Model theory of monadic predicate logic with the infinity quantifier

This paper establishes model-theoretic properties of ME∞, a variation of monadic first-order logic that features the generalised quantifier ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality (ME and M, respectively). For each logic L∈ { M, ME, ME∞} we will show the following. We provide syntactically defined fragments of L characterising four different semantic properties of L-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence φ to a sentence φp belonging to the corresponding syntactic fragment, with the property that φ is equivalent to φp precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for L-sentences

Directions In Generalized Quantifier Theory

This paper was inspired by the symposium on Generalized Quantifiers held at the 5th European Summer School in Logic, Language and Information in Lisbon, August 1993. We feel that the work presented there motivates a survey of recent research areas and research problems in the field of generalized quantifiers. The speakers at the symposium, Natasha Alechina, Jaap van der Does, Lauri Hella, Michal Krynicki, Michiel van Lambalgen, Kerkko Luosto, Marcin Mostowski, and Jouko Väänänen, have cooperated and made (oral and/or written) contributions and comments to this research survey which we gratefully acknowledge, and without which it would not have been written. But it is easier to produce a paper with two authors than with ten, and so the present two authors take full responsibility for the final formulation of the paper. In addition, we acknowledge comments received from some further colleagues, in particular, Dorit Ben-Shalom, Makoto Kanazawa, Victor Sanchez and Yde Venema. 2 Our purpose here is to indicate the direction of some of this recent research. We shall sketch a few major research areas and research problems. Such a condensed survey may be useful both for the practitioner in the field and for the interested logician, and also for the logic student who is looking around for something to set his or her teeth in. 1 At least, that is our intention. Moreover, through this unified presentation, we hope to illustrate, and to encourage the current confluence and interaction of more mathematical and more linguistic research lines in this area. After some background, the material is presented under ten distinct headings. This is for ease of exposition, but it will become clear that much of the work is interconnected and some of it belongs under more than one heading. 2. ..