In his intro to ( Skolem 1923a), Van Heijenoort (From Frege to Godel, p. 509) describes Skolem as giving “an alternative to the axiomatic approach” to proving a first-order formula. This is referring to the effective procedure Skolem gives for checking whether or not a first-order formula *U* has a solution of level n. A *solution of level n* is an assignment of truth-values to the atomic propositions of the nth level expansion. The *nth* level expansion of *U* is the conjunction of instances of *U* formed by dropping the quantifiers, letting the universal variables range over the domain of level *n-1*, and introducing new integers for the existential variables.


Skolem's procedure is as follow:


1. Form the expansions of *U* up the nth level.
2. At each level, write down all possible truth-value assignments to the atomic propositions.
3. If a truth-value assignment at level *m* has no continuations at level *m+1*, reject it.


*U* has a solution of level *n* if and only if there are truth-value assignments remaining when this procedure is carried out up to level *n*. If for some *n* there is no solution of level *n*, then we have shown that *U* is truth-functionally unsatisfiable.


VH says that this procedure “provides proofs that are cut free and have the subformula property”. I know what those properties are in the context of the sequent calculus, but I don’t understand what he means in this context.


References:


SKOLEM, THORALF


[1923a] Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre.
Matematikerkongressen i Helsingfords den 4–7 Juli 1922, Den
femte skandinaviska matematikerkongressen, Redogörelse . Helsinki:
Akademiska Bokhandeln, 1923, pp. 217–232. English translation in van
Heijenoort (ed.) [1967], pp. 290–231.


VAN HEIJENOORT, JEAN


[1967a] From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.



Skolem's method for checking truth-value assignments - a "cut-free proof procedure" for first-order logic?
The answer to the question can be found in Section I (especially p.11) of [this source](https://arxiv.org/abs/1904.10540) (it is a newly typeset version of Joseph E. Quinsey's 1980 Oxford doctoral thesis *Some Problems in Logic: APPLICATIONS OF KRIPKE’S NOTION OF FULFILMENT*).


