Four-particle correlations in fermion systems at finite temperatures a
re investigated with special attention to the formation of a condensat
e. Instead of the instability of the normal state with respect to the
onset of pairing described by the Gorkov equation, a new equation is o
btained which describes the onset of ''quartetting.'' Applying to symm
etric nuclear matter, below a critical density, the four-particle cond
ensation (alpha-like quartetting) is found to be favored over deuteron
condensation (tripler pairing). This pairing-quartetting competition
is expected to be a general feature of interacting fermion systems, su
ch as the exciton-biexciton system in excited semiconductors. Possible
experimental consequences are pointed out.