Connections with Model Theory
First-Order Languages and Signatures
The syntax of first-order logic as universal algebra needs it: languages with function and relation symbols, terms, formulas, and the distinction between free and bound variables.
Learning objectives
- Specify a first-order language from a signature
- Build terms and formulas by recursion
- Distinguish free from bound variables and sentences from formulas
Signatures and languages
A language L is determined by a set of function symbols with arities, a set of relation symbols with arities, and a set of constant symbols — together with the fixed logical apparatus: variables, the equality symbol, connectives ¬, ∧, ∨, →, and quantifiers ∀, ∃.
Algebras use only function and constant symbols. Adding relation symbols gives the more general structures, which is what Chapter V works with.
Chapters I–IV use only identities — universally quantified equations. Chapter V allows arbitrary quantification, negation and relation symbols. The expressive gain is large, and results proved for identities do not automatically transfer.
Terms and formulas
Variables and constant symbols are terms; if f is n-ary and t1,…,tn are terms then f(t1,…,tn) is a term.
An expression t1 ≈ t2 for terms ti, or r(t1,…,tn) for an n-ary relation symbol r.
Atomic formulas are formulas; if Φ and Ψ are formulas so are ¬Φ, Φ ∧ Ψ, Φ ∨ Ψ, Φ → Ψ, ∀x Φ and ∃x Φ.
Terms in this sense extend the Chapter II notion by allowing constant symbols from the language; formulas are the genuinely new layer.
Free and bound variables
An occurrence of a variable x in a formula is bound if it lies within the scope of a quantifier ∀x or ∃x; otherwise it is free.
A formula with no free variables.
Only sentences have a truth value in a structure. A formula with free variables defines a relation on the structure — the set of tuples satisfying it.
- Φ(<em>x</em><sub>1</sub>,…,<em>x<sub>n</sub></em>)
- a formula whose free variables are among the listed ones
- Sentence
- no free variables; true or false in a structure
- Open formula
- quantifier-free
- Universal sentence
- ∀x1…∀xn Φ with Φ open
- Existential sentence
- ∃x1…∃xn Φ with Φ open
Identities as a fragment
An identity p ≈ q corresponds to the sentence ∀x1…∀xn (p ≈ q). So equational logic is the fragment of first-order logic consisting of universally quantified atomic formulas.
| Fragment | Form | Class of models |
|---|---|---|
| Identities | ∀x (p ≈ q) | Varieties |
| Quasi-identities | ∀x (∧pi ≈ qi → p ≈ q) | Quasivarieties |
| Universal sentences | ∀x Φ, Φ open | Universal classes |
| Horn sentences | Clauses with at most one positive literal | Closed under reduced products |
| Arbitrary sentences | Any first-order formula | Elementary classes |
Each level corresponds to closure under different constructions. Birkhoff's theorem sits at the bottom; the preservation theorems of §2 identify the constructions matching each higher level.
Frequently asked questions
Why include equality as a logical symbol rather than a relation symbol?
Because its interpretation is fixed — it always means genuine identity — whereas relation symbols may be interpreted arbitrarily. Logic with this convention is called first-order logic with equality, and it is what universal algebra needs.
Can a language be infinite?
Yes. Modules over an infinite ring need one unary function symbol per ring element. Nothing in the theory requires finiteness of the language, though some results assume it.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section V.1, book pages 217-220.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.