The oxygen potentials have been measured by means of a thermogravimetric method in the range of $1200\,\sim\,1500^\circ C$ and $10^{-14}\,\leq\,p_{O_2}\,\leq\,10^{-3}$ atmosphere for pure $UO_2$, and $U_{1-y}Er_yO_{2\pmx}$ solid solutions with y=0.02, y=0.06 and y=0.20, respectively. Their oxygen partial pressures were maintained by controlling $CO_2$/CO mixture atmosphere, and the $p_{O_2}$ values corresponding to x of $U_{1-y}Er_yO_{2\pm x}$ solid solutions were measured with an electrolyte oxygen sensor. The lattice parameter of the stoichiometric $U_{1-y}Er_yO_2$ was also determined with use of X-ray diffraction data in the range of y=0.01 to y=0.33. The lattice parameter decreases linearly with an increase in the erbium content. The change of the lattice parameter can be expressed in a linear equation of y as a($\mbox{\AA}$)=5.4695-0.220y for $0\,\leq\,y\,\leq\,0.33$. The $\Delta\overline{G}_{O_2}$ has been found to undergo abrupt increase in the range of -270 to -360 kJ/mole for y=0.06 and -220 to -320 kJ/mole for y=0.20, respectively, in the temperature range of $1200\,\sim\,1500^\circ C$. The differences in $\Delta\overline{G}_{O_2}$ between the values at O/(U+Er)=1.999 and 2.001 decrease with increasing temperature. The $\Delta\overline{G}_{O_2}$ values as a function of temperature for the stoichiometric composition are expressed by $\Delta\overline{G}_{O_2}$(kJ/mole) = -423.43 + 0.089T for $U_{0.94}Er_{0.06}O_{2000}$ and $\Delta\overline{G}_{O_2}$(kJ/mole) = -422.00 + 0.125T for $U_{0.80}Er_{0.20}O_{2000}$. $\Delta\overline{G}_{O_2}$ increases with erbium content, but the effect of the dopant for x=0.01 is less significant than that for stoichiometry. Both $\Delta\overline{S}_{O_2}$ and $\Delta\overline{H}_{O_2}$ for $U_{1-y}Er_yO_{2\pm x}$ showed higher values as increasing y, and sharp maxima at the O/M ratio of approximately 2.005, irrespective of erbium content. The oxygen potentials for $UO_2$ and $U_{0.98}Er_{0.02}O_{2+x}$ can be approximately r...