HDS06: van Benthem, Johan (2014) Modal Correspondence Theory. Doctoral thesis, University of Amsterdam.
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Abstract
Modal Correspondence Theory
Johan van Benthem
Modal Correspondence Theory has for its subject the connections between modal formulas and formulas of classical logical systems, both viewed as means of expressing relational properties. Two main questions are treated in this dissertation: which modal formulas are definable in firstorder logic and which firstorder formulas are definable by means of modal formulas? As for the first, it is shown that a modal formula is firstorder definable if and only if it is preserved under ultrapowers. Moreover, two methods are developed, one using firstorder substitutions for secondorder quantifiers to show constructively that modal formulas satisfying certain syntactic conditions are firstorder definable, the other using the LöwenheimSkolem theorem to show that certain modal formulas are not firstorder definable. For the case of modal reduction principles, a class of modal formulas to which most betterknown modal axioms belong, these two methods yield a complete syntactic answer to the first question. As for the second question, there is a theorem by R.I. Goldblatt and S.K. Thomason about \Sigma\Deltaelementary classes of relational structures, characterizing the modally definable ones in terms of closure under four algebraic operations. A new proof of this result is given here, as well as a series of preservation results for the algebraic operations it involves. From these results it follows that a firstorder formula is modally definable only if it is equivalent to a "restricted positive" formula constructed from atomic formulas and the falsum (a constant denoting a fixed contradiction), using conjunction, disjunction and restricted quantifiers.
Item Type:  Thesis (Doctoral) 

Report Nr:  HDS06 
Series Name:  ILLC Historical Dissertation (HDS) Series 
Year:  2014 
Additional Information:  Originally published: February 1977 (Amsterdam). 
Subjects:  Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  11 Jan 2022 23:20 
Last Modified:  11 Jan 2022 23:20 
URI:  https://eprints.illc.uva.nl/id/eprint/1838 
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