The sentence %$(\forall x_{i}) (P(x_{i})\Rightarrow(\forall x_{j})\lnot P(x_{i},x_{j}))$% is true in the interpretation %$M$%, %\[M\models (\forall x_{i}) (P(x_{i})\Rightarrow(\forall x_{j})\lnot P(x_{i},x_{j}))\]% iff

for every sequence %$s$%, %\[M\models (\forall x_{i}) (P(x_{i})\Rightarrow(\forall x_{j})\lnot P(x_{i},x_{j}))[s]\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, %\[M\models (P(x_{i})\Rightarrow(\forall x_{j})\lnot P(x_{i},x_{j}))[s']\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either %\[M\not\models P(x_{i})[s']\]% or %\[M\models (\forall x_{j})\lnot P(x_{i},x_{j})[s']\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either %\[M\not\models P(x_{i})[s']\]% or for every sequence %$s''$% that agrees with %$s'$% except possibly at %$j$% %\[M\models \lnot P(x_{i},x_{j})[s'']\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either %\[M\not\models P(x_{i})[s']\]% or for every sequence %$s''$% that agrees with %$s'$% except possibly at %$j$% %\[M\not\models P(x_{i},x_{j})[s'']\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either it is not the case that %$s'^{*}(x_{i})$% is in %$P^{M}$% or for every sequence %$s''$% that agrees with %$s'$% except possibly at %$j$% %\[M\not\models P(x_{i},x_{j})[s'']\]% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either it is not the case that %$s'^{*}(x_{i})$% is in %$P^{1M}$% or for every sequence %$s''$% that agrees with %$s'$% except possibly at %$j$%, it is not the case that %$\langle s''^{*}(x_{i}),s''^{*}(x_{j})\rangle$% is in %$P^{2M}$% iff

for every sequence %$s$%, for every sequence %$s'$% that agrees with %$s$% except possibly at %$i$%, either it is not the case that the %$i$%-th member of %$s'$% is in %$P^{1M}$% or for every sequence %$s''$% that agrees with %$s'$% except possibly at %$j$%, it is not the case that %$\langle s''_{i},s''_{j}\rangle$% is in %$P^{2M}$% iff

for every member %$d$% of the domain of %$M$%, either it is not the case that %$d$% is in %$P^{1M}$% or for every member %$d'$% of the domain of %$M$%, it is not the case that %$\langle d,d'\rangle$% is in %$P^{2M}$%

-- ShaughanLavine - 15 Oct 2009