One says that \(t>0\) is an increase time for the path \(\omega\) of a real- valued process if \(\omega\) is below level \(\omega (t)\) immediately before time \(t\) and above this level immediately after time \(t\). According to a celebrated result of Dvoretzky, Erdős and Kakutani, almost every Brownian path has no increase times. The author of the present paper has, in other recent publications, given criteria for the existence or otherwise of increase times for particular categories of Lévy processes [Stochastics Stochastics Rep. 37, No. 4, 247-251 (1991; Zbl 0739.60065); Lévy processes that can creep downwards never increase (to appear in Ann. Inst. Henri Poincaré)]. In the present article he establishes the following result: for a strictly stable process \(X_ t\), \(t \geq 0\), the probability that there exist increase times is 0 or 1 according as \(P(X_ 1 \geq 0)\) is \(\leq {1 \over 2}\) or \(>{1 \over 2}\). To prove this the author uses results of fluctuation theory for the maxima and minima of stable processes which he applies to a measure of the ``nearly'' increase times.