What's the definition of equational theory? Why is $\lambda$ logic free? (compare to equational logic)

Question

Henk Barendregt says that "$\lambda$ is logic free: it is an equational theory."

But I don't understand the "logic free" and "equational theory". Can you help me?

Answer 1

You can see Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007), Ch.5 : Quantifier-free logic, page 101-on.

It is used a first-order language without quantifiers and it is studied a natural deduction calculus for quantifier-free formulas :

Atomic formulas of the form $(s = t)$ are called equations, and the symbol ‘$=$’ is known as equality or identity.

The natural deduction rules for qf formulas are exactly the same as for $LP$ [propositional logic], except that we also have introduction and elimination rules ($=$I), ($=$E) for equality.

Thus, basically, we can construct derivations "made of" equations.

Added

See Henk Barendregt, The Lambda Calculus. Its Syntax and Semantics (2nd revised ed - 1985), page 6 :

The theory $\lambda$ [of pure lambda calculus] has as terms the set $\Lambda$ ($\lambda$-terms) built up from variables using application and abstraction. The statements of $\lambda$ are equations between $\lambda$-terms [...].

See page 23 :

Note that $\lambda$ is logic free [emphasis added]: it is an equational theory. Connectives and quantifiers will be used in the informal metalanguage discussing about $\lambda$.

From these quotations, the intent of the author is clear. The mathematical theory of pure lambda calculus is based on a syntax of terms, where the set $\Lambda$ of it is built up form variables ($v_0, v_1, ...$), the parentheses and the abstractor $\lambda$ [see Def.2.1.1, page 22], and the formulas of the calculus are equations between terms of the form $M=N$ where $M,N \in \Lambda$ [see Def.2.1.4, page 23].

The formulas have no logical connectives : $\lnot$, $\land$, $\rightarrow$, nor the "usual" quantifiers : $\forall$, $\exists$; in this sense it is "logic free".

Expressions like :

$M = N \implies N = M$

or :

$\forall M \, (\lambda x.x) M = M$

with connectives and quantifiers, are not formulas of the language, but statements in the meta-language describing the $\lambda$ calculus.

Answer 2

Consider a broad class of "formal systems": each comes with a set of symbols, a set of statements or well-formed formulas, axioms and rules how to define new formulas from axioms. The derivable formulas in a given formal system called "the theory of the system".

Then "propositional logic" or "first order logic" refer to a subclass of formal systems, their set of symbols contain operators that can be called "logical", and either the axioms or the rules will contain "logical" reasoning; we are talking about conjunction, disjunction, implication, logical equivalence, negation, forall, exists (not necessarily all of them).

(Note that there are often several equivalent ways to get reasoning: axiomatic ("Hilbert style"), where there are "logical" axioms and the only inference rule is Modus Ponens, or in Natural Deduction or Sequent Calculus where we get reasoning through the inference rules and the only axioms are the "non-logical" ones)

The lambda-calculus is a formal system where the only well-formed formulas are equations. Therefore, the rules serve to derive new equations from existing ones, the theory is "equational".

In order to use lambda calculus for logical reasoning, one can add "logical" connectors in the form of constants, with axioms (equations) giving these connectors their expected meaning. See the article by Farmer "The seven virtues of simple type theory" or the book by Andrews "An Introduction to Mathematical Logic and Type Theory"