part of my propositional calculus homework is to answer the following question:
Give a minimum size structure which shows that the following sentence is not valid.
$\forall x \forall y \forall z((x < y ∧ y < z) ⊃ x < z)$
My first question is that how could this sentence possibly be invalid? I mean surely given the antecedent the consequent must hold. I'm thinking of the naturals and reals for universes but this sentence always holds. Second, what does a minimum sized structure mean?
Can somebody help me come up with a model such that the sentence is invalid?
Minimum-sized is asking you to find a structure with the smallest possible number of elements in which the sentence is not valid. Just because the relation is given the name $<$ doesn't mean that it has to have the usual properties of an ordering relation. Take a model with 2 elements $a$ and $b$ say and define $x < y$ to hold iff $x \neq y$. Then $a < b$ and $b < a$, but $\lnot a < a$. I'll leave it to you to show that the sentence holds for any binary relation on a singleton set (there are only two such relations to check) (and hence that 2 is the minimum size).