An analysis of the physical significance of the Griffith criterion σG from the viewpoint of the first law of thermodynamics (thermodynamic approach) and the viewpoint of the thermofluctuation-theory of strength (kinetic approach) makes it possible to conclude that the criterion is not the fracture criterion it is usually thought to be. It is actually a safe stress and, thus, is preferably treated as a criterion of the indestructibility of a material. This conclusion is supported by calculations of the quantity σG (Griffith formula) and the safe σG and critical σc stresses (formulas of thermofluctuation theory) for a thin plate containing transverse edge cracks and loaded uniaxially. Organic glass (polymethylmethacrylate), inorganic glass (silicate plate glass), and a crystalline polymer (capron) are used as examples to show that the value of σG is equal or close to the safe stress σ0 but lower than the critical stress σc by one or two orders of magnitude.