Inflation supported by a real massless scalar inflaton field $\varphi$ whose potential is identically equal to zero is described. Assuming that inflation takes place after the Plank scale (after quantum gravity effects are important), zero potential is concomitant with an initial condition for $\varphi$ that is exponentially more probable than an initial condition that assumes an initial inflaton potential of order of the Planck mass. The Einstein gravitational field equations are formulated on an eight-dimensional spacetime manifold of four space dimensions and four time dimensions. The field equations are sourced by a cosmological constant $\Lambda$ and the real massless scalar inflaton field $\varphi$. Two solution classes for the coupled Einstein field equations are obtained that exhibit temporal exponential \textbf{deflation of three of the four time dimensions} and temporal exponential inflation of three of the four space dimensions. For brevity this phenomenon is sometimes simply called ``inflation." We show that \textbf{the extra time dimensions do not generally induce the exponentially rapid growth of fluctuations of quantum fields.} Comoving coordinates for the two \textbf{unscaled} dimensions are chosen to be $(x^4 , x^8 )$ (unscaled means a constant scale factor equal to one). The $x^4$ coordinate corresponds to our universe's observed physical time dimension, while the $x^8$ coordinate corresponds to a new spatial dimension that may be compact. $\partial_{x^8}$ terms of $\varphi$ and the metric are seen to play the role of an effective inflaton potential in the dynamical field equations. In this model, after ``inflation" the observable physical macroscopic world appears to a classical observer to be a homogeneous, isotropic universe with three space dimensions and one time dimension.