The aim of logic is to characterize the forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter; thus its concerns combine semantical and inferential notions in an essential way. Up to now most proposed characterizations of logicality of sentence generating operations have been given either in semantical or inferential terms. This paper offers a combined semantical and inferential criterion for logicality (improving one originally

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... ed by Jeffery Zucker) and shows that any quantifier that is to be counted as logical according to that criterion is definable in first order logic. The aim of logic is to characterize the forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter. The sentences involved are analyzed according to their logical (as opposed to grammatical) structure, i.e. how they are compounded from their parts by means of certain operations on propositions and predicates, of which the familiar ones are the connectives and quantifiers of first order logic. To spell this out in general, one must explain how the truth of compounds under given operations is determined by the truth of the parts, and characterize those forms of rules of inference for the given operations that insure preservation of truth. The so-called problem of "logical constants" (Gomez-Torrente 2002) is to determine all such operations. That has been pursued mostly via purely semantical (qua set-theoretical) criteria on the one hand⎯stemming from Tarski (1986)⎯and purely inferential criteria on the other⎯stemming from Gentzen (1935) and pursued by Prawitz (1965), among others⎯even though on the face of it a combination of the two is required. 2 What is offered here is such a combined criterion for quantifiers, whose semantical part is provided by Lindström's (1966) generalization of quantifiers, and whose inferential part 1 The main body of material for this article was first presented for a talk at the ESSLLI Workshop on Logical Constants, Ljubljana, Aug. 9, 2011. A second presentation was made on May 23, 2012 at a conference at CUNY in honor of Sergei Artemov, on the occasion of his 60 th birthday. 2 Some further contributions to the semantical approach are Sher (1991) and McGee (1996) , and to the inferential approach is Hacking (1979); Gomez-Torrente (2002) provides a useful survey of both approaches. I have critiqued the semantical approach as given by set-theoretical criteria in Feferman (2000, 2010) where, in conclusion, I called for some combined criterion.