0
$\begingroup$

Assume the only parameters in the language are the quantifier symbol and a two-place predicate symbol $P$. And assume the language does not have the equality symbol. Find a sentence that is satisfiable, but has no models of size $3$ or less. In some article I found the following decision:

$$ \forall xPxx \land \exists y \exists z \exists w \exists v(\neg Pyz∧ \land \neg Pyw \land \neg Pzw \land \neg Pyv \land \neg Pzv \land \neg Pwv) $$

If this sentence is true in some structure, then there exist some $a, b, c, d$ in that structure such that $(a,b), (a,c), (a,d), (b,c), (b,d), (c,d)\notin R$. $R$ is the binary relation - the interpretation of $P$. And $(x,x)\in R$ for any $x$. From here is deduced that $a, b, c, d$ are all distinct. Why are they distinct?

$\endgroup$
1
  • $\begingroup$ Suppose, for instance, $a=b$. What could you say about $(a,b)$ then? $\endgroup$
    – Wojowu
    Commented Feb 15, 2019 at 14:02

1 Answer 1

Reset to default
1
$\begingroup$

I have just understood: if, for instance, a=d, then (a,d)=(d,d) is not in R which is impossible. Some blindness.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .