While its non-rivalrous property says that no one should be excluded from the enjoyment of a public good (since the marginal cost of benefiting from it is zero), non-excludability implies that no one can be excluded. This too has important implications: it means that knowledge cannot be provided privately. For assume someone produced, say, a theorem. Assume the theorem is valuable in providing insights into how to solve practical problems. But assume also that the theorem cannot be kept secret and must be immediately available. Then, since anyone can immediately enjoy the theorem, the individual could make no profit from it. Competition would drive its price to zero. At any positive price, it would pay someone to get the information (which by assumption he could do) and undercut the seller.
Some forms of knowledge are (or can be made) "excludable." For example, in some industries, such as metallurgy, trade secrets are used. To be sure, the firms run a risk: a competitor, observing the new alloy, could analyze its composition and infer the mix of metals (and with modern techniques, even the relative proportion of the atoms). The firm might have a hard time inferring precisely how the alloy is made, but there is no way that rivals can be excluded from knowledge of the chemical composition and the properties of the alloy. By the same reasoning, when a firm discovers that consumers love, say, yogurt, others cannot be excluded from using that knowledge to put on the market their own yogurts.
Patents provide the exclusive right to the inventor to enjoy the fruits of his innovative activity over a limited period of time (17 years), but in return, the inventor must disclose the details of his invention. The fact of the invention, let alone the details provided in the patent application, make an enormous amount of knowledge freely available. The development of rayon provided other researchers with enormous information: it demonstrated the feasibility of a synthetic fiber - knowledge which itself was of enormous commercial value and which enhanced incentives for others to look for other synthetic fibers. Indeed, research in chemicals often consists of looking for slight variations of the original chemical.
It is precisely because of the high value of the knowledge disclosed through the patent process (and the limited duration of the patent) that some firms prefer the seemingly less protective route of trade secrecy.
But because the returns to some knowledge can, to some extent, be appropriated (there is some degree of non-excludability) knowledge is often thought of as an impure public good.