# John Nolt - Logics, chp. 11-12

I am taking an advanced logic class this semester. Som of the reading material has been posted in our internal system. I'll post it here so that others may get good use of it as well. The text in question is John Nolt's *Logics* chp. 11-12. I remade the pdfs so that they ar smaller and most of the text is copyable making for easyer quoting. Enjoy John Nolt - Logics, chp 11-12

**Edit** - A comment to the stuff (danish)

Dær står i Nolt kap. 12, at:

”We have said so far that the accessibility relation for all forms of alethic

possibility is reflexive. For physical possibility, I have argued that it is transitive as

well. And for logical possibility it seems also to be symmetric. Thus the accessibil-

ity relation for logical possibility is apparently reflexive, transitive, and symmetric.

It can be proved, though we shall not do so here, that these three characteristics

together define the logic S5, which is characterized by Leibnizian semantic... That

is, making the accessibility relation reflexive, transitive, and symmetric has the

same effect on the logic as making each world possible relative to each.” p. 343

mæn min entuisjon sagde maj strakes, at dette var forkert, altså, at de er forkert at

R1. For any world, that world relates to itself. (Reflexsive)

R2. For any world w1 and for any world w2, if w1 relates to w2, then w2 relates to w1. (Symmetry)

R3. For any world w1, for any world w2, and for any world w3, if w1 relates to w2, and w2 relates to w3, then w1 relates to w3. (Transitive)

er ækvivalænt mæd

R4. For any world w1 and for any world w2, w1 relates to w2. (Omni-relevans)

Mæn de er ganske rægtigt, at ves man prøver at køre diverse beviser igænnem, så kan man godt bevise fx (◊A→☐◊A) vha. en modæl som er reflexsive, symmetrical og transitive (jaj valgte 1r1, 2r2). Mæn stadig er dær någet galt. Di forskællige verdener er helt isolerede, modsat vad di er i givet R4. Jaj googlede de, og andre har osse bemærket de:

”Requiring the accessibility relation to be reflexive, transitive and symmetric is to require that it be an equivalence relation. This isn’t the same as saying that every world is accessible from every other. But it is to say that the class of worlds is split up into classes within which every world is accessible from every other; and there is no access between these classes. S5, the system that results, is in many ways the most intuitive of the modal systems, and is the closest to the naive ideas with which we started.”