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We present a method of transferring Tarski's technique of classifying finite order concepts by means of truth-definitions into finite model theory. The other considered question is the problem of representability relations on words or natural numbers in finite models. We prove that relations representable in finite models are exactly those which are of degree ≤ 0 . Finally, we consider theories of sufficiently large finite models. For a given theory T we define sl(T ) as the set of all sentences true in almost all finite models for T . For theories of sufficiently large models our version of Tarski's technique becomes practically the same as the classical one. We investigate also degrees of undecidability for theories of sufficiently large finite models. We prove for some special theory ST that its degree is stronger than 0 but still not more than Σ 0 2 .Mathematics Subject Classification: 03B15, 03C13, 03C85, 03D25, 03D40.

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We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second-order logic. Two arguments for this thesis are formulated. First, we show that Barwise's so-called test of negation normality works properly only when assuming our main thesis. Second, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they are bounded by second-order existential properties. Moreover, there are known examples of everyday language sentences that are the most difficult in this class ( NPTIME-complete).

The paper gives a survey of known results related to computational devices (finite and push-down automata) recognizing monadic generalized quantifiers in finite models. Some of these results are simple reinterpretations of descriptive-feasible correspondence theorems from finite-model theory. Additionally a new result characterizing monadic quantifiers recognized by push down automata is proven.The aim of the work is presentation of the state of knowledge and main research problems in computational approach to finite interpretations of monadic generalized quantifiers. We will concentrate on purely logical approach -that is we will consider mainly structures without standard linear orderings. However proper results for linearly ordered structures will be mentioned when relevant.Let us observe that monadic quantifiers -being the first natural class of quantifiers investigated from computational point of view -are not systematically treated in logical literature. In the first attempt [6] of surveying the subject of generalized quantifiers from computational point of view, monadic quantifiers are only shortly mentioned. GeneralitiesBy monadic generalized quantifiers we mean Lindström's quantifiers (see [5]) of types of the form (1, 1, . . . , 1); if the number of ones in the type signature is n then we say that the corresponding quantifier is a monadic * The research reported here has been supported by the research grant of Polish National Commitee of Scientific Research (KBN) 433/H01/95/08.