Table 4 Properties of the has_grain relation (in first order logic)

Irreflexivity

∀x ~(x has_grain x)

Assume that x has_grain x. But then, by definition, x has_proper_part x. This is impossible, thus the proposition follows.

Symmetry

∀x ∀y (x has_grain y ↔

~(y has_grain x))

Assume both that x has_grain y and that y has_grain x. But then, by definition, x has_proper_part y, as well as y has_proper_part x. This is impossible, thus the proposition follows.

Non-Transitivity

~∀x ∀y ∀z (x has_grain y ⋀

y has_grain z ⊃ x has_grain z)

Proof by non-transitive example: A galaxy is a star collection and a star is a molecule collection, but a galaxy is not a molecule collection.

Propagation I

∀x ∀y ∀z (x has_part y ⋀

y has_grain z ⊃ x has_part z)

Follows from the transitivity of has_part and the fact that has_grain implies has_proper_part.

Propagation II

∀x ∀y ∀z (x has_grain y ⋀

y has_part z ⊃ x has_part z)

Follows from the transitivity of has_part and the fact that has_grain implies has_proper_part.