The law of demand asserts that as the price of a good rises, ceteris paribus (other things being equal), the quantity demanded falls, and as the price of a good falls, ceteris paribus, the quantity demanded rises. That is, the price and quantity demanded are negatively inclined ceteris paribus.
The ceteris paribus assumption entails that all other factors except price are held constant: incomes, prices of other goods, fashions, expectations, information, preferences/tastes, population, the weather, etc.
What is the epistemological status of such a proposition? If analytic a priori, it must be regarded as true by virtue of the meanings of the terms used. But that entails that we simply define the “law of demand” to mean that as the price of a good rises, ceteris paribus, the quantity demanded falls, etc.
Such a necessary analytic a priori statement would be tautologous and informationally vacuous, and would tell us nothing necessarily true of the real world.
It follows that, if the law of demand were really understood this way, it could only be asserted of the real world by transformation into a synthetic a posteriori statement (as pure geometry is transformed into synthetic a posteriori propositions when asserted as empirical statements about real space).
But, at that point, one must ask: how is such an assertion proven? How do we prove that the law of demand is empirically true?
The history of how neoclassical economics has unsuccessfully attempted to prove the law of demand is told by Steve Keen in Debunking Economics: The Naked Emperor Dethroned? (rev. edn. 2011). pp. 38–73, though I will leave a review of this for another post.
Keen, Steve. 2011. Debunking Economics: The Naked Emperor Dethroned? (rev. and expanded edn.). Zed Books, London and New York.