Medicinal compositions for treating, ameliorating or preventing diseases with sensitivity to 3,6-anhydrogalactopyranose are represented by formula (I): These compositions include foods, drinks, cosmetics, etc. containing as the active ingredient at least one such compound, its aldehyde, its hydrate and 2-O-methylated derivatives thereof and soluble sugar compounds containing the above compound. These compounds show an apoptosis-inducing activity, carcinostatic activity and inhibitory activities on the production of active oxygen, lipid peroxide radicals and NO, which makes them useful also as the active ingredient of antioxidants and preservatives.