Random Graph: Stronger logic but with the zero one law

Abstract

We find a logic really stronger than first order for the random graph with edge probability $\frac{1}{2}$ but satisfies the 0-1 law. This means that on the one hand it satisfies the 0-1 law, e.g. for the random graph ${\mathcal{G}}_{n,1/2}$ and on the other hand there is a formula $\phi (x)$ such that for no first order $\psi (x)$ do we have: for every random enough ${\mathcal{G}}_{n,1/2}$ the formulas $\phi (x),\psi (x)$ equivalent in it.