Gillian Russell: Barriers to entailment Hopefully we can translate the structure of the proof into knitting, line by line.
"Suppose Γ is a satisfiable set of R-preserved sentences and is R-fragile."
Γ is the rows of some scarf that have already been knitted, while tells us about some arbitrary set of any rows at all.
"Let M be a model which satisfies Γ"
Let M be any scarf with the rows Γ already knitted.
"Either is true in M or it isn’t."
Either the rows described by will be added to M, or they won't.
"If it isn’t, then M is a counterexample showing that Γ⊭ "
If the rows are not added to M, the the rows Γ could not have led us to conclude that they would be added.
"But if is true in M, then since is R-fragile there is some M' such that R(M,M') and is not true in M'."
But if the rows are added to the scarf, then since they might not have been added (they are fragile), there is some other scarf M' such that the rows were not added.
"Since each member of Γ was particular, each member of Γ is also true in M'."
Since the rows Γ have already been knitted, they are the same in both scarves. M' also has the rows Γ
"Hence M' is our counterexample, and Γ⊭ ."
In which case, the other scarf M' has the rows Γ but not the rows , and so again, the rows Γ could not have led us to conclude that the rows described by would be added.
Clear as mud? There was a bit of trouble with the parsing, such that I had to use mathjax for the delta but not the Gamma. Odd. Let me know if it doesn't parse well.