mlbook-full-02-02.dvi (1)

mlbook-full-02-02.dvi

Metadata

• Author: cs112.org
• Title: mlbook-full-02-02.dvi

Page Notes

to-process

Highlights

to-process

• Modal Distribution:(p→q)→(p→q) — Updated on 2024-04-27 18:55:35

  • $square(p->q)->(squarep->squareq)$

• Themodal depth md(φ)ofaformulaφis the maximal length of a nested sequence of modal operators. — Updated on 2024-04-27 19:07:47

• Veridicality — Updated on 2024-04-27 19:12:05

  • $boxphi->phi$

• labeled modalities — Updated on 2024-05-03 17:14:47

  • This is something that might be fun to think about! How would modalities with labels function, and how could we think of them semantically?

• M=(W, R, V) — Updated on 2024-05-03 17:18:32

  • Possible worlds models ($bold(M)$) can represent a modal logic problem. The worlds $s,tinW$, accessibility relation $sRt$, and valuation of propositions at a world $V(p,s)$.

• In thefollowing process graph, some states have a unique outgoingarrow (theprocess is “deterministic” there), in others, there are several (the pro-cess is then “non-deterministic”), while there is also a “dead-lock” statewithout any outgoing arrows at all: — Updated on 2024-05-03 17:20:21

  • Deterministic worlds have access to exactly one world. Dead-lock worlds have access to no worlds.

• Amodalformulaφisvalid,writtenas|=φ,ifM,s|=φfor all models and worlds. — Updated on 2024-05-03 17:27:08

  • Modal validity:

A modal formula $phi$ is valid, written $tack.doublephi$, if $bold(M),sstack.doublephi$ for all models and all worlds.

So when we say that an argument is valid in modal logic, i.e. the argument has modal validity, it must be valid in all models and in all worlds.