Consider the following formal axiomatic system
Undefined terms ag, kow, trad
Axiom 1 Every ag kows at least two trads.
Axiom 2 There is at least one trad that every ag kows.
Axiom 3 For each trad, at least one ag kows it.
Axiom 4 The set of trads is non-empty.
- Construct a model showing that this axiomatic system is consistent.
- Is each axiom independent of the other axioms?
I understand that to construct a model I must find a set of examples for the undefined terms such that the axioms are true. Any further direction would be appreciated, because I don't really understand the concept of formal axiomatic systems...
Furthermore, how would you go about proving that there are at least two trade? Would you use the model you constructed to prove it, or go back to using the undefined terms? Do independent axioms factor into the proof?
HINT
Formalize your system :
Ax.1) $\forall x [Ag(x) \rightarrow \exists y \exists z(Tr(y) \land Tr(z) \land (y \ne z) \land Kow(x,y) \land Kow(x,z)]$
Ax.2) $\exists x[Tr(x) \land \forall y (Ag(y) \rightarrow Kow(y,x))]$
Ax.3) $\forall x[Tr(x) \rightarrow \exists y(Ag(y) \land Kow(y,x))]$
Ax.4) $\exists xTr(x)$
About 2 : independency - clearly Ax.4 follows form Ax.2, because if we know that there exists a $Trad$ such that ..., for sure there exists a $Trad$.
About 1 : consistency - we can try with a stupid but simple model with domain $D = \{ a, b \}$ such that both objects $a,b$ are simultaneously $Trad$ and $Ag$.
Ax.4 is clearly satisfied.
Ax.3 implies that, e.g. : $Kow(a,a)$ and $Kow(a,b)$
Ax.2 implies that, e.g. : $Kow(a,a)$ and $Kow(b,a)$
Ax.1 implies that : $Kow(a,a)$ and $Kow(a,b)$, $Kow(b,a)$ and $Kow(b,b)$