Past exam of the logic and set theory course of the University of Oxford / 2007 / A / 3

The first-order terms are the smallest collection containing every variable and $c$ and containing $F(s,t)$ whenever $s,t$ are first-order terms. The atomic formulas are exactly $s\approx t$ for first-order terms $s,t$. Logical formulas are then obtained from atomic formulas by the logical connectives and quantifiers: if $\theta ,\psi$ are logical formulas and $x$ a variable, then $\neg \theta$, $(\theta \wedge \psi )$, $(\theta \vee \psi )$, $(\theta \to \psi )$, $\forall x\theta$ and $\exists x\theta$ are logical formulas. These closure rules, with no additional strings admitted, define the syntax; a presentation using fewer primitive connectives treats the others as abbreviations.

Semantic entailment is the conditionThe same first-order variable assignment is used for all premises and the conclusion. This matters when free variables occur; for first-order sentences the truth value is independent of the first-order variable assignment.

For independence, construct four first-order structures, each satisfying exactly the three required remaining axioms and failing the chosen one.

To fail associativity, use the carrier $\{0,1,2\}$, interpret $c$ as $0$, and define the commutative operation by its table:The first row and column give a two-sided identity, and every element is its own inverse because the diagonal entries are $0$. Symmetry gives commutativity. Butso associativity fails.

To fail the identity axiom, take $\{0,1\}$, $c=0$, and the constant operation $F(x,y)=0$. It is associative and commutative, and every element has a two-sided partner producing $c$; but $F(1,c)=0\ne 1$.

To fail the inverse axiom, use natural numbers under addition with $c=0$. Addition is associative and commutative and has the required identity, but $1$ has no additive inverse in this carrier.

To fail commutativity, use the symmetric group $S_3$ under permutation composition with $c$ its identity. Associativity, identity and inverses are group properties, while $(12)(23)\ne (23)(12)$. Each countermodel witnesses the failure of semantic entailment from the other three to the failed first-order sentence. Hence the four source axioms are independent.