A Neighbourhood Frame for T with No Equivalent Relational Frame

✍ Scribed by Martin Gerson

• Book ID: 102485311
• Publisher: John Wiley and Sons
• Year: 1976
• Tongue: English
• Weight: 378 KB
• Volume: 22
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19760220104

✦ Synopsis

A NEIGHBOUREOOD FRAME FOR T WITH NO E QUIVALEXT RELATIONAL FRAME by M A B ~ GERSON in Burnaby (Canada)I)

We present a normal neighbourhood frame with the property that no relational frame is equivalent to it; i.e., no relational frame models exactly the same formulae.

This answers the question presented by SEOERBERC [3] and by HmssoN and GARDEN-FORS [2]: whether the neighbourhood or SCOTT-MONTAGUE semantics has the same strength or depth with respect to normal logics as the relational or KRIPKE semantics.

Our definitions of neighbourhood frames, assignments on neighbourhood frames, and validity in neighbourhood frames will be as in GERSON [l] : definitions of relational frames, assignments on and validity in relational frames, as in THOMASON [4]. A modal logic is normal if and only if u ( p --t q) --+ (up --t nq) is provable and the rule of necessitation holds. A neighbourhood frame is normal if and only if the set of all neighbourhoods of any point is a filter. T is the smallest normal modal logic in which up -+ p is provable.