A proof of Graves' Theorem

I wrote a formal proof for Robert Graves' statement: “If there is no money in poetry, neither is there poetry in money.” I have also included the English version of symbolic statements where I think they would be helpful for the reader, although a formal proof would not include such English statements. These statements are denoted by asterisks and are written in gray. You can highlight gray text to make it easier to read. 

Proposition:

“If there is no money in poetry, neither is there poetry in money.”

Proof:

We begin by defining a few sets, which will be useful in completing the proof:

              

Let M = {x : x form of money}

                      m = {x : x monetary]

                       P = {x : x is a form of poetry}

                       p = {x : x is poetic}

We state an axiom necessary for the proof:

               Poetry completeness axiom- k ∈ p, ⇒ ∃t ∈ P such that t is written about k

*That is, if k is poetic, then there exists a poem that has been written about k*

Thus, we can translate Graves’s statement into the logical statement:

∀x ((x ∈ m) ⇒ ¬ (x ∈ P)) ⇒∀x ((x ∈ p) ⇒ ¬ (x ∈ M))

*That is, if it is the case that, for all x, if x is monetary, then x is not a form of money, then it is also the case that, for all x, if x is poetic, x is not a form of money. *

Notice that, by the properties of existential quantifiers and logical operators,

 (∀x ((x ∈ m) ⇒ ¬ (x ∈ P)) ⇒ ∀x ((x ∈ p) ⇒ ¬ (x ∈ M))) ⇔ (¬∃x ((x ∈ m) ∧ (x ∈ P)) ⇒ (¬∃x ((x ∈ p) ∧ (x ∈ M)))

Furthermore, since every statement is logically equivalent to its contrapositive,

 (¬∃x ((x ∈ m) ∧ (x ∈ P)) ⇒ (¬∃x ((x ∈ p) ∧ (x ∈ M))) ⇔ (∃x ((x ∈ p) ∧ (x ∈ M)) ⇒ (∃x ((x ∈ m) ∧ (x ∈ P)))

Thus, 

(∀x ((x ∈ m) ⇒ ¬ (x ∈ P)) ⇒ ∀x ((x ∈ p) ⇒ ¬ (x ∈ M))) ⇔ (∃x ((x ∈ p) ∧ (x ∈ M)) ⇒ (∃x ((x ∈ m) ∧ (x ∈ P)))

So, it suffices to prove that:

∃x ((x ∈ p) ∧ (x ∈ M)) ⇒ (∃x ((x ∈ m) ∧ (x ∈ P)))

*That is, if it is the case that there exists an x such that x is poetic and x is a form of money, then there must also exist an x such that x is monetary and x is a form of poetry. *

We prove this conditional statement directly: 

*That is, we assume the hypothesis and derive the conclusion*

Assume that 

∃x ((x ∈ p) ∧ (x ∈ M))

*This is, the hypothesis of the conditional statement we are trying to prove*

Hence, some form of money is poetic, so by the poetic completeness axiom, a poem must be written about that form of money.

Since, by definition, ∀x((x ∈ M) ⇒ x ∈ m)), it follows that M ⊆ m   

*That is, everything that is a form of money relates to money, and is thus monetary*

So, such a poem must be monetary.

This yields: (∃x ((x ∈ m) ∧ (x ∈ P)))

*This is, the conclusion of the conditional statement we are trying to prove*

Thus ∃x ((x ∈ p) ∧ (x ∈ M)) ⇒ (∃x ((x ∈ m) ∧ (x ∈ P))), so the proof is complete.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Q.E.D.


- Kevin