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Universal First-Order Definability in Modal Logic

✍ Scribed by R. E. Jennings; D. K. Johnston; P. K. Schotch

Publisher
John Wiley and Sons
Year
1980
Tongue
English
Weight
198 KB
Volume
26
Category
Article
ISSN
0044-3050

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✦ Synopsis

UPiIVERSAL FIRST-ORDER DEFINABILITY I N MODAL LOGIC by R. E. JENNIXGS and D. K. JOHNSTON in Burnaby, British Columbia (Canada) and P. K. SCHOTCH in Halifax, Nova Scotia (Canada)l)

In [ l ] R. I. GOLDBLATT presents a model theoretic characterization of the class of modal sentences determined by first-order conditions upon frames. I n this paper we extend these results to a wider class of relational model structures, in which the specified arity of R is at least 2 .

Where possible we have adopted the definitions of [l]. However, we adopt a different notion of a frame and a correspondingly wider notion of first-order definability. D e f i n i t i o n 1. A modal f r a m e s = ( U , R) consists of a set U + 0 on which an n-ary relation R is defined. 9 is said to be an n-ary frame, provided that R is n-ary.

Valuations on 9, an n + 1-ary frame, are defined in the usual way for PC formulae.

For modal formulae

As in [l] we say that a is valid in 9 or 9 is a model of o( (9 k a ) iff V ( a ) = U for every valuation V on 9. D e f i n i t i o n 2. A formula a is n-adically first-order definable iff there is an n-adic first-order sentence a* such that for every n-ary frame 9, 9 k 01 iff 9 k 01* in the first-order sense. D e f i n i t i o n 3. A formula 01 is universally first-order definable iff for every n, 01 is D e f i n i t i o n 4. A formula a is universally iirst-order undefinable iff for every 92,

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