2 Aug 2016

Nolt (12.1) Logics, ‘Kripkean Modal Logic,’ summary

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]

Summary of

John Nolt

Logics

Part 4: Extensions of Classical Logic

Chapter 12: Kripkean Modal Logic

12.1 Kripkean Semantics

Brief summary:

Kripkean semantics allows us to model certain logical ideas and principles in modal logic that we are unable to model using Leibnizian semantics. The main problem is that Leibnizian semantics will make certain arguments valid (or invalid) when they should not be for a certain type of modality. For example, physical possibility does not behave the same way as logical possibility. Take for instance the fact that accelerating an object faster than the speed of light is logically possible but not physically possible. So we need to change the way we make models for physical possibility. One way of thinking about this is by comparing what is physically possible in each world. In our world (world 1), objects in space can have either circular or elliptical orbits. Now suppose that in world 2, they only have circular orbits. So it is physically impossible according to the laws of physics in world 2 for objects to have elliptical orbits. Now suppose further we take the perspective of world 2, where elliptical orbits are physically impossible. Were we to consider world 1 from world 2’s perspective, we would say that world 1 is a physically impossible world, because its laws of physics do not obey our own. However, were we to look at world 2 from our perspective, we would say that it is a physically possible world, because it does not break any of our physical laws (it just is more physically restricted than ours). So in order to model physical possibility, we can specify this relation of world relativity. It is called relative possibility, accessibility, or alternativeness. In our example, we would say that world 2 is possible relative to world 1, or that world 2 is an alternative to world 1, or that world 2 is accessible to world 1. However, we cannot invert these formulations. We write world y is accessible to world x as xy. We can diagram this with circles and arrows, with an arrow going from a first circle to a second meaning that the second is accessible from the first. Our orbit worlds would be diagramed as:

(Nolt 337)

As we can see, each world is accessible to itself, because each world follows its own physical laws. However, other sorts of modality, like deontic modality, do not guarantee this reflexive self accessibility. So when we use Kripkean semantics, we must stipulate the world relativities by making a set that lists ordered couples of the form <x, y> where y is accessible to (possible relative to) x. So for our example above:

ℛ = {<1, 2>, <1, 1>, <2, 2>}

A Kripkean model is defined in the following way:

DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of modal predicate logic consists of the following:

1. A nonempty set Wv of objects, called the worlds of v.

2. A relation ℛ, consisting of a set of pairs of worlds from Wv.

3. For each world w in Wv a nonempty set Dw of objects, called the domain of w.

4. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:

i. If σ is a name, then v(σ) is a member of the domain of at least one world.

ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F.

iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .

iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.

(337)

And these are the valuation rules for Kripkean semantics:

Valuation Rules for Kripkean Modal Predicate Logic

Given any Leibnizian [or Kripkean?] valuation v, for any world w in Wv:

1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then

v(Φα, w) = T iff v(α) ∈ v(Φ, w);

v(Φα, w) = F iff v(α) ∉ v(Φ, w).

2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then

v(Φα1, ... , αn, w) = T iff <v1), ... , vn)> ∈ v(Φ, w);

v(Φα1, ... , αn, w) = F iff <v1), ... , vn)> ∉ v(Φ, w).

3. If α and β are names, then

v(α = β, w) = T iff v(α) = v (β);

v(α = β, w) = F iff v(α) ≠ v (β).

For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;
v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.

For the next two rules, Φα/β  stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.

9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d)α/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d)α/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of all objects d in Dw, v(α,d)α/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d)α/β , w) ≠ T;
11′ .
v(□Φ, w) = T iff for all worlds u such that wu, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u, wu and v(Φ, u) ≠ T;
12′ .
v(◊Φ, w) = T iff for some world u, wu and v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u such that wu, v(Φ, u) ≠ T.
We define validity in Kripkean semantics in the following way:

A sequent is valid relative to a given set of models (valuations) iff there is no model in that set containing a world in which the sequent's premises are true and its conclusion is not true. To say that a sequent is valid relative to Kripkean semantics in general is to say that it has no counterexample in any Kripkean model, regardless of how ℛ is structured.

(340)

[Here the models in the set are restricted in accordance with the sort of world relativity being modeled].

Summary

Nolt notes that “There is among modal logicians a modest consensus that Leibnizian semantics accurately characterizes logical possibility, in both its formal and informal variants” (Nolt 334). Nonetheless, Leibnizian modal logic is not perfect, as it “does not specify which worlds to rule out as embodying informal contradictions” (Nolt 334). [Nolt says we learned this in section 11.3. There Nolt distinguished logical possibility (meaning that there is no logical contradiction with some entity having certain properties in one world and certain properties in another) and metaphysical possibility (meaning that there are no contradictions in essence when something has certain different properties in different worlds). But then there are two sorts of logical possibility. There is formal logical possibility, which means that there are no contradictions on the syntactical level, and informal logical possibility, which means that there are no contradictions on the semantic level. Nolt is saying that Leibnizian semantics does not tell us which worlds to rule out on account of them having informal contradictions. He gave as one example something having both the properties of being red and being colorless.] Nonetheless, Nolt says that “the semantic rules of Leibnizian logic as laid out in Section 11.2 and the inference rules of Section 11.4 do arguably express correct principles of both formal and informal logical possibility” (334).

But [as we noted above], things that are logically possible still could be metaphysically impossible, and so “logical possibility, whether formal or informal, is wildly permissive” (334). Nolt also says that things that are metaphysically possible do not need to be physically possible. So for example, “It seems both logically and metaphysically possible, for example, to accelerate an object to speeds greater than the speed of light. But this is not physically possible” (334). Nolt then notes another difference between types of possibilities: what is physically possible is not necessarily practically possible. His example is that it is physically possible to destroy all the weapons used for warfare in the world, but that does not mean it is practically possible. Nolt then mentions other types of non-alethic possibilities, like epistemic possibility, moral permissibility, and temporal possibility. But we still wonder if Leibnizian semantics can deal with all of these or if instead some types of modalities would need a different kind of semantic system (335).

Nolt then shows how a sequent that is valid in certain types of modal logic is not valid in others. [Recall from section 11.1 the different types of modalities. The ones for possibility and necessity are the alethic modalities. There are also the deontic or ethical modalities, like ‘it ought to be the case that’. There are propositional attitudes, like ‘believes that’ and ‘knows that’. And there are tense modalities, like ‘it was the case that’ and so on.] The alethic formula he mentions is:

□Φ ⊦ Φ

, which was a metatheorem that he proved in section 11.2.2 (see p.320). [He said previously that this means, “what is necessary is the case” (320).] He says that “This seems right for all forms of alethic possibility. What is logically or metaphysically or physically or practically necessary is in fact the case” (335). But what about for the other sorts of modalities? For the epistemic modality, we would render this formula to mean ‘s knows that Φ; so Φ’, notated as:

KΦ ⊦ Φ

(Nolt 335)

Nolt says that this is valid. [That seems odd to say that whatever we know must itself be true; for surely we are often wrong. I wonder if this is using the notion of knowledge as being justified true belief, and thus if person K knows something, it must be true by definition of knowledge.] For temporal modalities, this would be formulated as, “It has always been the case that Φ; so Φ’, written as

HΦ ⊦ Φ

(Nolt 335)

Nolt says that this is invalid; for “What was may be no longer” (335). [Perhaps the idea is that just because something always was the case does not mean that it still is now or that it is generally the case.] For deontic modalities, this would be formulated as “It is obligatory that Φ; so Φ”, or

OΦ ⊦ Φ

(Nolt 335)

This also is invalid; for “what ought to be often isn’t” (335).

Nolt shows this inconsistency again now with the sequent:

□Φ ⊦ □□Φ

(Nolt 335)

The alethic interpretation is “It is necessary that Φ; so it is necessarily necessary that Φ” (335). This of course is valid in Leibnizian semantics. [Later he says that it is plausible in its alethic modality. It seems that the modality is one thing, and the semantics is a way to interpret or maybe model situations under that modality, but I am not sure. ] The epistemic version,

sKΦ ⊦sKsKΦ

or “s knows that Φ; so s knows that s knows that Φ.” This would only be valid if we ruled out unconscious knowledge, so its validity is “dubious” (336). The temporal version is

HΦ ⊦ HHΦ

or “It has always been the case that Φ; so it has always been the case that it has always been the case that Φ.” Nolt says that this is plausible (336). And the deontic version is

OΦ ⊦ OOΦ

(Nolt 335)

or “It is obligatory that Φ; so it is obligatory that it is obligatory that Φ” (335). Nolt says that like the epistemic version, the obligatory one is also dubious. “the deontic version expresses a kind of moral absolutism: The fact that something ought to be the case is not simply a (morally) contingent product of individual choice or cultural norms, but is itself morally necessary” (Nolt 336). Nolt says that the epistemic thesis that rules out unconscious knowledge and this one just quoted for the deontic interpretation are “controversial  theses”  and we should be suspicious of a semantics that validates them (336).

[His next point seems to be that certain things that Leibnizian semantics deems valid can also be physically impossible. Nolt explains that something is physically possible if it obeys the laws of physics and physically necessary if it is required by those laws. But there is debate over whether or not the laws of physics should be the same in all possible world. Some philosophers think that the laws can be different in other worlds, as they might even have more laws, for example. This means that the laws of physics are world relative, as each world can have its own set of laws. However, Leibnizian semantics considers possibility as something absolute, which means that from the perspective of any given world, the others are possible. So furthermore, Leibnizian semantics assumes that if another world is possible, it is possible from the perspective of any world. However, if the physics are different between the worlds, then from the perspective of one world it would say that the others are physically impossible. Thus, although P ⊦ □⋄P is valid in Leibnizian semantics, it is not so obviously valid for the case of physical possibility. For, this expression says that if something is physically true in our world, then it must be physically possible in other worlds, but we just said that other worlds might deem that situation physically impossible. What confuses me a little here is that we learned in section 11.2.2 that when something is possible, it is true in at least one world. So I would think that even in other worlds with different physical laws, would not the fact that in our world something is physically the case mean it is possible in other worlds? Perhaps the idea is that when in Leibnizian semantics we say something is possible in world 1 because it is the case in world 2, maybe we are implying that it could conceivably happen in world 1 also. But for physical possibility, something that physically the case in world 2 might be physically impossible to ever happen in world 1. The example he gives in the third paragraph in the following quotation seems to be saying something like that.]

In fact, Leibnizian semantics seems inadequate even for some forms of alethic modality. Consider the sequent ‘P ⊦ □⋄P’ with respect to physical possibility. (This sequent is valid given a Leibnizian semantics; see problem 5 of Exercise 11.2.2.)

What does it mean for something to be physically possible or physically necessary? Presumably, a thing is physically possible if it obeys the laws of physics and physically necessary if it is required by those laws. But are the laws of physics the same in all worlds? Many philosophers of science believe that they are just the regularities that happen to hold in a given world. Thus in a more regular world there would be more laws of physics, in a less regular world fewer. If so, then the laws of physics-and physical possibility – are world – relative. Leibnizian semantics treats possibility as absolute; all worlds are possible from the point of view of each. But our present reflections suggest that physical possibility, at least, is world-relative.

To illustrate, imagine a world, world 2, in which there are more physical laws than in the actual world, which we shall call world 1. In world 2, not only do all of our physical laws hold, but in addition it is a law that all planets travel in circular orbits. (Perhaps some novel force accounts for this.) Now in our universe, planets move in either elliptical or circular orbits. Thus in world 1 it is physically possible for planets to move in elliptical orbits (since some do), but in world 2 planets can move only in circular orbits. Since world 2 obeys all the physical laws of world 1, what happens in world 2, and indeed world 2 itself, is physically possible relative to world 1. But the converse is not true. Because what happens in world 1 violates a physical law of world 2 (namely, that planets move only in circles), world 1 is not possible relative to world 2. Thus the very possibility of worlds themselves seems to be a world-relative matter!

(Nolt 336)

[The idea seems to be the following. Previously with Leibnizian possible world semantics, something can be the case in another possible world, and thus be possible in our world (or some other possible world). Now we will consider a different sort of possible world semantics, namely, Kripkean semantics, where one world is not possible in an absolute sense but rather only possible in relation to some other world. This is relative possibility, which is also called alternativeness or accessibility. I am not sure how to make this more concrete, but suppose we have worlds with different physical laws. Perhaps another world is possible with respect to our world if its physical laws are compatible (or identical). And perhaps further a world that is not physically possible relative our world can be physically possible to some third world. World 1 being possible relative world 2 is notated as 2ℛ1. World relativity can be diagramed with circles and arrows going away from circles and arriving upon the same or upon other circles. Each circle represents a world, and a number is placed inside it to indicate which world it is. An arrow going from one world to another (or to the same) means that “the world it points to is relative to the world it leaves”. Now, since each world “obeys the same laws which hold within it,” that means “Each world is also possible relative to itself;” and this is depicted in the diagram with an arrow leaving a world and circling back down upon it again.]

Kripkean semantics takes the world-relativity of possibility seriously. Within Kripkean semantics, various patterns of world-relativity correspond to different logics, and this variability enables the semantics to model a surprising variety of modal conceptions.

The fundamental notion of Kripkean semantics is the concept of relative possibility (which is also called alternativeness or accessibility). Relative possibility is the relation which holds between worlds x and y  just in case y is possible relative to x. The letter ℛ is customarily used to express this relation in the metatheory. Thus we write |

xy

to mean “y is possible relative to x” or “y is an alternative to x” or “y is accessible from x.” (These are all different ways of saying the same thing.) So in the example just discussed it is true that 1ℛ2 (“world 2 is possible relative to world 1”), but it is not true that 2ℛ1. Each world is also possible relative to itself, since each obeys the laws which hold within it. Hence we have 1ℛ1 and 2ℛ2. The structure of this two-world model is represented in the following diagram, where each circle stands for a world and an arrow indicates that the world it points to is possible relative to the world it leaves:

(Nolt 336-337)

Nolt then explains that a Kripkean model is just like a Leibnizian one, however, we need also to specify which worlds are possible with regard to which other worlds. This we do by providing a set of couples where the second term is possible relative to the first term.

A Kripkean model is in most respects like a Leibnizian model, but it contains in addition a specification of the relation ℛ – that is, of which worlds are possible relative to which. This is given by defining the set of pairs of the form <x, y> where y is possible relative to x. In the example above, for instance, ℛ is the set

{<1,2>,<1, 1>,<2,2>}

(Nolt 337)

Nolt then defines a Kripkean model or valuation in nearly the exact same way as a Leibnizian model, with the exception of rule 2 for the accessibility relation [see section 11.2.1 for the definition of a Leibnizian model].

DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of modal predicate logic consists of the following:

1. A nonempty set Wv of objects, called the worlds of v.

2. A relation ℛ, consisting of a set of pairs of worlds from Wv.

3. For each world w in Wv a nonempty set Dw of objects, called the domain of w.

4. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:

i. If σ is a name, then v(σ) is a member of the domain of at least one world.

ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F.

iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .

iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.

(Nolt 337, with uppercase script ‘V’ being notated as italics lower-case v; with uppercase script ‘W’ being notated as uppercase italics W; and with uppercase script ‘D’ being notated as uppercase italics D)

[Recall the valuation rules from section 11.2.1, especially rules 11 and 12, which were:
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T;
12 .
v(◊Φ, w) = T iff for some worlds u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for some worlds u in Wv, v(Φ, u) ≠ T.
(Nolt 315-316)

The problem now with Kripkean semantics is that necessity and possibility are both world relative.]

The addition of ℛ brings with it a slight but significant change in the valuation rules for ‘□’ and ‘⋄’. Necessity at a world w is no longer simply truth in all worlds, but truth in all worlds that are possible relative to w. Likewise, possibility in w is truth in at least one world that is possible relative to w. Thus, instead of the valuation rules 11 and 12 for Leibnizian semantics (Section 11.2), Kripkean semantics has the modified rules:

11′ .
v(□Φ, w) = T iff for all worlds u such that wu, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u, wu and v(Φ, u) ≠ T;
12′ .
v(◊Φ, w) = T iff for some world u, wu and v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u such that wu, v(Φ, u) ≠ T.
(Nolt 338)

[In the following, I will place all the valuation rules for Kripkean semantics in combination, so that they are in one place. But since they are not listed this way in Nolt’s text, I cannot confirm that I am presenting them correctly. (He does write “No other valuation rules are changed” (338).)

Valuation Rules for Kripkean Modal Predicate Logic

Given any Leibnizian [or Kripkean?] valuation v, for any world w in Wv:

1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then

v(Φα, w) = T iff v(α) ∈ v(Φ, w);

v(Φα, w) = F iff v(α) ∉ v(Φ, w).

2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then

v(Φα1, ... , αn, w) = T iff <v1), ... , vn)> ∈ v(Φ, w);

v(Φα1, ... , αn, w) = F iff <v1), ... , vn)> ∉ v(Φ, w).

3. If α and β are names, then

v(α = β, w) = T iff v(α) = v (β);

v(α = β, w) = F iff v(α) ≠ v (β).

For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;
v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.

For the next two rules, Φα/β  stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.

9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d)α/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d)α/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of all objects d in Dw, v(α,d)α/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d)α/β , w) ≠ T;
11′ .
v(□Φ, w) = T iff for all worlds u such that wu, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u, wu and v(Φ, u) ≠ T;
12′ .
v(◊Φ, w) = T iff for some world u, wu and v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u such that wu, v(Φ, u) ≠ T.
]

Nolt now has us consider a Kripkean model in propositional logic. We can ignore the domains of the worlds [I think because it is not predicate logic, where the domains of the arguments need to be specified.]

P = planets move in elliptical orbits.

Wv = {1, 2}

ℛ = {<1,2>,<1, 1>,<2,2>}

(Nolt 338)

So we see this is the same situation we diagrammed above. [Recall also the idea of the elliptical orbits and world relativity. In our own universe, call it 1, planets move either in elliptical or circular orbits. In world 2, however, planets move only in circular orbits. Now, this means that everything that is physically possible in world 2 is also possible in world 1, since in world 1 planets also move in circular orbits. However, not everything that is possible in world 1 is possible in world 2, as there are elliptical orbits in world 1 but not in 2. So this means the statement “Planets move in elliptical orbits” is true in world 1 but not in world 2. Now we think further about the the sequent ‘P ⊦ □⋄P’. We said that this is valid in Leibnizian semantics. But it will not be valid in Kripkean semantics. Let us try to work through this a little. Nolt defined validity in Leibnizian semantics (section 11.2.2) as: “A formula is valid iff it is true in all worlds on all of its valuations” (319). I do not follow very well how Nolt shows this to be invalid in Kripkean semantics, because I would have thought that we would show that it does not hold for world 2, where the planets do not move in elliptical orbits. But instead, Nolt will show that it is invalid, because v(‘□⋄P’, 1) ≠ T. Here perhaps we can also think of invalidity in terms of the premises being true but the conclusion false, but I am not sure. We will have that v(‘P’, 1) = T and v(‘□⋄P’, 1) ≠ T, so here we have the premise true and the conclusion false in the sequent: ‘P ⊦ □⋄P’ (at least with respect to world 1, which provides a counterexample). So let me just assume for now that we need to show how in Kripkean semantics the premise can be true but the conclusion false (for at least one world). I am not exactly sure how to evaluate when there are two operators, because the rules only explain what to do when there is one. Let me also assume that we do it in two steps. So we will first evaluate the following:

v(‘□⋄P’, 1) = ?

And we will begin with the possibility operator, which means we might formulate the valuation as:

v(‘⋄P’, 1) = T iff for for some world u, 1ℛu and v(P, u) = T;

v(‘⋄P’, 1) = F iff for all worlds u such that 1ℛu, v(P, u) ≠ T.

Now, 1ℛu means, ‘world u is possible relative to world 1”. Now, although P is false in world 2, it is true in world 1. And since world 1 has access to world one, then there is at least one world that is possible relative to world 1 (namely, world 1 itself) where it is true, and so:

v(‘⋄P’, 1) = T

We also will need to ask:
v(‘⋄P’, 2) = ?
We might formulate our evaluation as:
v(‘⋄P’, 2) = T iff for some world u, 2ℛu and  v(P, u) = T;
v(‘⋄P’, 2) = F iff for all worlds u such that 2ℛu, v(P, u) ≠ T.

Now, the only world that world 2 has access to is world 2. (That is to say, the only world that is possible relative to world 2 is world 2 itself.) Now, this would seem to mean that for all worlds related to 2, v(P, u) ≠ T (because P is not true in world 2, and world 2 has no access to any other world but itself), and therefore

v(‘⋄P’, 2) = F
Now we have the valuation for ‘⋄P’ in both worlds, so we can move on to the evaluation of:
v(‘□⋄P’, 1) = ?
We would formulate our evaluation perhaps as:
v(‘□(⋄P)’, 1) = T iff for all worlds u such that 2ℛu, v(⋄P, u) = T;
v(‘□(⋄P)’, 1) = F iff for some world u, 2ℛu and v(⋄P, u) ≠ T;

(Added the extra parentheses around ‘⋄P’ in order to emphasize that we are evaluating it for the necessity operator.) Now, world 2 is possible relative to world 1, but in world 2, v(⋄P, u) ≠ T. Therefore,

v(‘□⋄P’, 1) = F

So to summarize, in order to determine ‘P ⊢ □⋄P’ as invalid, we needed to find a counterexample. Such a counterexample would have the premises true and the conclusion false. We then showed that for world one, this condition holds, and thus it is invalid.]

Suppose further that

v(‘P’, 1) = T

v(‘P’, 2) =F

as in that example. (That is, planets move in elliptical orbits in world 1 but not in world 2.) Now the sequent ‘P ⊦ □⋄P’, which was valid on Leibnizian semantics, is invalid on this Kripkean model. For v(‘P’, 1) = T, but v(‘□⋄P’, 1) ≠ T. That is, world 1 provides a counterexample.

We can see that v(‘□⋄P’, 1) ≠ T as follows. Note first that the only world in Wv accessible from world 2 is 2 itself; in other words, the only world u in Wv such that 2ℛu is world 2. Moreover, v(‘P’, 2) ≠ T. Hence for all worlds u in Wv  such that 2ℛu, v(‘P’, u) ≠ T. So by rule 12′, v(‘⋄P’, 2) ≠ T. Therefore, since 1ℛ2, there is some world x in Wv (namely, world 2) such that 1ℛx and v(‘⋄P’, x) ≠ T. It follows by rule 11′ that v(‘□⋄P’, 1) ≠ T. We restate this finding as a formal metatheorem:

METATHEOREM: The sequent ‘P ⊢ □⋄P’ is not valid on Kripkean semantics.

PROOF: As given above.

(338)

Nolt then shows that

neither of the other sequents mentioned in this section – ‘□P ⊢P’ and ‘□P ⊢□□P’ – is valid, either. Let’s take ‘□P ⊢ P’ first.

METATHEOREM: The sequent ‘□P ⊢P’ is not valid on Kripkean semantics.

PROOF: Consider the following Kripkean model for propositional logic. Let the set Wv of worlds be {1, 2} and the accessibility relation ℛ be the set {<1, 2>, <2, 2>}, and let

v(‘P’, 1) = F

v(‘P’, 2) = T

Now v(‘P’, 2) = T and 2 is the only world possible relative to 1; that is, 2 is the only world u such that 1ℛu. Hence for all worlds u such that 1ℛu, v(‘P’, u) = T. Therefore by rule 11′, v(‘□P’, 1) = T. But v(‘P’, 1) ≠ T. Therefore ‘□P ⊢ P’ is not valid on Kripkean semantics.   QED

(Nolt 339)

Nolt then explains a problem with this proof.  First note that the deontic formulation of ‘□P ⊢ P’  should be invalid, contrary to what the proof indicates [In the deontic modality, this would be written is ‘OP ⊢ P’, which we said means, “It is obligatory that P; so P.”]. [We also said that it should be invalid for the temporal interpretation. See p.335.] But the structure ‘□P ⊢P’ intuitively should be valid for the alethic and epistemic interpretations [see again p.335], and so the proof is problematic for these two sorts of interpretions [because the proof says the structure is invalid].  [Recall the epistemic and alethic formulations. Epistemic: ‘KΦ ⊦ Φ’ or ‘s knows that Φ; so Φ’. The alethic we know in terms of necessity.]

[Nolt then explains why the deontic interpretation should be invalid. Nolt first notes a sort of moral statement that is true in a morally perfect world (world 2) but not in our world, which is not morally perfect (world 1), namely, the statement, “Everything is morally perfect”.  Previously we understood the accessibility relation xy as meaning that a world y is physically possible in relation to world x, which means furthermore that all of world x’s physical laws are included among those of world x. Now in this deontic context we will understand the accessibility relation as meaning the relation of permissibility or moral possibility. I am not sure how to grasp this notion just yet, however. I will guess that “world x is morally permissible relative to world y” means that “world x in actuality follows all the moral laws of world y, regardless of whether or not y follows those laws”. I say this, because Nolt writes that world 1, our morally imperfect world, does not have this accessibility relation to itself, because all kinds of bad, immoral things go on in it. However, the perfect world has accessibility to itself, because “what is morally perfect is surely morally permissible”. I really do not follow any of this very well, but in the way I am proposing to understand it, the perfect world actually does follow all its own moral laws. I do not know how else to understand why our world is identical to itself but it is not morally permissible or morally possible in relation to itself. What I am saying is that it has laws which do not permit certain moral actions, but nonetheless they are still committed. Furthermore, our imperfect world is morally permissible in relation to the perfect world. Again, under my proposed understanding, this is because the perfect world obeys all the laws of our imperfect world, even though our imperfect world does not follow those laws. So as we can see, ‘OP ⊢ P’ meaning, “It is obligatory that P; so P,” is not valid, because it does not hold in our world.]

The reasoning for the deontic interpretation is straightforward. Think of world 1 as the actual world, world 2 as a morally perfect world, and P as expressing the proposition Everything is morally perfect. Then, of course, P is true in world 2 but not in world 1. Moreover, think of ℛ as expressing the relation of permissibility or moral possibility. Now world 2 is morally permissible, both relative to itself and relative to world 1 (because what is morally perfect is surely morally permissible!). But world 1 is not morally permissible, either relative to itself or relative to world 2, because all kinds of bad (i.e., morally impermissible) things go on in it. Our model, then, looks like this:

Now since in this model every world that is morally permissible relative to the actual world is morally perfect (since there is, in the model, just one such world, world 2), it follows (by the semantics for , i.e., formally, rule 11′) that it ought to be the case in world 1 that everything is morally perfect, even though that is not the case in world 1. Thus, when we interpret as it ought to be the case that, we can see how □P ⊢ P can be invalid. Kripkean semantics, then, seems | right for the deontic interpretation, but wrong for the epistemic, temporal, and alethic interpretations.

(339-340)

[I do not understand why Nolt says that Kripkean semantics is wrong for temporal interpretations. We said that ‘HΦ ⊢ P’ should be invalid, and the proof above indicates that.]

Nolt then explains that these issues can be resolved in Kripkean semantics, because we can make validity world relative too. In order to understand this notion, we should look at the proof in terms of alethic modality.

The mistake from the alethic point of view lies in the way we specify the accessibility relation. [The idea might be that, as we saw (and for reasons I could not with certainly provide), from the deontic point of view, world relativity need not be reflexive. Perhaps the idea is that what is morally permissible in a world need not actually be what is actually done in that world. At any rate,] for the alethic modality, the access relation must be reflexive, meaning that any world is possible relative to itself. [So if something actually is so in a world, it is also possible in that world.] This means that

The only admissible models – the only models that count – for the alethic interpretation are models whose accessibility relation is reflexive. This is also true for the epistemic modalities, but not for the deontic or temporal ones.

(Nolt 340)

[So in other words, the proof itself is not valid for the alethic modality, because the model constructed in making that proof is not a valid model for the alethic modality.]

This means that we might deal with these problems by stipulating the admissible models for each type of modality. So

Each of the various modalities is to be associated with a particular set of admissible models, that set being defined by certain restrictions on the relation ℛ. Validity, then, for a sequent expressing a given modality is the lack of a counterexample among admissible models for the particular sorts of modal operators it contains. Other semantic notions (consistency, equivalence, and the like) will likewise be defined relative to this set of admissible models, not the full range of Kripkean models. In this way we can custom-craft a different semantics for each of the various modalities.

(340).

We then define validity in the following way:

A sequent is valid relative to a given set of models (valuations) iff there is no model in that set containing a world in which the sequent's premises are true and its conclusion is not true. To say that a sequent is valid relative to Kripkean semantics in general is to say that it has no counterexample in any Kripkean model, regardless of how ℛ is structured.

(340)

[The idea seems to be that were we using a certain type of modality, we are only using models that are admissible, but I am not sure if that is to be included in the above formulation.] Nolt will then show that sequents taking the form □Φ ⊢ Φ are valid for models where worlds are always reflexive.

METATHEOREM: All sequents of the form □Φ ⊢ Φ are valid relative to the set of models whose accessibility relation is reflexive.

PROOF: [see pp.340-341. Nolt does a reductio proof.]

(Nolt 340-341)

Thus

We may say, then, that all sequents of the form □Φ ⊢ Φ are valid when ‘□’  is interpreted as an alethic or epistemic operator, but not if we interpret it as a deontic or temporal operator of the sort indicated earlier. But the validity of all sequents of this form is the same thing as the validity of the T rule introduced in Section 11.4. Thus we may conclude that the T rule is valid for some modalities but not for others.

...

Accessibility in Leibnizian semantics is therefore automatically reflexive. But Kripkean semantics licenses accessibility relations that do not link each world to each, thus grounding the construction of logics weaker in various respects than Leibnizian logic.

(Nolt 341)

Nolt then explains that in Kripkean semantics we can make other requirements for the accessibility relation to accommodate different modal principles (341).

Nolt then has us consider the principle □Φ ⊢ □□Φ, which was suitable for temporal and alethic modalities, but dubious for deontic and epistemic modalities. He says that it is really just rule S4 that we learned in section 11.4. We know then that it is valid in Leibnizian semantics, but Nolt shows it is invalid in Kripkean semantics.

METATHEOREM: The sequent □Φ ⊢ □□Φ is not valid on Kripkean semantics.

PROOF: [see pp. 341-342 for details.]

(Nolt 341-342)

However, this principle is valid in models where the accessibility relation is transitive, that is, if y is accessible from x (xy) and z is accessible from y (yz), then z is accessible from x (xz) (342). Nolt then proves this.

METATHEOREM: All sequents of the form □Φ ⊢ □□Φ are valid relative to the set of models whose accessibility relation is transitive.

PROOF: [see p. 342 for details. Nolt makes a reductio proof.]

(Nolt 342)

Nolt then turns to the principle behind rule B from section 11.4, namely, □Φ ⊢ □◊Φ. This of course is valid in Leibnizian semantics. But it seems invalid for physical possibility: “The fact that planets move in elliptical orbits does not mean that it is necessarily possible that planets move in elliptical | orbits, for there are physically possible worlds in which planetary orbits are necessarily circular and hence in which elliptical orbits are impossible” (342-343). What is needed to make this argument valid is the property of symmetry in the accessibility relation. This would mean that if y is access from x (xy) then x is access from y (yx). Nolt then explains that logical possibility involves symmetry, but not physical possibility:

The accessibility relation for physical possibility is not symmetric, since a world with our physical laws plus some "extra'' laws would be physically possible relative to our world, but ours would not be physically possible relative to it (since our world violates its "extra" laws). Logical possibility, however, presumably does have a symmetric accessibility relation-assuming (as is traditional) that the laws of logic are the same for all worlds.

(Nolt 343)

Nolt then proves this sequent.

METATHEOREM: All sequents of the form Φ ⊢ □◊Φ are valid relative to the set of models whose accessibility relation is symmetric.

PROOF: [see p. 343 for details. Nolt makes a reductio proof.]

(Nolt 342)

So we see that “the accessibility relation for logical possibility is apparently reflexive, transitive, and symmetric” (342). [Recall that the rules of Leibnizian modal logic include all the rules from propositional logic along with the identity rules and the seven additional rules given in section 11.4, namely, DUAL, K, T, S4, B, N, and □=. And the rules other than =I, =E, and = (that is, the set of purely propositional rules) makes up a logic called S5. Nolt now says that reflexivity, transitivity, and symmetry in the accessibility relation “define the logic of S5, which is characterized by Leibnizian semantics”.]

these three characteristics together define the logic S5, which is characterized by Leibnizian semantics. 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.

(Nolt 343)

A Kripkean semantics where in all models each world is accessible from each [including from themselves] would then be Leibnizian semantics (343). When accessibility is like this, it is called universal, which also means that it is reflexive, transitive, and symmetric (343).

Physical possibility is best modeled when we drop symmetry. This gives us a logic that is weaker than S5 called S4.

Nolt concludes with the observation that Kripkean semantics, by allowing the freedom to modify the accessibility relation, allows us to conceive different sorts of philosophical issues.

Logics for the other modalities involve other principles and other properties of ℛ, many of which are disputed. The chief merit of Kripkean semantics is that it opens up new ways of conceiving and interrelating issues of time, possibility, knowledge, obligation, and so on. For each we can imagine a relevant set of worlds (or moments) and a variety of ways an accessibility relation could structure this set and define an appropriate logic. This raises intriguing questions that, were it not for Kripke's work, we never would have dreamed of asking.

(344)

From:

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

.