  To give an example, suppose that the language we are considering is LAr+. Since both the relation of being less than and the relation of being the immediate predecessor of are irreflexive over the domain of the natural numbers, the sentence “∀x(Nx→¬Mxx)” would be a logical consequence of every set of premises on criterion (F): no replacement of the non-logical constants “N” and “M” by other non-logical constants of LAr+ turns that sentence into a falsehood. But clearly “∀x(Nx→¬Mxx)” is not a logical consequence of, say, “N0”. This can be justified, e.g., keeping fixed the usual interpretation of “0” and “N” but observing that “M” can be interpreted by means of the reflexive relation of being less than or equal to; under this interpretation, “∀x(Nx→¬Mxx)” is false, although “N0” is true. (Tarski’s remark that the supposition that all objects have names in the language can never be realized can be justified, for example, by observing that there are non-denumerably many sets of natural numbers, but in the languages he considers there are only denumerably many constants.)