Is it true that all processes where entropy increases are permitted by the second law, if the system is isolated? It is possible to deduce that in a thermodynamic  process for an isolated system $\mathrm{d}S$ has to be greater than zero, from this it follows trivially that $ \Delta S > 0$. 
It is usually said then that in an isolated system, thermodynamic processes always increase entropy between the initial and final states. My question is: is the converse true? Meaning, is it true that if $\Delta S > 0 $ then the process is permitted by the second law? 
 A: Spontaneity of a thermodynamic process can be analysed from the perspective of free energy. 
Lets look at the Gibbs' free energy:
$$dG=dH-TdS$$
For a process to be spontaneous, $dG<0$. Likewise, a non-spontaneous process is equivalent to $dG>0$. At equilibrium, we have $dG=0$.
Consider a couple of scenarios:


*

*$dH>0$ & $TdS>0$: Then $dG<0$ or $dG>0$ depending on the relative size of $dH$ vs $TdS$. If $dH>TdS$, then $dG>0$ and is non-spontaneous. If $dH<TdS$, then $dG<0$ and is spontaneous.

*$dH<0$ & $TdS>0$: Then $dG<0$ and is always spontaneous.

*$dH>0$ & $TdS<0$: Then $dG>0$ and is always non-spontaneous.

*$dH<0$ & $TdS<0$: Then $dG<0$ or $dG>0$ depending on the relative size of $dH$ vs $TdS$. If $dH<TdS$, then $dG>0$ and is non-spontaneous. If $dH>TdS$, then $dG<0$ and is spontaneous.


So to answer your question: No, it is not necessary to have $dS>0$ for your process to be 'permitted'. Instead you could have the same situation as case 4. However in any process the total entropy change has to be $dS_{tot}\ge0$ by the second law of thermodynamics.
