An alternative semantics for knowledge.
Gregory Mellema · Notre Dame Journal of Formal Logic · 1979
In 1962, with the publication of Knowledge and Belief ([6]), Jaakko Hintikka proposed an ingenious model set theoretic semantics for knowledge, KB.A somewhat unusual feature of KB, according to which 'Kap -> p 9 is valid but i {x)KaFx -* (x)Fx 9 is not, has been widely criticized. 1 It has been felt that by reading ί {x)KaFx 9 as ''Everything known to a is known by a to be F" (as Hintikka does) rather than "Everything is known by a to be F" (which entails "Everything is F") an unnatural restriction is placed upon quantifiers ranging into epistemic contexts.Consequently, a number of modifications of KB, have in recent years been proposed which manage to avoid this feature. 2In what follows I shall propose still another modification KB* of KB which lacks the Restricted Range feature (as I shall call the invalidity of ( (x)KaFx -> (x)Fx 9 in a situation where ζ Kap -» p 9 is valid).I shall argue that KB* avoids a number of troublesome theorems which show up in these other proposed systems and that the intuitions underlying KB*, while slightly different from those underlying KB, are every bit as natural.Consider:It can easily be shown that all of these formulas are theorems in KB.Now in KB (l)-( 4) are read, respectively, as:(5) Everyone known to a is known by a to be self-identical, (6) For everyone known to a there is someone known to a such that they are known by a to be identical, (7) Any two persons known to a are, if identical, known by a to be identical (8) Everyone known to a is known by a to be either F or not-F.(Here for purposes of simplicity we assume that we are dealing with domains of persons.)It may be a matter of dispute whether these readings