I was discussing with a student today the nature of the non-commutativity of cup-product at the level of cochains. In trying to explain what happens, I came up with the following statement.

Theorem.Fix a commutative ring $R$. Suppose $F:\mathrm{Top}^{\mathrm{op}}\to \mathrm{cDGA}_R$ is a contravariant functor from spaces to commutative DGAs over $R$, such that $X\mapsto H^*(F(X))$ is ordinary cohomology with coefficents in $R$. Then $R$ contains $\mathbb{Q}$ as a subring.

I'm pretty sure this "Theorem" is true. But I don't have a proof at hand.

The question is: does anyone have a proof, or know of one in the literature? I'm particularly interested in seeing a proof which is relatively "elementary", in the sense of not requiring much more heavy machinery than is needed in order to make the statement.

*Added.* Tyler points out in his answer that this can't be true as stated. We should add a hypothesis, such as: F takes homotopy pushouts of spaces to homotopy pullbacks of chain complexes.