S. Comer, Classes without the amalgamation property, Pacific Journal of Mathematics, 28(1969), pp. 309-318.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases. In 1957, W. Craig proved the interpolation theorem for first order logic [Cra57] Comer [Com69] showed that the property fails for all finite variable fragments except the onevariable fragment. The n variable fragment of first order logic for short L n contains all first order formulas using just n variables and containing only predicate symbols of arity not higher that n (we assume ....

Comer, S. Classes without the amalgamation property. Pacific Journal of Mathematics, 28:309--318, 1969.

....fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. It s use is exemplified in a wide range of cases. In 1957, W. Craig proved the interpolation theorem for first order logic [Cra57] Comer [Com69] showed that the property fails for all finite variable fragments except the onevariable fragment. The n variable fragment of first order logic for short L n contains all first order formulas using just n variables and containing only predicate symbols of arity not higher than n (we assume ....

Comer, S. Classes without the amalgamation property. Pacific Journal of Mathematics, 28:309--318, 1969.

....lemmas. Logics with the interpolation property include classical (and intuitionistic) propositional calculus, and the modal logics K, K4 and S5. Examples of logics without this property are the nite variable fragments of rst order logic, with the exception of the one variable fragment, Com69] Another important reason for investigating interpolation, is that it can be seen as an intermediary step in proving the Beth de nability property. In a slogan, the Beth property states that implicit de nability equals explicit de nability. Generally, this may be regarded as an indication that ....

S. Comer. Classes without the amalgamation property. Pacic Journal of Mathematics, 28:309-318, 1969.

....(such as local niteness, and the amalgamation property) which Df 2 does not preserve. It was already noticed by Tarski that Df 2 is not locally nite (see, e.g. Henkin, Monk and Tarski [11] Halmos [10] and below) That Df 2 does not have the amalgamation property was rst noticed by Comer [7] (see also Sain [22] and Marx [19] On the other hand, it is known that Df 2 is nitely approximable. Two di erent frame theoretical proofs of this can be found e.g. in Shehtman [23] and Marx [18] Here we give a sketch of an algebraic proof. Theorem 10 Df 2 is nitely approximable. Proof. ....

S. Comer, Classes without the amalgamation property, Pacic Journal of Mathematics, 28(1969), pp. 309-318.

....in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. S. Comer showed that for finite n larger than 1, the embedding property fails in any class of cylindric type algebras of dimension n lying between RCA n and CA n (Comer, 1969). On the other hand, the proof method employed by I. N emeti in (N emeti, 1985) can be used to show that for any ff, the class NCA ff (defined by all CA ff axioms except commutativity of the cylindrifications) and its subclass D ff of cylindric relativised set al..gebras have the strong ....

....and the class of all representable diagonal free cylindric algebras by RDf ff . For finite dimensional cylindric type algebras we use n instead of ff. Axiom C 4 is called the commutativity axiom for the cylindrifications. It corresponds to the law 9x9y 9y9x in first order logic. Remark 0. 1 In (Comer, 1969), the similarity type of diagonal free cylindric algebras is 3 called cylindrification type, a full diagonal free cylindric set al..gebra is called a full cylindrification set al..gebra, etc. He denotes the class Df ff by Cy ff , and RDf ff by RCy ff . Here we use the, nowadays standard, terminology ....

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Comer, S. (1969). Classes without the amalgamation property. Pacific Journal of Mathematics, 28:309--318.

....fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases. In 1957, W. Craig proved the interpolation theorem for first order logic (Craig, 6] Comer [5] showed that the property fails for all finite variable fragments except the onevariable fragment. The n variable fragment of first order logic for short L n contains all first order formulas using just n variables and containing only predicate symbols of arity not higher that n (we assume ....

Comer, S. Classes without the amalgamation property. Pacific Journal of Mathematics, 28:309--318, 1969.

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S. Comer, Classes without the amalgamation property, Pacific Journal of Mathematics, 28(1969), pp. 309-318.

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