Provide a satisfying and falsifying interpretation for the following WFF:
∀x∀y(P(x,y) ↔ P(y,x))
my attempt:
x,y are numbers
P(x,y): x > y
falsifying intepretation:
x: 5
y: 3
so the function would be false because it would not work both ways for the ↔ connective.
Is my interpretation sound? or am I completely missing the question?
We need to produce (1) a structure in which the given sentence is true and (2) a structure in which it is false. By structure one means a non-empty set together with an interpretation of the binary predicate symbol $P$.
For true, let the underlying set be the set $\{1,2,3\}$ and let the interpretation of $P$ be the equality relation. If we call this structure $M$, it is clear that our sentence is true in $M$.
For false, let the underlying set be the same, and let the interpretation of $P$ be the ordinary $\lt$ relation. The given sentence is false in this structure.