<p>Here is a sketch (massively simplifying for reasons of
pedagogy):</p>

<p><strong>Setting the scence.</strong> You can think of logic as a tool, an API in computing parlance,
to reason about things, where reasoning means deriving true
statements from true assumptions. The things that we reason about
using logic is called a <strong>model</strong>.  What makes logic useful is that
this API, in particular its dominant strain called first-order
logic (of course there are other logics), is completely generic, 
meaning we can use it to reason
about teddy bears, cars, bridges over rivers, stars, computation,
whatever you want. Because logic is so generic, we can build up a
generic reasoning infrastructure that can be reused in many
domains. Example include interactive theorem provers, SAT
solvers, textbooks, social conventions such as learned journals,
career structures in the sciences, conventions about when we deem to have
lost/won an argument etc. So logic is a great labour saving
device, because we don't have to reinvent these core tools again
and again for everything that we want to reason about.</p>

<p><strong>Is logic all we need to reason?</strong> Naturally not. Because logic is generic, we have to inject domain
specific assumptions, usually called axioms, to cater for domain specific
knowledge (e.g. ZFC or Peano arithmetic to found mathematics,
deontic axioms for reasoning about ethics etc). </p>

<p><strong>What has all of   this to do  with Kripke semantics?</strong>
Often we want to reason about things that change over
time (e.g. traffic at a traffic light, concurrent processes using
shared memory communication, knowledge of observers).  We can do
this by injecting suitable axioms into first-order logic for each
domain separately.  However, just as with first-order logic, it
turns out that a lot of the axioms that we need to reason about
change are similar regardless of what changes over time.  So why
not 'factor out'    this similarity and create an API for reasoning
about change, just as we did with first-order logic as an API for
general reasoning? Let this new logic/API be called modal logic.</p>

<p>Now we come back to the notion of model, because models are what
we reason about.</p>

<p><strong>What is this API called modal logic about?</strong> Well, things that change.
Let's call them <strong>worlds</strong>. Now ,
the worlds change into each other over time.  Would a description of 
this change not be a 
binary relation on the set of world? The set of worlds together with the
relation is the model.</p>

<p>Voila, Kripke semantics.</p>

<p><strong>Warning:</strong> the description above is simplistic, and terminology is
not uniform.  Moreover, for most modal logics the model must also 
account for variables.
In more neutral terminology (and still simplifying), a model is a
triple $( W, R, \sigma )$ where</p>

<ul>
<li><p>$W$ is a non-empty set
(elements of W are called nodes or worlds or points).</p></li>
<li><p>$R$ is a binary relation on $W$, and is known as the
accessibility relation which regulates how 'time' flows.</p></li>
<li><p>$\sigma$ is a map from variables (in the ambient logic) to the
powerset of W. The idea behind $\sigma$ is that $\sigma$
controls at what points/worlds the variables are true.</p></li>
</ul>

<p>If you want to know even more, maybe check the Wikipedia entries 
on <a href="https://en.wikipedia.org/wiki/Kripke_semantics" rel="nofollow">Kripe semantics</a> or <a href="https://en.wikipedia.org/wiki/Modal_logic" rel="nofollow">modal logic</a> or any good book on 
the subject such as <a href="http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/modal-logic" rel="nofollow">Modal Logic</a>, by Blackburn, de Rijke and Venema.</p>

CC BY-SA 3.0
2013-08-08T13:25:17.237Z
Added note on the existence of more than one kind of logic.
Kile Kasmir Asmussen
CC BY-SA 3.0
2013-08-08T13:45:18.067Z
@KarlDamgaardAsmussen that's right, there are many logics. First-order logic is somewhat canonical though (e.g. the meta-theory of most logics is developed in FOL), and it might be helpful at for the original poster not to worry too much about other logics.
Martin Berger
CC BY-SA 3.0
2013-08-08T15:24:07.347Z
I am not talking about First-Order/Second-Order, I mean Intuitionistic/Classical and weaker systems.
Kile Kasmir Asmussen
CC BY-SA 3.0
2013-08-09T19:39:47.947Z
@MartinBerger, thank you, but as D.W. says this contradicts with the notion that states are equal to worlds. In particular, this seems to say that a collection of states is the world, which looks to me like the world is the same as the model. Could you clarify in the last part how models fit in to the story? Thanks for the explanation so far though :-)
Mythio
CC BY-SA 3.0
2013-08-15T12:00:28.543Z
@D.W. You are right. I'm sorry, the terminology is not uniform (partly because
there are so many different modal logics), and confusingly I have
made terminological choices that are unusual. I added some explanation. I hope it's more clear now.
Martin Berger
CC BY-SA 3.0
2013-08-08T10:00:13.800Z
2013-08-15T11:58:38.443Z
2013-08-15T11:58:38.443Z
Martin Berger
<p>Can someone explain to me how the concepts of states, models and worlds work together in Kripke semantics?</p>

<p>I've been trying to piece together how the parts work are linked together but cannot figure it out.</p>

<p>From what I understand, a model $M$ consists of worlds. 
Now as far as I can gather from that, states would be the same as worlds.
However, I cannot find this confirmed anywhere.</p>

<p>Could someone explain to me how these concepts work?
Obviously, I've googled them, but most texts are quite complicated to grasp with little experience on the subject. Even textbooks remain quite vague. </p>

CC BY-SA 3.0
2013-08-08T00:33:49.980Z
You can understand a state as a world and a set of states as a set of worlds. The different terms are used by different communities. If you are interested in logical and meta-logical characterisations, then the notion of a world may suffice. If you want to analyse a single model and that model is generated by a program, the structure of the state becomes important.
Vijay D
<modal-logic>
Explanation of states, models and worlds in Kripke semantics?
Mythio
# Explanation of states, models and worlds in Kripke semantics?

(Asked by: username_0 on 2013-08-07)

Can someone explain to me how the concepts of states, models and worlds work together in Kripke semantics?

I've been trying to piece together how the parts work are linked together but cannot figure it out.

From what I understand, a model $M$ consists of worlds.
Now as far as I can gather from that, states would be the same as worlds.
However, I cannot find this confirmed anywhere.

Could someone explain to me how these concepts work?
Obviously, I've googled them, but most texts are quite complicated to grasp with little experience on the subject. Even textbooks remain quite vague.

**Tags:** modal-logic

> **Comments:**
> **username_1**: You can understand a state as a world and a set of states as a set of worlds. The different terms are used by different communities. If you are interested in logical and meta-logical characterisations, then the notion of a world may suffice. If you want to analyse a single model and that model is generated by a program, the structure of the state becomes important.  

---

## 1 answers

### Answer by username_2 on 2013-08-08. Score: 5

Here is a sketch (massively simplifying for reasons of
pedagogy):

**Setting the scence.** You can think of logic as a tool, an API in computing parlance,
to reason about things, where reasoning means deriving true
statements from true assumptions. The things that we reason about
using logic is called a **model**. What makes logic useful is that
this API, in particular its dominant strain called first-order
logic (of course there are other logics), is completely generic,
meaning we can use it to reason
about teddy bears, cars, bridges over rivers, stars, computation,
whatever you want. Because logic is so generic, we can build up a
generic reasoning infrastructure that can be reused in many
domains. Example include interactive theorem provers, SAT
solvers, textbooks, social conventions such as learned journals,
career structures in the sciences, conventions about when we deem to have
lost/won an argument etc. So logic is a great labour saving
device, because we don't have to reinvent these core tools again
and again for everything that we want to reason about.

**Is logic all we need to reason?** Naturally not. Because logic is generic, we have to inject domain
specific assumptions, usually called axioms, to cater for domain specific
knowledge (e.g. ZFC or Peano arithmetic to found mathematics,
deontic axioms for reasoning about ethics etc).

**What has all of this to do with Kripke semantics?**
Often we want to reason about things that change over
time (e.g. traffic at a traffic light, concurrent processes using
shared memory communication, knowledge of observers). We can do
this by injecting suitable axioms into first-order logic for each
domain separately. However, just as with first-order logic, it
turns out that a lot of the axioms that we need to reason about
change are similar regardless of what changes over time. So why
not 'factor out' this similarity and create an API for reasoning
about change, just as we did with first-order logic as an API for
general reasoning? Let this new logic/API be called modal logic.

Now we come back to the notion of model, because models are what
we reason about.

**What is this API called modal logic about?** Well, things that change.
Let's call them **worlds**. Now ,
the worlds change into each other over time. Would a description of
this change not be a
binary relation on the set of world? The set of worlds together with the
relation is the model.

Voila, Kripke semantics.

**Warning:** the description above is simplistic, and terminology is
not uniform. Moreover, for most modal logics the model must also
account for variables.
In more neutral terminology (and still simplifying), a model is a
triple $( W, R, \sigma )$ where

* $W$ is a non-empty set
  (elements of W are called nodes or worlds or points).
* $R$ is a binary relation on $W$, and is known as the
  accessibility relation which regulates how 'time' flows.
* $\sigma$ is a map from variables (in the ambient logic) to the
  powerset of W. The idea behind $\sigma$ is that $\sigma$
  controls at what points/worlds the variables are true.

If you want to know even more, maybe check the Wikipedia entries
on [Kripe semantics](https://en.wikipedia.org/wiki/Kripke_semantics) or [modal logic](https://en.wikipedia.org/wiki/Modal_logic) or any good book on
the subject such as [Modal Logic](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/modal-logic), by Blackburn, de Rijke and Venema.

> **Comments:**
> **username_3**: Added note on the existence of more than one kind of logic.  
> **username_2**: @KarlDamgaardAsmussen that's right, there are many logics. First-order logic is somewhat canonical though (e.g. the meta-theory of most logics is developed in FOL), and it might be helpful at for the original poster not to worry too much about other logics.  
> **username_3**: I am not talking about First-Order/Second-Order, I mean Intuitionistic/Classical and weaker systems.  
> **username_0**: @MartinBerger, thank you, but as D.W. says this contradicts with the notion that states are equal to worlds. In particular, this seems to say that a collection of states is the world, which looks to me like the world is the same as the model. Could you clarify in the last part how models fit in to the story? Thanks for the explanation so far though :-)  
> **username_2**: @D.W. You are right. I'm sorry, the terminology is not uniform (partly because there are so many different modal logics), and confusingly I have made terminological choices that are unusual. I added some explanation. I hope it's more clear now.  

---

