Consider(normal)modallogics.LetIsBlue(Pen),IsInPocket(Pen)bepossiblefacts,letAnne,Jim beagentsandletKAnne,KJimdenotethemodaloperatorsAnneknowsthat...andJimknows that...respectively. Foreachofthefollowingfacts,sayifitistrueorfalse.

(a)IfKAnneIsBlue(Pen)andKAnne(IsBlue(Pen)KJimIsBlue(Pen))hold,thenKAnneKJimIsBlue(Pen)) holds

(b)IfKJimIsBlue(Pen)andIsBlue(Pen)IsInPocket(Pen)hold,thenKJimIsInPocket(Pen)holds

(c)IfKAnneIsBlue(Pen)holds,thenKAnneIsBlue(Pen)holds

(d)IfKAnneIsBlue(Pen)holds,thenKAnneIsBlue(Pen)holds

Consider (normal) modal logics. Let IsInPocket(Pen), IsBlue(Pen) be possible facts, let Jim, Anne be agents and let KJim, KAnne denote the modal operators Jim knows that... and Anne knows that... respectively. For each of the following facts, say if it is true or false.

(a) If KJimIsInPocket(Pen) holds, then KJimIsInPocket(Pen) holds

(b) If KJimIsInPocket(Pen) holds, then KJimIsInPocket(Pen) holds

(c) If KJimIsInPocket(Pen) and KJim(IsInPocket(Pen) KAnneIsInPocket(Pen)) hold, then KJimKAnneIsInPocket(Pen)) holds

(d) If KAnneIsInPocket(Pen) and IsInPocket(Pen) IsBlue(Pen) hold, then KAnneIsBlue(Pen) holds