alphabet/Vocabulary = 1+2
• Logic = 1+2+3 ?
• Signature/Language $L$ = 4 OR Logic (implicitly)+4 ?
• Theory = 5 OR Logic (implicitly)+5 ?
• $L$-Theory = 4 (implicitly)+5 OR Logic (implicitly)+4 (implicitly)+5 ?

Also, I've seen Logic being defined as an ordered pair consisting of a function mapping alphabets to sets of formulae and a so-called satisfaction relation... or by means of institutions in category theory... I'm just confused as to what a logic ultimately really is, because even the most rigorous logic texts I could find begin their definitions at languages and vocabularies. Or, if not a universal definition, maybe at least what a logic consists of, i.e. foregoing technicalities, what needs to be defined to have a certain logic.

Perhaps the difficulty is that Rules of Inference are more Proof-theoretic and the other stuff more Model-Theoretic, but maybe you can help me anyway, it'd be much appreciatetd.

Answer 1

We start with Symbols : either logical : $\to, \lnot, \land$ etc. (the connectives), the quantifiers and equality ($=$) and some auxiliary symbols (like parentheses).

In addition we need propositional variables : $p_0,p_1,\ldots$ for propositional logic as well as predicate symbols : $P,Q,\ldots$ and individual variables : $x_0,x_1,\ldots$ for predicate logic.

This is enough for the languages of "pure" logic : propositional calculus, predicate calculus.

If we want to develop some mathematical theory, we have to enlarge the language of predicate logic with specific mathematical symbols : $\in$ for set theory and $0,s(x),+,\times$ for arithmetical theory.

Usually, we call the list of the non-logical symbols of a language : signature.

Up to now, we have considered only symbols, i.e. the alphabet of the language. In order to have a language, we need also syntax, i.e. the rules to produce finite strings of symbols that are meaningful : the so-called well-formed formulas.

For propositional logic, $(p_0 \to p_1)$ is a wff while $\land p_0$ is not.

For predicate logic, $\forall x (Px \land Qx)$ is an example of wff.

Having defined the syntax of the language, we add the semantics to it, based on the concept of interpretation.

---

Up to now we have covered the Language : your points 1 and 4.

In order to develop a logical calculus, we need rules of inference : at least one (usually Modus Ponens) but often more than one, as well as some (maybe zero) logical axioms : this corresponds to your point 2 and 3.

We may call propositional calculus and predicate calculus with the general term : Logic.

---

Finally, in order to develop a mathematical theory, we have to add to the predicate calculus some specific mathematical axioms, like e.g. first order version of Peano's axioms for arithemtic or Zermelo-Fraenkel axioms for set theory (and this is point 5).

---

If we denote with $\text L$ a specific language, e.g. the language of predicate calculus with the mathematical symbol $\in$, we can use $\text L$-theory to denote a theory that uses language $\text L$.

---

---

For a more abstract definition, see e.g. Heinz-Dieter Ebbinghaus et alii Mathematical logic (Springer, 2nd ed.1994), Ch.XIII.1 Logical Systems :

In the following definition of a "logical system" we collect several properties which are shared by the logics we have considered so far. As we are mainly interested in semantic aspects, we shall speak of a logical system as soon as we have the following: We are given, for every symbol set $S$, an "abstract" set whose elements play the role of $S$-sentences, and in addition, a relationship between structures and such sentences which corresponds to the satisfaction relation, and determines whether an "abstract" sentence holds in a structure.

Definition. A logical system $\mathcal L$ consists of a function $L$ and a binary relation $\vDash_{\mathcal L}$. $L$ associates with every symbol set $S$ a set $L(S)$, the set of $S$-sentences of $\mathcal L$. The following properties are required: [...]