First order logic with empty structures

Studia Logica 48 (2):169 - 177 (1989)
For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem
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DOI 10.1007/BF02770510
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Quantification Theory and Empty Individual-Domains.Theodore Hailperin - 1953 - Journal of Symbolic Logic 18 (3):197-200.
Some Impredicative Definitions in the Axiomatic Set-Theory.Andrzej Mostowski - 1951 - Journal of Symbolic Logic 16 (4):274-275.

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