When the neutral interstellar medium is exposed to extreme-ultraviolet and soft X-ray radiation, the argon atoms in it are far more susceptible to being ionized than the hydrogen atoms. We make use of this fact to determine the level of ionization in the nearby warm neutral medium. By analyzing Far-Ultraviolet Spectroscopic Explorer observations of ultraviolet spectra of 44 hot subdwarf stars a few hundred parsecs away from the Sun, we can compare column densities of Ar I to those of O I, where the relative ionization of oxygen can be used as a proxy for that of hydrogen. The measured deficiency [ArI/OI]=-0.427+/-0.11dex below the expectation for a fully neutral medium implies that the electron density n(e){approx}0.04/cm3 if n(H)=0.5/cm3. This amount of ionization is considerably larger than what we expect from primary photoionizations resulting from cosmic rays, the diffuse X-ray background, and X-ray emitting sources within the medium, along with the additional ionizations caused by energetic secondary photoelectrons, Auger electrons, and photons from helium recombinations. We favor an explanation that bursts of radiation created by previous, nearby supernova remnants that have faded by now may have elevated the ionization, and the gas has not yet recombined to a quiescent level. A different alternative is that the low-energy portion of the soft X-ray background is poorly shielded by the H I because it is frothy and has internal pockets of very hot, X-ray emitting gases.