Satisfaction of a Formula in a Structure (Open Logic Project, 2026)

This is the complete official five-page Open Logic Project module Satisfaction of a Formula in a Structure, revision 9620cc7 dated 2026-07-12 under CC BY 4.0. All five pages were read from native text and visually inspected at 200 dpi. The archived file is byte-identical to the official build; SHA-256 is 5e0e37837c938c3c8fd411a99cd134d6d965a449b1c4018ee7d615e5fc106376.

How truth is computed #

A variable assignment is a total map from variables to the structure's domain. Term values follow the term tree: constants use their interpretation, variables use the assignment, and compound terms apply the interpreted function to recursively obtained argument values.

Formula satisfaction follows the formula tree. Atomic predicates test tuple membership, equality compares term values, connectives use classical two-valued clauses, and quantifiers range over point updates of one assignment coordinate. The notation $s[m/x]$ is a semantic update by a domain element; it is not the syntactic substitution $\varphi[t/x]$.

This distinction explains why quantifiers cannot be defined merely by substituting written names: a domain element need not have any term naming it.

Reconstructed example and correction #

The source's four-element model was independently enumerated. Its term values and principal truth judgments are correct, including the falsity of

$$\exists x\bigl(R(a,x)\land\forall y\,R(x,y)\bigr).$$

The printed proof of that last judgment is not correct as written. It says the same value $y=4$ defeats both candidate values $x=1$ and $x=2$, but the displayed relation contains $(2,4)$. For $x=2$, either $y=1$ or $y=2$ is a valid counterexample. The conclusion survives; the witness argument must be repaired. The source also writes “for all $n$” where its case variable is $m$.

Scope boundary #

The module defines satisfaction relative to both a structure and an assignment. It does not prove the substitution lemma, the theorem that truth depends only on free variables, or the assignment-independence of sentences, and it does not define validity, consequence, theories, or modelhood. Those results cannot be attributed to this five-page source alone.

Its opening claim that every quantified formula's immediate body has a free variable is also too strong: the quantified variable may be absent and the body may already be closed. The formal clauses themselves do not rely on that overstatement.