by Corry Shores

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[Priest, *Introduction to Non-Classical Logic*, entry directory]

[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*

2. Basic Modal Logic

2.3. Modal Semantics

Brief summary:

In our modal semantics, we add to our propositional language two modal operators, □ for ‘necessarily the case that’ and ◊ for ‘possibly the case that’. An *interpretation* in our modal semantics takes the form ⟨*W*, *R*, *v*⟩, with *W* as the set of worlds, *R* as the accessibility relation, and *v* as the valuation function. *‘uRv’* can be understood as either, “world *v *is accessible from *u,*” “in relation to *u*, situation *v* is possible,” or “world *u* access world *v*.” Negation, conjunction, and disjunction are evaluated (assigned 0 or 1) just as in classical propositional logic, except here we must specify in which world the valuation holds.

ν(¬_{w}A) = 1 ifν(_{w}A) = 0, and 0 otherwise.

ν(_{w}A∧B) = 1 ifν(_{w}A) =ν(_{w}B) = 1, and 0 otherwise.

ν(_{w}A∨B) = 1 ifν(_{w}A) = 1 orν(_{w}B) = 1, and 0 otherwise.(21)

A formula is possibly true in one world if it is also true in another world that is possible in relation to the first. A formula is necessarily true in a world if it is also true in all worlds that are possible in relation to it.

For any world

w∈W:

ν(◊_{w}A) = 1 if, for somew′∈Wsuch thatwRw′,ν(_{w′}A) = 1; and 0 otherwise.

ν(□_{w}A) = 1 if, for allw′ ∈Wsuch thatwRw′,ν(A) = 1; and 0 otherwise._{w′}(22)

Given these definitions, we can conclude that if a world has no other related worlds, then any ◊*A *formulation will be false in that world (for, it cannot be true in any related world, as there are none), and any □*A *formulation will be true (for, it is the case that it is true in every accessible world, as there are no accessible worlds). We can diagram the interpretation. Consider this example of an interpretation:

W= {w_{1},w_{2},w}_{3}

w_{1}Rw_{2},w_{1}Rw_{3},w_{3}Rw_{3}

v(_{w1 }p) = 0,v(_{w1 }q) = 0;

v_{w2 }(p) = 1,v_{w2 }(q) = 1,

v_{w3 }(p) = 1,v_{w3 }(q) = o,

This is depicted as:

xxxxxxxxxxxx

w_{2}xxpxxqxxxxxxxxxx↗

¬

px¬qxxw_{1}xxxxxxxxxx↘x↷

xxxxxxxxxxxx

w_{3}xxpxx¬q

Each world (*w*_{1}, *w*_{2}, *w _{3}*) is given its own place on the diagram. Arrows from one world to another indicate the accessibility of the first to the second. The rounded arrow (high above

*w*) thus means the accessibility of a world to itself. And all the true propositions in a world are listed in that world’s place on the diagram (so if a formula is valuated as 0, its negation is listed). Then, on the basis of our rules, we can infer the following other formulas for each world:

_{3}xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxx¬◊q¬

pxxxxx¬qxxxxxw_{1}◊

p∧qxxx◊□pxxxxx↘x↷xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxx□

pxxxxxxxxxx

¬◊*A *at any world is equivalent to □¬*A*. And, ¬□*A *at any world is equivalent to ◊¬*A*. An inference is valid (as a semantic consequence) if it is truth-preserving in all worlds of all interpretations, (that is, if in all worlds in all interpretations, whenever the premises are true, so too is the conclusion). A logical truth (or tautology) is a formula that is true in all worlds of all interpretations.

Σ ⊨Aiff for all interpretations ⟨W,R,v⟩ and allw∈W: ifν(_{w}B) = 1 for allB∈ Σ, thenν(_{w}A) = 1.⊨

Aiffφ⊨A, i.e., for all interpretations ⟨W,R,v⟩ and allw∈W,ν(_{w}A) = 1.

(23)

Summary

2.3.1

[We will add to our propositional language the modal operators □ for ‘necessarily the case that’ and ◊ for ‘possibly the case that’.]

We will add two operators to our propositional language: □ and ◊. [See section 1.2 for a description of this language.] Priest writes:

Intuitively, □

Ais read as ‘It is necessarily the case thatA’; ◊Aas ‘It is possibly the case thatA’.(21)

2.3.2

[In our grammar, any wff with a modal operator added in front is also a wff.]

[Recall the grammar in section 1.2.2:

The (well-formed) formulas of the language comprise all, and only, the strings of symbols that can be generated recursively from the propositional parameters by the following rule:

If A and B are formulas, so are ¬A, (A ∨ B), (A ∧ B), (A ⊃ B), (A ≡ B).(Priest 4)

] Priest says we will augment the grammar of section 1.2.2 with this new rule:

If

Ais a formula, so are □Aand ◊A.(21)

2.3.3

[The *interpretation* in our modal semantics will involve the truth-valuation function *v*, but it also requires that we specify the world in question (from the set of worlds, *W*); and for the modal operators, we specify the accessibility relation (*R*) of the worlds in question. *‘uRv’* means either, “world *v *is accessible from *u*” or “in relation to *u*, situation *v* is possible” (or “world *u* accesses world *v.*”) The interpretation then takes the form: ⟨*W*, *R*, *v*⟩. ]

[Recall from section 1.1.5 that an interpretation can be understood “crudely” as “a way of assigning truth values. And recall from section 1.3.1 that: “An *interpretation* of the language is a function, *ν*, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as *v*(*p*) = 1 and *v*(*q*) = 0” (5). Priest will define the interpretation for our modal logic. It will have the valuation function. But now we need to specify two additional things. The first is the world we are referring to. The formula true or false in which possible world? The other is world relativity or access. I am not certain yet, but this might only be an issue for formulations taking the modal operators. The idea is that one world is accessible from another, meaning that it is possible in relation to it. Nolt in section 12.1 of his *Logics* discusses this relation. We write: *uRv*, and we say either, “world *v *is accessible from *u*” or “in relation to *u*, situation *v* is possible”. (Here it seems that ‘situation’ means ‘possible world’, but I am not sure.) (I have the impression from parts below that *uRv *can also be read as “world *u* accesses world *v.*”]

An

interpretationfor this language is a triple ⟨W,R,v⟩.Wis a non-empty set. Formally,Wis an arbitrary set of objects. Intuitively, its members are possible worlds.Ris a binary relation onW(so that, technically,R⊆W×W). Thus, ifuandvare inW,Rmay or may not relate them to each other. If it does, we will writeuRv, and say thatvisaccessiblefromu. Intuitively,Ris a relation of relative possibility, so thatuRvmeans that, relative tou, situationvis possible. υ is a function that assigns a truth value (1 or 0) to each pair comprising a world,w, and a propositional parameter,p. We write this asν(_{w}p) = 1 (orν(_{w}p) = 0). Intuitively, this is read as ‘at worldw,pis true (or false)’.(21)

[Let us try to understand part reading: “*R *is a binary relation on *W *(so that, technically, *R *⊆ *W*×*W*).” The subset relation was defined in section 0.1.6 in the following way: “A set, *X*, is a subset of a set, *Y*, if and only if every member of *X* is a member of *Y*. This is written as* X* ⊆ *Y*” (xxviii). Next recall from section 01.1.10 the idea of Cartesian Product: “Given *n *sets *X*_{1}, . . . , *X _{n}*, their

*cartesian product*,

*X*

_{1}×· · ·×

*X*, is the set of all

_{n}*n*-tuples, the first member of which is in

*X*

_{1}, the second of which is in

*X*

_{2}, etc.” (xvii). Later in section 01.1.11, an example is given, and Priest writes: “Let

*N*be the set of numbers. Then

*N*×

*N*is the set of all pairs of the form ⟨

*n*,

*m*⟩, where

*n*and

*m*are in

*N*. If

*R*= {⟨2, 3⟩, ⟨3, 2⟩} then

*R*⊆

*N*×

*N*and is a binary relation between

*N*and itself.” So the idea with the Cartesian product seems to be that it would list every possible combination from the one set to the other, but I am not sure. But supposing that to be the case, then

*R*, which would be a set of couples relating one world to another, would then be in the Cartesian product, since the Cartesian product contains all possible combinations of worlds, including each world with itself.]

2.3.4

[The valuation function for negation, conjunction, and disjunction works the same as in classical propositional logic, except now we must specify in which world the valuation holds.]

The valuation function for negation, conjunction, and disjunction operates like we saw for classical propositional logic (section 1.3.2). But now we need to specify in which world the valuation holds.

ν(¬_{w}A) = 1 ifν(_{w}A) = 0, and 0 otherwise.

ν(_{w}A∧B) = 1 ifν(_{w}A) =ν(_{w}B) = 1, and 0 otherwise.

ν(_{w}A∨B) = 1 ifν(_{w}A) = 1 orν(_{w}B) = 1, and 0 otherwise.(21)

2.3.5

[A formula is possibly true in one world if it is also true in another world that is possible in relation to the first. A formula is necessarily true in a world if it is also true in all worlds that are possible in relation to it.]

The valuation function for the modal operators, however, will make use of worlds and their relativities.

For any world

w∈W:

ν(◊_{w}A) = 1 if, for somew′∈Wsuch thatwRw′,ν(_{w′}A) = 1; and 0 otherwise.

ν(□_{w}A) = 1 if, for allw′ ∈Wsuch thatwRw′,ν(A) = 1; and 0 otherwise._{w′}(22)

[Recall *‘uRv’* means either, “world *v *is accessible from *u*” or “in relation to *u*, situation *v* is possible”. So let us look at the first formulation. “*ν _{w}*(◊

*A*) = 1 if, for some

*w′*∈

*W*such that

*wRw*′,

*ν*(

_{w′}*A*) = 1; and 0 otherwise.” So a formula is possibly true in one world if it is true in another world that is possible in relation to the first. Now: “

*ν*(□

_{w}*A*) = 1 if, for all

*w*′ ∈

*W*such that

*wRw*′,

*ν*(A) = 1; and 0 otherwise.” A formula is necessarily true if it is true in all worlds that are possible in relation to it.]

_{w′}In other words, ‘It is possibly the case that

A’ is true at a world,w, ifAis true at some world, possible relative tow. And ‘It is necessarily the case thatA’ is true at a world,w, ifAis true at every world, possible relative tow.(22)

2.3.6

[If a world has no other related worlds, then any ◊*A *formulation will be false in that world, and any □*A *formulation will be true.]

Priest then makes the following note. [Something is not possibly true in a world if there are no other worlds possible in relation to it where it is true. This means if there are no other worlds that are possible in relation to the first, then no statement with a possibility operator can be true. The next point is a bit tricky. The informal wording for the necessity operator was: “‘It is necessarily the case that *A*’ is true at a world, *w*, if *A *is true at every world, possible relative to *w*.” Now suppose there are no worlds relative to *w*, and with regard to *w* we have □*A*. We ask, is *A* true in all worlds relative to *A*? The answer here is, yes, because there are no worlds relative to *A*. If we stick literally to the valuation’s formulation, then all formulas of the form □*A *are true in *w* when it accesses no other world.]

Note that if w accesses no worlds, everything of the form ◊

Ais false atw– ifwaccesses no worlds, it accesses no worlds at whichAis true. And ifwaccesses no worlds, everything of the form □Ais true atw– ifwaccesses no worlds, then (vacuously) at all worlds thatwaccessesAis true.(22)

2.3.7

[We can diagram the interpretation by giving each world it place in the diagram, where the formulations that are true in it are listed, and by using arrows to indicate world accessibility.]

Priest will now show us how we can make a diagram to represent an interpretation where there is a finite number of possible worlds involved. The diagram will show the accessibility relations, and it will state the formulas that are true. [So if the valuation function says a certain formula is false, then we write its negation in the diagram. It seems each world gets its own line or at least a spatial place, and we use arrows for the accessibility relation. Suppose you have an arrow going from world 1 to world 2. This means that world 2 is possible in relation to world 1, or that world 2 is accessible from world 1.] Here is our interpretation:

W= {w_{1},w_{2},w}_{3}

w_{1}Rw_{2},w_{1}Rw_{3},w_{3}Rw_{3}

v(_{w1 }p) = 0,v(_{w1 }q) = 0;

v_{w2 }(p) = 1,v_{w2 }(q) = 1,

v_{w3 }(p) = 1,v_{w3 }(q) = o,

[Note, in the last three lines, where it is formulated for example like *v _{w1 }*, the numeral is given as an additional subscript, which I cannot reproduce here. It looks like:

.] Here is how we would diagram it:

xxxxxxxxxxxx

w_{2}xxpxxqxxxxxxxxxx↗

¬

px¬qxxw_{1}xxxxxxxxxx↘x↷

xxxxxxxxxxxx

w_{3}xxpxx¬q

[Note the rounded arrow above world 3. It is set a bit high, but it means that world 3 has access to itself.]

2.3.8

[Using the valuation rules, we can write other formulas that are true for the depicted worlds in the diagram.]

The diagram can show truth values for complex formulas, notably ones with modal operators. Let us keep our model from above.

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗

¬

pxxxxx¬qxxxxxw_{1}xxxxxxxxxxxxxxxx↘x↷

xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

We notice that *p *and *q* are true in world 2. That means *p*∧*q* is true in world 2 as well. So we write that formula in the area for world 2.

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxxxxxx¬

pxxxxx¬qxxxxxw_{1}xxxxxxxxxxxxxxxx↘x↷

xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

But, world 2 is possible in relation to world 1: *w*_{1}*Rw*_{2}. That means ◊*p*∧*q *holds in world 1. So let us write that formula in the area for world 1.

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxxxxxx¬

pxxxxx¬qxxxxxw_{1}◊

p∧qxxxxxxxxxxx↘x↷xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

[One thing we should note is that while maybe in most cases a world should be able to access itself,] the only world that is possible in relation to itself here is world 3. [Recall the rule for necessity:

ν(□_{w}A) = 1 if, for allw′ ∈Wsuch thatwRw′,ν(A) = 1; and 0 otherwise._{w′}

] Since *p* is true in all worlds possible in relation to world 3, that means it is necessary in world 3. So let us note that.

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxxxxxx¬

pxxxxx¬qxxxxxw_{1}◊

p∧qxxxxxxxxxxx↘x↷xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxx□

pxxxxxxxxxx

Now, although world 1 is not possible in relation to world 3 (that is, world 3 does not access world 1), world 3 is nonetheless possible in relation to world 1 (*w*_{1}*Rw*_{3}, that is, world 1 accesses world 3). So since it is necessary that *p* in world 3, that means *p *is possible that it is necessarily true in world 1. [I am not sure what that would mean for something to be possible that it is necessary. I guess the idea is that in a world that is possible in relation to our own, it is necessary, so it could be necessary in our own too, even though it is not.] We will write that in for world 1, then:

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxxxxxx¬

pxxxxx¬qxxxxxw_{1}◊

p∧qxxx◊□pxxxxx↘x↷xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxx□

pxxxxxxxxxx

Note that world 2 accesses no other world and not even itself. [Again recall how we evaluate possibility:

ν(◊_{w}A) = 1 if, for somew′∈Wsuch thatwRw′,ν(_{w′}A) = 1; and 0 otherwise.

Since there are no other worlds such that world 2 has access to them, that means there are no formulas that are true in accessible worlds. That furthermore means that formula with a possibility operator will be false.] This means that in world 2, ◊*q* is false. [But we only write true formulas, so] that means we write its negation in the area for world 2.

xxxxxxxxxxxxxxxxxxx

w_{2}xxxxxpxxxxxqxxxxxxxxxxxxxxxx↗xxxxxxxx

p∧qxxx¬◊q¬

pxxxxx¬qxxxxxw_{1}◊

p∧qxxx◊□pxxxxx↘x↷xxxxxxxxxxxxxxxxxx

w_{3}xxxxxpxxxxx¬qxxxxxxxxxxxxxxxxxxxxxxxx□

pxxxxxxxxxx

2.3.9

[¬◊*A *at any world is equivalent to □¬*A*.]

The next point is that ¬◊*A *in some world is equivalent □¬*A*. [I will quote the reasoning below, as I will probably missummarize it. Intuitively, the idea might be that if some formula is not possible in some world, that means it is not true in any other accessible world. That would seem to mean that its negation is true in all these other worlds. That then means that in all accessible worlds, the formula’s negation is true. That therefore means that the negation of the formula is necessarily true in the first world, since it is so in all accessible worlds. But let us look at the formulation, and I will try to work through it below.]

Observe that the truth value of ¬◊

Aat any world,w, is the same as that of □¬A. For:v(¬◊_{w}A) = 1 iffv(◊_{w}A) = 0iff for all

w′ such thatwRw′, v(_{w′}A) = 0iff for all

w′ such thatwRw′, v(¬_{w′}A) = 1iff

v(□¬_{w}A) = 1(23)

[We first have:

v(¬◊_{w}A) = 1 iffv(◊_{w}A) = 0

I am not sure, but it might be derived from this rule:

ν(¬_{w}A) = 1 ifν(_{w}A) = 0, and 0 otherwise.

Next is:

iff for all

w′ such thatwRw′, v(_{w′}A) = 0

So maybe the idea here is that if *v _{w}*(◊

*A*) = 0, that means there is no other related world where

*A*is true, thus it is false in all other related worlds.

iff for all

w′ such thatwRw′, v(¬_{w′}A) = 1

If *A* is false in other related worlds, then perhaps by the same negation rule we can say that in these other worlds, ¬*A *is true.

iff

v(□¬_{w}A) = 1

And if ¬*A *is true in all these other worlds, then it is necessarily true in the first world.]

2.3.10

[¬□*A *at any world is equivalent to ◊¬*A*.]

“Similarly, the true value of ¬□*A *at a world is the same as that of ◊¬*A*” (23).

2.3.11

[An inference is valid (as a semantic consequence) if it is truth-preserving in all worlds of all interpretations, (that is, if in all worlds in all interpretations, whenever the premises are true, so too is the conclusion). A logical truth (or tautology) is a formula that is true in all worlds of all interpretations.]

Priest will now define validity, semantic consequence, and logical truth. [I may not summarize this correctly, so see the quotation to follow. The first notion is valid inference. He says that an inference is valid if it preserves truth in all worlds. I would assume that this is a matter then of semantic validity (see section 1.1.5), and I am not entirely sure if this notion of valid inference is in any way different from the notion of semantic consequence formulated below. What is interesting to note is that an inference is valid only if it (semantically) preserves truth in all worlds of all interpretations and not for example just in worlds that are possible in relation to one another. The way semantic consequence is defined formally is something like the following. We have a premise or premises, and a conclusion. If in all worlds of all interpretations, whenever the premises are true the conclusion is true, then the conclusion is a semantic consequence of the premises. And a formula is logically true (tautological, see section 1.3.4) if it is true in all worlds of all interpretations. (I do not understand the part reading, “⊨ *A* iff φ ⊨ *A*). In section 0.1.4, *φ *was defined as the empty set. I discuss this issue in section 1.3.4. There I wondered if the idea was the following. If no valuation can make a formula false, then it can never be that any premises can be true while that formula, understood as a conclusion, is false. So in other words, on the basis of no premises at all (the empty set), we could derive the formula.)]

An inference is valid if it is truth-preserving at all worlds of all interpretations. Thus, if Σ is a set of formulas and

Ais a formula, then semantic consequence and logical truth are defined as follows:Σ ⊨Aiff for all interpretations ⟨W,R,v⟩ and allw∈W: ifν(_{w}B) = 1 for allB∈ Σ, thenν(_{w}A) = 1.⊨

Aiffφ⊨A, i.e., for all interpretations ⟨W,R,v⟩ and allw∈W,ν(_{w}A) = 1.(23)

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

Also cited:

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

.

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