Priest (3.2) An Introduction to Non-Classical Logic, ‘Semantics for Normal Modal Logics’, summary

by Corry Shores

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and¥ boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part I. Propositional Logic

 

3. Normal Modal Logics

 

3.2 Semantics for Normal Modal Logics

Brief summary:

We distinguish the types of modal logic by subscripting their name to the turnstile, as for example: ⊨_K. There are different classes of modal logics. Normal logics are the most important class, and K is the most basic of them. The different modal logics are defined according to certain constraints on the accessibility relation, R, including:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w₁, w₂, if w₁Rw₂, then w₂Rw₁.

τ (tau), transitivity: for all w₁, w₂, w₃, if w₁Rw₂ and w₂Rw₃, then w₁Rw₃.

η (eta), extendability: for all w₁, there is a w₂ such that w₁Rw₂.

An interpretation in which R satisfies conditions ρ (or  σ, etc.) is a ρ-interpretation (or a σ-interpretation, etc.). A logic defined in terms of truth preservation over all worlds of all ρ-interpretations is called Kρ (or Kσ, etc). The consequence relation of such a logic is written ⊨_Kρ (or ⊨_Kσ, etc.). So we would say for example that Σ ⊨_Kρ A if and only if for all ρ-interpretations ⟨W, R, v⟩, and all w ∈ W, if v_w(B) = 1 for all B ∈ Σ, then v_w(A) = 1. We can combine the R conditions to get additional sorts of interpretations, like a ρσ-interpretation for example. Then, the logic Kστ is the consequence relation defined over all στ-interpretations. There are some conventional names for certain various such logics, like S5 for Kρστ, S4 for Kρτ, and B for Kρσ. In nearly all cases, the conditions on R are independent, and they can be mixed and matched at will. Every normal modal logic, L, is an extension of K, in the sense that if Σ ⊨_K A then Σ ⊨_L A. A restricted K modal logic will have fewer interpretations than K on account of many of the K interpretations not meeting the restriction’s criterion. However, these restrictions also happen to allow the restricted K logics to make more inferences valid. Thus there is an inverse relation between inferences and interpretations with respect to the effects of the restrictions. For this reason Kρσ is an extension of Kρ; Kρστ is an extension of Kρσ, and so on.

 

 

 

 

 

Summary

3.2.1

[To distinguish the types of modal logic, we subscript their name to the turnstile, as for example: ⊨_K.]

 

As there are numerous systems of modal logic, whenever there might be confusion between them,

we subscript the turnstile (⊨ or ⊢) used. Thus, the consequence relation of K is written as ⊨_K.
(36)

3.2.2

[Normal logics are the most important class of modal logics, and K is the most basic normal logic.]

 

Priest writes: “The most important class of modal logics is the class of normal logics. The basic normal logic is the logic K” (36).

 

 

3.2.3

[The different modal logics are defined according to certain constraints on the accessibility relation, R, including: ρ (rho), reflexivity: for all w, wRw. σ (sigma), symmetry: for all w₁, w₂, if w₁Rw₂, then w₂Rw₁. τ (tau), transitivity: for all w₁, w₂, w₃, if w₁Rw₂ and w₂Rw₃, then w₁Rw₃. And η (eta), extendability: for all w₁, there is a w₂ such that w₁Rw₂.]

 

[Recall from section 2.3.11 how validity was defined for basic modal logic.

An inference is valid if it is truth-preserving at all worlds of all interpretations. Thus, if Σ is a set of formulas and A is a formula, then semantic consequence and logical truth are defined as follows:

Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w ∈ W: if ν_w(B) = 1 for all B ∈ Σ, then ν_w(A) = 1.
⊨ A iff φ ⊨ A, i.e., for all interpretations ⟨W, R, v⟩ and all w ∈ W, ν_w(A) = 1.

(23)

I am not yet certain about the next idea, so please consult the quotation below. For now I will guess that the basic idea here is the following. There are different properties that the accessibility relation, R, can take, for example, reflexivity, symmetry, etc. Perhaps the different normal modal logics will be defined according to which relations are involved somehow in truth preservation.]

Other normal modal logics are obtained by defining validity in terms of truth preservation in some special class of interpretations. Typically, the special class of interpretations is one containing all and only those interpretations whose accessibility relation, R, satisfies some constraint or other. Some important constraints are as follows:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w₁, w₂, if w₁Rw₂, then w₂Rw₁.

τ (tau), transitivity: for all w₁, w₂, w₃, if w₁Rw₂ and w₂Rw₃, then w₁Rw₃.

η (eta), extendability: for all w₁, there is a w₂ such that w₁Rw₂.

(36)

 

 

3.2.4

[An interpretation in which R satisfies conditions ρ (or  σ, etc.) is a ρ-interpretation (or a σ-interpretation, etc.). A logic defined in terms of truth preservation over all worlds of all ρ-interpretations is called Kρ (or Kσ, etc). The consequence relation of such a logic is written ⊨_Kρ (or ⊨_Kσ, etc.). So we would say for example that Σ ⊨_Kρ A if and only if for all ρ-interpretations ⟨W, R, v⟩, and all w ∈ W, if v_w(B) = 1 for all B ∈ Σ, then v_w(A) = 1.]

 

[Let me quote first, as I may misinterpret this next part.]

We term any interpretation in which R satisfies condition ρ a ρ-interpretation. We denote the logic defined in terms of truth preservation over all worlds of all ρ-interpretations, Kρ, and write its consequence relation as ⊨_Kρ. Thus, Σ ⊨_Kρ A iff, for all ρ-interpretations ⟨W, R, v⟩, and all w ∈ W, if v_w(B) = 1 for all B ∈ Σ, then v_w(A) = 1. Similarly for σ, τ and η.

(37)

[Let me go part by part.

We term any interpretation in which R satisfies condition ρ a ρ-interpretation.

(37)

I am not certain, but it might mean the following. Suppose the model we create has a number of possible worlds, and every world in the model has access to itself. Then we would say that this is a ρ-interpretation.

We denote the logic defined in terms of truth preservation over all worlds of all ρ-interpretations, Kρ, and write its consequence relation as ⊨_Kρ. Thus, Σ ⊨_Kρ A iff, for all ρ-interpretations ⟨W, R, v⟩, and all w ∈ W, if v_w(B) = 1 for all B ∈ Σ, then v_w(A) = 1. Similarly for σ, τ and η.

(37)

I am again not sure what this means. Here is my guess. We have a model with a set of worlds, and all of them have access to themselves. We also have formulas shared by all the worlds. And we have the sequent: Σ ⊨ A. Now, if in all of these self-accessible worlds, whenever the formulas of Σ are 1, then A is 1 too, then we say Σ ⊨_Kρ A. And the logic of this system is called Kρ, because in it truth is preserved in all worlds that can access themselves. Perhaps all this can be formulated in the following way. The structure of our model is:

⟨W, R, v⟩

W = w₁, w₂, w₃

R = w₁Rw₁, w₂Rw₂, w₃Rw₃

Since in this interpretation R satisfies condition ρ, then it is a ρ-interpretation. We now give a logic called Kρ. It stipulates that for example that in all these worlds that access themselves, whenever all the premises of some Σ ⊨ A are true, so too is the conclusion. I am not sure it is as simple as that when I consider the other types of interpretations. Suppose we have this model now.

⟨W, R, v⟩

W = w₁, w₂, w₃

R = w₁Rw₂, w₂Rw₁, w₃Rw₄, w₄Rw₃

I am not sure, but perhaps this could be a σ-interpretation, because all worlds that access another world involve that other world accessing the first. Now we give a logic called Kσ. What do we say that for some Σ ⊨ A? Do we say that it preserves truth in worlds 1, 2, 3, and 4? Or do we say that if it preserves truth across worlds with symmetrical access? In other words, do say that the following is possible? Some Σ ⊨ A preserves truth in both world 1 and world 2, but not both world 3 and world 4.]

 

 

3.2.5

[We can combine the R conditions to get additional sorts of interpretations, like a ρσ-interpretation for example. Then the logic Kστ is the consequence relation defined over all στ-interpretations. There are some conventional names for certain various such logics, like S5 for Kρσ τ, S4 for Kρτ, and B for Kρσ.]

 

Priest notes that “The conditions on R can be combined. Thus, for example, a ρσ-interpretation is one in which R is reflexive and symmetric; and the logic Kστ is the consequence relation defined over all στ-interpretations” (37). He also notes that conventionally we give the following names for these logics:

Kρ = T

Kη = D

Kρσ = B

Kρτ = S4

Kρστ = S5

[Note, regarding S5, we saw in Nolt in his Logics section 12.1 say something similar:

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 accessibility 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 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)

]

3.2.6

[In nearly all cases, the conditions on R are independent, and they can be mixed and matched at will.]

 

There is one R constraint that entails another, namely, when a world has reflexivity (it accesses itself), then it also has extendability (it accesses some world or another, in this case, itself). But besides this, all the other conditions on R are independent. So we can mix and match them at will. [Recall from section 2.3 the notation for modal logic diagrams. An arrow from one world to another means the first accesses the second, and so a rounded arrow above one world means it accesses itself. Priest gives a diagram for a relation that is symmetric and reflexive, but not transitive:]

x

↷xx↷xx↷

w₁⇄w₂⇄w₃

x

[Note, the reflexive arrow is set a bit high, but it belongs to the world written directly below it.] Priest notes another exception:

The exception is that σ, τ and η, together, give ρ.¹

(37)

1. Consider any world, w, By η, wRw′ for some w′. So, by σ, w′ Rw, and, by τ, wRw.
(37)

3.2.7

[Every normal modal logic, L, is an extension of K, in the sense that if Σ ⊨_K A then Σ ⊨_L A.]

 

[We noted in 3.2.2 above that K is the most basic normal modal logic. The other normal logics are more restricted than K. As we will see later in sections 3.3.3, 3.3.4, and 3.3.5, these increased restrictions allow the other logics to validate the same inferences as K but even more in addition to that. Thus they are “proper extensions” of K, even though they are more restricted than K.]

Every normal modal logic, L, is an extension of K, in the sense that if Σ ⊨_K A then Σ ⊨_L A. For if truth is preserved at all worlds of all interpretations, a fortiori it is preserved at all worlds of any restricted class of interpretations.

(37)

 

 

3.2.8

[A restricted K modal logic will have fewer interpretations than K on account of many of the K interpretations not meeting the restriction’s criterion. However, these restrictions also happen to allow the restricted K logics to make more inferences valid. Thus there is an inverse relation between inferences and interpretations with respect to the effects of the restrictions.]

 

[(This part is a bit tricky, and certain ideas here are still unclear to me. It becomes clearer as we proceed into other sections of the book, so I will return to make revisions in the future. For now, see primarily the quotation below.) (The first distinction we need to make is between interpretations and valid formulas/inferences. Recall that in section 2.3.3 we said that “An interpretation for this language is a triple ⟨W, R, v⟩. W is a non-empty set. Formally, W is an arbitrary set of objects. Intuitively, its members are possible worlds. R is a binary relation on W (so that, technically, R ⊆ W×W)” (p.21, section 2.3.3). At this point I will make some guesses. Suppose we have no restrictions on the world relations. That opens a wide range of possible interpretations. In one, for example, we may have

Unrestricted 1:

W = w₁, w₂

R = {w₁Rw₂}, {w₂Rw₁}

Unrestricted 2:

W = w₁, w₂

R = {w₁Rw₂}

But in comparison, a set of interpretations constrained by symmetry (σ) cannot have the second case. It can only have the first.

Symmetry restricted 1:

W = w₁, w₂

R = {w₁Rw₂}, {w₂Rw₁}

Symmetry restricted 2:

W = w₁, w₂

R = {w₁Rw₂}

So this possibly can give us a sense for why additional constraints reduce the number of possible interpretations. And this would seem to be because many interpretations will be disallowed on account of them not fulfilling the additional criteria. Priest shows this with a diagram.

_______________________________

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxI_Xxxxxx_xxx|_xxxxxxI_Yxxx|xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

_______________________________

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Here we think of the I_X as being the set of all K interpretations, and the I_Y  contains a more restricted class of interpretations, like one of the normal modal logics we have discussed in section 3.2.3 above. As we saw with our possible illustration above, K would include all interpretations that a more restricted modal logic has, but the more restricted one would lack some from K. But since all of interpretations in the restricted logic are also in K, that means that whenever truth is preserved in all worlds of all interpretations of K, then it is preserved for all worlds of all interpretations of the restricted logic. For, it would hold for that restricted class of interpretations shared by both K and the restricted logic. The next idea is about how having the greater restrictions allows for more inferences to be valid. We noted this in section 3.2.7 above, and we will see some illustrations in sections 3.3.3, 3.3.4, and 3.3.5. This inverse relationship is shown with a second diagram (see the quotation below.)]

This is an important kind of argument that we use a number of times, so let us pause over it for a moment. Consider the following diagram:

_______________________________

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxI_Xxxxxx_xxx|_xxxxxxI_Yxxx|xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

_______________________________

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Suppose that the outer box contains all interpretations of a certain kind (in our case, all K interpretations), and that the inner box contains some more | restricted class of interpretations (in our case, those appropriate for the logic L). Then if truth (from premise to conclusion) is preserved in all worlds of all interpretations in I_X, then it is preserved in all worlds of all interpretations in I_Y . Hence, the logic determined by the class of interpretations I_Y is an extension of that determined by the class I_X.

 

In other words, if V_X and V_Y are the sets of the inferences that are valid in the two logics, they are related as in the following diagram:

_______________________________

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxV_Yxxxxx_xxx|_xxxxxxV_Xxxx|xxxx|

|xxxxxxxxxxxxxx|xxxxxxxxx|xxxx|

|xxxxxxxxxxxxxx___________xxxx|

|xxxxxxxxxxxxxxxxxxxxxxxxxxxxx|

_______________________________

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Note that the relationship between I_X and I_Y is inverse to that between and V_X and V_Y : fewer interpretations, more inferences. (Or, to be more precise, no less. It is possible to have fewer interpretations with the same set of valid inferences. We will have an example of this in 3.5.4. Thus, V_Y may be a degenerate (improper) extension of V_X , namely V_X itself.)

(37-38)

 

 

3.2.9

[For this reason (see above) Kρσ is an extension of Kρ; Kρστ is an extension of Kρσ, and so on.]

 

 

For exactly this reason, Kρσ is an extension of Kρ; Kρστ is an extension of Kρσ, and so on.
(38)

 

 

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

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