  A general method to prove strong laws of large numbers for random fields is
given. It is based on the H\'ajek - R\'enyi type method presented in Nosz\'aly
and T\'om\'acs \cite{noszaly} and in T\'om\'acs and L\'ibor \cite{thomas06}.
Nosz\'aly and T\'om\'acs \cite{noszaly} obtained a H\'ajek-R\'enyi type maximal
inequality for random fields using moments inequalities. Recently, T\'om\'acs
and L\'ibor \cite{thomas06} obtained a H\'ajek-R\'enyi type maximal inequality
for random sequences based on probabilities, but not for random fields. In this
paper we present a H\'ajek-R\'enyi type maximal inequality for random fields,
using probabilities, which is an extension of the main results of Nosz\'aly and
T\'om\'acs \cite{noszaly} by replacing moments by probabilities and a
generalization of the main results of T\'om\'acs and L\'ibor \cite% {thomas06}
for random sequences to random fields. We apply our results to establishing a
logarithmically weighted sums without moment assumptions and under general
dependence conditions for random fields.
