Pirates & Revolutionaries

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[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 unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

Summary of

Graham Priest

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

Part I:

Propositional Logic

11.

Fuzzy Logics

11.2

Sorites Paradoxes

Brief summary:

(11.2.1) Priest first illustrates the sorites paradox. A person begins at age five and is thus a child. One second after that the person is still a child. Thus also one second after that new second the person is still a child. No additional second will cause the child to definitively cease being a child and start being an adult. However, after 30 years, we know that the person is now an adult. (11.2.2) The sorites paradox results from vague predicates like “is a child,” where , “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221). (11.2.3) Many other vague predicates, like “is tall,” “is drunk,” “is red,” “is a heap,” and even “is dead,” can all be used to construct sorites paradoxes. (11.2.4) We can structure the sorites paradox as a chain of modus ponens inferences where we say that something begins at a certain state at a certain time, and next that if something is so at that time it is so in the next second, and we repeat that indefinitely, never arriving upon the state we know it will change into.

11.2.1

[Sorites’ Paradox Age Illustration]

11.2.2

[Vague Predicates Causing the Sorites Paradox]

11.2.3

[Other Vague Predicates]

11.2.4

[The Inferential Structure of Sorites Paradoxes]

Summary

11.2.1

[Sorites’ Paradox Age Illustration]

[Priest first illustrates the sorites paradox. A person begins at age five and is thus a child. One second after that the person is still a child. Thus also one second after that new second the person is still a child. No additional second will cause the child to definitively cease being a child and start being an adult. However, after 30 years, we know that the person is now an adult.]

[Priest first illustrates the sorites paradox. (He gives a similar example in his book Logic: A Very Short Introduction, chap.10.) Here we have a graduated transition between two distinct and exclusive states, where there seems to be no way to know when along the transition it can be said that the change of state happens; or rather, we cannot on the basis of the addition of each small alteration induce that the alternate state will ever be attained.]

Suppose that Mary is aged five, and hence is a child. If someone is a child, they are a child one second later: there is no second at which a person turns from a child to an adult. (We are talking about biological childhood here, not legal childhood. The latter does terminate at the instant someone turns eighteen, in many jurisdictions.) So in one second’s time, Mary will still be a child. Hence, one second after that, she will still be a child; and one second after that; and one second after that ... Hence, Mary will be a child after any number of seconds have elapsed. But this is, of course, absurd. After an appropriate number of seconds have elapsed, so have thirty years, by which time Mary is thirty-five, and so certainly not a child.

(221)

11.2.2

[Vague Predicates Causing the Sorites Paradox]

[The sorites paradox results from vague predicates like “is a child,” where , “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221).]

Priest now explains the problem involved in the sorites paradox. Mary has the predicate “is a child.” But this predicate is vague; for, “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221). [This notion of vagueness as understood here is still a bit unclear to me, so I will not try to further comment until learning more in forthcoming sections.]

The argument of 11.2.1 is known as a sorites paradox. It arises because the predicate ‘is a child’ is vague in a certain sense. Specifically, very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate.

(221)

11.2.3

[Other Vague Predicates]

[Many other vague predicates, like “is tall,” “is drunk,” “is red,” “is a heap,” and even “is dead,” can all be used to construct sorites paradoxes.]

Priest then gives other examples of vague predicates that can result in sorites paradoxes:

In fact, most of the predicates we commonly use are vague in this sense: ‘is tall’, ‘is drunk’, ‘is red’, ‘appears red’, ‘is a heap of sand’ (‘sorites’ comes from the Greek soros meaning ‘heap’) – even ‘is dead’ (dying takes time: one nanosecond makes no difference). One can construct sorites arguments for all such predicates.

(222)

11.2.4

[The Inferential Structure of Sorites Paradoxes]

[We can structure the sorites paradox as a chain of modus ponens inferences where we say that something begins at a certain state at a certain time, and next that if something is so at that time it is so in the next second, and we repeat that indefinitely, never arriving upon the state we know it will change into.]

Priest lastly provides an inferential structuration for sorites paradoxes. We chain together a series of modus ponens inferences where we say that “if the thing is of a certain state at a certain time, then it is still in that state one second later.” We also affirm the antecedent and infer the consequent. We repeat the same procedure on the predication for that next second, and do so indefinitely.

Sorites arguments can often be put in the form of a sequence of modus ponens inferences. Thus, if M_i is the sentence ‘Mary is a child after i seconds’, then the sorites of 11.2.1 is just:

M_{0 xxx}M₀→ M₁

____________

xxxxxM₁ M₁→ M₂

xxxxx____________

xxxxxxxxxM₂

xxxxxxxxx.

xxxxxxxxxx.

xxxxxxxxxxx.

xxxxxxxxxxxxxxM_k-1 M_k-1→ M_k

xxxxxxxxxxxxxx____________

xxxxxxxxxxxxxxxxxxxxM_k

where k is some very large number.

(222)

From:

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

Priest (11.1) An Introduction to Non-Classical Logic, ‘Introduction [to ch.11, “Fuzzy Logics”],’ summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

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

Summary of

Graham Priest

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

Part I:

Propositional Logic

11.

Fuzzy Logics

11.1

Introduction

Brief summary:

(11.1.1) In this chapter we examine fuzzy logic, which assigns to sentences truth values of any real number between 0 and 1. (11.1.2) We will also discuss vagueness, which is one of the main philosophical motivations for fuzzy logic, and we will discuss fuzzy logic’s relation to relevant logics. (11.1.3) We also examine fuzzy conditionals, including how modus ponens fails in fuzzy logic.

11.1.1

[Fuzzy Logic’s Truth Values]

11.1.2

[Vagueness and Fuzzy Logic’s Relation to Relevant Logics]

11.1.3

[Fuzzy Conditionals]

Summary

11.1.1

[Fuzzy Logic’s Truth Values]

[In this chapter we examine fuzzy logic, which assigns to sentences truth values of any real number between 0 and 1.]

Priest in this chapter will cover fuzzy logic, which is “logic in which sentences can take as a truth value any real number between 0 and 1” (221).

11.1.2

[Vagueness and Fuzzy Logic’s Relation to Relevant Logics]

[We will also discuss vagueness, which is one of the main philosophical motivations for fuzzy logic, and we will discuss fuzzy logic’s relation to relevant logics.]

Also in this chapter Priest will discuss one of the most prevalent philosophical motivation for fuzzy logic, namely, vagueness, and in addition to that he will examine “the connections between fuzzy logic and relevant logics” (221).

11.1.3

[Fuzzy Conditionals]

[We also examine fuzzy conditionals, including how modus ponens fails in fuzzy logic.]

Additionally we will examine conditionals in fuzzy logic, noting how modus ponens may fail in it. Conditionals in fuzzy logic are called “fuzzy conditionals” (221).

From:

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

4 Apr 2018

Priest (4.3) An Introduction to Non-Classical Logic, ‘Tableaux for Non-Normal Modal Logics,’ summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

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

Summary of

Graham Priest

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

Part I:

Propositional Logic

Non-Normal Modal Logics; Strict Conditionals

4.3

Tableaux for Non-Normal Modal Logics

Brief summary:

(4.3.1) The tableau rules for non-normal modal logics N are mostly the same as for normal modal logics K. We need however to add the following exception: “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i is activated only when i = 0 or i is □-inhabited” (65). (4.3.2) The ◊-rule (that if you have ◊A,i on one node you can obtain that there is an accessible world where A holds) applies only to normal worlds, because possibility in non-normal worlds does not require it holding in an accessible world; rather, simply all possibilities are true regardless of other worlds. (4.3.3) Priest gives an example showing how the ◊-rule is applied when dealing with world 0. (4.3.4) Priest gives another example where we see that as world 1 is not □-inhabited, we do not apply the ◊-rule to a case in world 1 where there is the possibility operator. (4.3.5) We form counter-examples while keeping in mind which worlds are non-normal. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign v_{w_i}(p) = 1. And for nodes of the form ¬p, i, we assign v_{w_i}(p) = 0. If there are neither of these two, then v_{w_i}(p) can be given any value we want . (4.3.6) Priest gives an example of a counter-model. When depicting non-normal worlds, we place the world designator in a box but write the true formulas in that world above the box. (4.3.7) Tableaux for Nρ, Nρτ, etc. use the same additional rules as for Kρ, Kρτ,etc. (4.3.8) “The tableaux for N and its extensions are sound and complete” (67).

4.3.1

[Additional Notions and Rules for Non-Normal Modal Logic Tableaux]

4.3.2

[The Rationale for the ◊-Rule]

4.3.3

[Example 1]

4.3.4

[Example 2]

4.3.5

[Counter-Model Formation]

4.3.6

[Counter-Model Example]

4.3.7

[Rules for N Extensions]

4.3.8

[The Soundness and Completeness of N]

Summary

4.3.1

[Additional Notions and Rules for Non-Normal Modal Logic Tableaux]

[The tableau rules for non-normal modal logics N are mostly the same as for normal modal logics K. We need however to add the following exception: “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i (2.4.4) is activated only when i = 0 or i is □-inhabited.”]

[We will now learn the tableau procedures for non-normal modal logics, N, whose semantics we learned in the previous section, 4.2. Let us first review the tableau rules for normal modal logics from section 2.4, as we will be building from them. Our tableaux’s nodes have one of two structures: A,i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r) (section 2.4.1). We test for validity by setting the premises to true in world 0 and the negation of the conclusion to true in world 0 (section 2.4.2). We indicate worlds on our tableaux, and the branches inherit the world indicators from above (section 2.4.3). Here are the rules (note, the names are my own. I made them following David Agler (see section 4.2 of Agler’s Symbolic Logic text. Their purpose is so I can follow the reasoning behind the steps of the trees.):

Double Negation

Development (¬¬D)

¬¬A,i

↓

A,i

Conjunction

Development (∧D)

A ∧ B,i

↓

A,i

B,i

Negated Conjunction

Development (¬∧D)

¬(A ∧ B),i

↓

¬A ∨ ¬B,i

Disjunction

Development (∨D)

A ∨ B,i

↙ ↘

A,i B,i

Negated Disjunction

Development (¬∨D)

¬(A ∨ B),i

↓

¬A,i

↓

¬B,i

Conditional

Development (⊃D)

A ⊃ B, i

↙ ↘

¬A, i B, i

Negated Conditional

Development (¬⊃D)

¬(A ⊃ B), i

↓

A, i

↓

¬B, i

Negated Necessity

Development (¬□D)

¬□A,i

↓

◊¬A,i

Negated Possibility

Development (¬◊D)

¬◊A,i

↓

□¬A,i

Relative Necessity

Development (□rD)

□A,i

irj

↓

A,j

(both □A,i and irj must occur somewhere on the same branch, but in any order or location)

Relative Possibility

Development (◊rD)

◊A,i

↓

irj

A,j

(j must be new: it cannot occur anywhere above on the branch)

(p.24 section 2.4.4)

Branches close when there are contradictions in the same world (section 2.4.5). An inference is valid if it makes a completed closed tree, and invalid if it makes a completed open tree. We make counter-models using completed open branches. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p,i we assign v_{w_i}(p) = 1. And for nodes of the form ¬p,i, we assign v_{w_i}(p) = 0. If there are neither of these two, then v_{w_i}(p) can be given any value we want (section 2.4.7). Priest says that the tableau procedures are the same except for some other complications that we will see illustrated later. We now have the notion of a branch being “□-inhabited.” It seems that if we have a node of the form □B,i then it is □-inhabited. It seems we need this designation in order to know when to use the rule for ◊A,i

Relative Possibility

Development (◊rD)

◊A,i

↓

irj

A,j

(j must be new: it cannot occur anywhere above on the branch)

(p.24, section 2.4.4)

Priest says that this rule “is activated only when i = 0 or i is □-inhabited.”]

A tableau technique for N is obtained by modifying the technique for K as follows. If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i (2.4.4) | is activated only when i = 0 or i is □-inhabited. Otherwise, details are the same as for K.

(65-66)

4.3.2

[The Rationale for the ◊-Rule]

[The ◊-rule (that if you have ◊A,i on one node you can obtain that there is an accessible world where A holds) applies only to normal worlds, because possibility in non-normal worlds does not require it holding in an accessible world; rather, simply all possibilities are true regardless of other worlds.]

[Priest next gives the rationale for the ◊-rule stated above in section 4.3.1. I will not summarize these ideas well, so just skip to the quotation. My best guess is the following, but I will revise it later when I understand it better. In section 4.2.5, we defined semantic validity for non-normal modal logics in the following way:

Logical validity is defined in terms of truth preservation at normal worlds, thus:

∑ ⊨ A iff for all interpretations ⟨W, N, R, v⟩ and all w ∈ N: if v_w(B) = 1 for all B ∈ ∑ then v_w(A) = 1.

⊨ A iff φ ⊨ A, i.e., iff for all ⟨W, N, R, v⟩ and all w ∈ N, v_w(A) = 1.

(p.65, section 4.2.5)

As we see, even though we are dealing with non-normality, validity is a matter of truth preservation in just normal worlds. Let us try to comment now on the current paragraph. “the tableau is a search for a normal world where the premises are true and the conclusion is false.” So our notion of proof-theoretic validity (see section 1.3.3) seems to match what we said about semantic validity, with respect to normal worlds. He also says that for this reason, “If i = 0, i must be a normal world”. I did not follow that, but maybe the idea is that when testing for validity, there is this idea that the truth preservation holds for all interpretations. So maybe the idea is that we are free to think of an inference under the conditions that it holds in some normal world, in order to ultimately determine if it is valid in some logical system. I do not know. I am not sure if this is possible for what we are doing, but what if there are no normal worlds in a particular interpretation? Is it irrelevant because we are supposed to also include other interpretations with normal worlds? Or would we instead say that validity should still hold for that situation where there are only non-normal worlds? If so, I wonder then if the point would be that the formula would have to be valid, but vacuously so, because it is the case that there are no normal worlds where the premises are true and the conclusion false (since there are no normal worlds to begin with). At any rate, his point is that when we set up our tableau, the original 0 world for one reason or another will designate a normal world. And recall from section 4.3.1 above that “The rule for ◊A,i (2.4.4) is activated only when i = 0 or i is □-inhabited.” So since we set our original formulas to world 0, which is normal, that can activate the ◊-rule. The next idea is “If i > 0, it can be assumed to be non-normal as long as the branch of the tableau is not □-inhabited.” This is also tricky, and I will guess. We said that in non-normal worlds, nothing is necessary. And “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch.” So for some reason, if there is no indication that there is a true necessity in a world, then we can assume it is non-normal. I am not sure why, but maybe a lack of such an indication is enough to know there are no other such cases of necessity, or maybe it is enough for us to arbitrarily stipulate that there are no other such cases, but I am not sure. So again recall the ◊-rule: “The rule for ◊A,i is activated only when i = 0 or i is □-inhabited.” So when i > 0, it fulfills neither criteria (being equal to zero or being necessity inhabited), then this rule is not activated. But I am still not sure why all this is the rationale for the rule itself. Maybe the idea is the following, but my guesses are getting wilder. It might be that we need to explain why is it that the ◊-rule is only applied when “i = 0 or i is □-inhabited.” We said that in both cases, they are normal worlds. So what would be the problem with the ◊-rule being used in non-normal worlds? The rule is the following:

Relative Possibility

Development (◊rD)

◊A,i

↓

irj

A,j

(j must be new: it cannot occur anywhere above on the branch)

(24)

It is saying that if something is possible in some world, then there must be another world where it is the case. Maybe we next suppose that were something possible in a non-normal world, that would not mean there is another world where it is the case. My best guess for this is that we have rejected the normal semantic rule for evaluating possibility in non-normal worlds. In normal worlds, if something is possible, it must be the case in some accessible possible world. But we do not have that standard of evaluation for non-normal worlds, thus our tableau rule should only apply to normal worlds. Sorry for not helping here; this is the quotation:]

The rationale for the new ◊-rule is, roughly, as follows. If i = 0, i must be a normal world (since the tableau is a search for a normal world where the premises are true and the conclusion is false), and so the ◊-rule is applied in the usual way. If i > 0, it can be assumed to be non-normal as long as the branch of the tableau is not □-inhabited. Nothing, then, needs to be done. But as soon as i is □-inhabited, it can no longer be non-normal (since nothing of the form □A is true at a non-normal world), and so the standard rule for ◊ must be applied. The next two subsections give example tableaux for N.

(66)

4.3.3

[Example 1]

[Priest gives an example showing how the ◊-rule is applied when dealing with world 0.]

[Priest will give some examples. Here is the first one.]

⊢_N □(A ⊃ B) ⊃ (□A ⊃ □B):

⊢_N □(A ⊃ B) ⊃ (□A ⊃ □B)

7a.

7b.

10.

¬(□(A ⊃ B) ⊃ (□A ⊃ □B)),0

↓

□(A ⊃ B),0

↓

¬(□A ⊃ □B),0

↓

□A,0

↓

¬□B,0

↓

◊¬B,0

↓

0r1

¬B,1

↓

A ⊃ B,1

↓

A,1

↙ ↘

¬A,1 B,1

× ×

1¬⊃D

3¬⊃D

5¬□D

6◊rD

2,7a□rD

4,7a□rD

8⊃D

(9)(7b)

The ◊-rule is applied to ◊¬B,0, because we are dealing with world 0.

(66, enumeration and step-accounting are my additions)

4.3.4

[Example 2]

[Priest gives another example where we see that as world 1 is not □-inhabited, we do not apply the ◊-rule to a case in world 1 where there is the possibility operator.]

[Here is the second example.]

⊬_N□(p ⊃ □(q ⊃ q)):

⊬_N□(p ⊃ □(q ⊃ q))

3a.

3b.

¬□(p ⊃ □(q ⊃ q)),0

↓

◊¬(p ⊃ □(q ⊃ q)),0

↓

0r1

¬(p ⊃ □(q ⊃ q)),1

↓

p,1

↓

¬□(q ⊃ q),1

↓

◊¬(q ⊃ q),1

1¬□D

2◊rD

3b¬⊃D

5¬□D

(open)

| On the (only) branch of the tableau, world 1 is not □-inhabited. Consequently, the ◊-rule is not applied to the last line, and the tableau ends open.

(67-68, enumeration and step-accounting are my additions)

4.3.5

[Counter-Model Formation]

[We form counter-examples while keeping in mind which worlds are non-normal. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign v_{w_i}(p) = 1. And for nodes of the form ¬p, i, we assign v_{w_i}(p) = 0. If there are neither of these two, then v_{w_i}(p) can be given any value we want .]

[Recall from section 4.3.2 that our tableau seeks a normal world where the premises are true and the conclusion false. (Priest has us bear in mind our comments from that section, but I am not sure exactly what is most relevant here. It may not be what I note above.) Next recall the counter-model method for modal logics from section 2.4.7 :

Counter-models can be read off from an open branch of a tableau in a natural way. For each number, i, that occurs on the branch, there is a world, w_i; w_iRw_jiff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_{w_i}(p) = 1, if ¬p, i occurs on the branch, v_{w_i}(p) = 0 (and if neither, v_{w_i}(p) can be anything one wishes).

(p.27, section 2.4.7)

]

Bearing in mind the comments of 4.3.2, it is easy to see how a countermodel for an inference can be read off from an open tableau branch. The method is exactly the same as for K, except that world 0 is always normal, and all other worlds are non-normal, unless they are □-inhabited.

(67)

4.3.6

[Counter-Model Example]

[Priest gives an example of a counter-model. When depicting non-normal worlds, we place the world designator in a box but write the true formulas in that world above the box.]

[Priest now gives an example where in the non-normal world 1 from the example above, we assign p as 1. Unfortunately I have not yet figured out how this makes □(p ⊃ □(q ⊃ q)) be false. My best wild guess at the moment is that it has something to do with all necessity formulations being false in world 1 and that somehow leading to the original formula being false in the normal world 0. If I learn the reasoning here, I will revise this section.]

Thus, in the counter-model determined by the tableau of 4.3.4, W = {w₀, w₁}; N = {w₀}; w₀Rw₁; and v is such that v_{w_i}(p) = 1. If we indicate that a world is non-normal by putting it in a box, the interpretation can be depicted thus:

w₀xxxx→xxxx__p__

w₀xxxx→xxxx|xw_1x|

w₀xxxx→xxxx|____|

(p.67, in the original, there is closed box around w₁.)

4.3.7

[Rules for N Extensions]

[Tableaux for Nρ, Nρτ, etc. use the same additional rules as for Kρ, Kρτ,etc.]

[Recall from section 3.3 (especially section 3.3.2) the extra tableau rules for the extensions of K. The will be the same for N.]

Tableaux for Nρ, Nρτ, etc. are obtained by adding the extra tableau rules for ρ, ρτ, etc., as for K (3.3).

(67)

4.3.8

[The Soundness and Completeness of N]

[“The tableaux for N and its extensions are sound and complete”.]

Priest ends by noting that:

The tableaux for N and its extensions are sound and complete with respect to their respective semantics. The proof can be found in 4.10.

(67)

From:

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

2 Apr 2018

Priest (4.2) An Introduction to Non-Classical Logic, ‘Non-Normal Worlds’, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

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

Summary of

Graham Priest

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

Part I

Propositional Logic

Non-Normal Modal Logics; Strict Conditionals

4.2

Non-Normal Worlds

Brief summary:

(4.2.1) We will first examine the technical elements of non-normality. (4.2.2) Our interpretations of non-normal modal logics take the structure ⟨W, N, R, v⟩. W is the set of worlds. R is the accessibility relation. v is the valuation function. And N is the set of normal worlds, with all the remaining worlds in W being non-normal ones. (4.2.3) The semantics are the same for non-normal worlds, except that at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. For, in non-normal worlds, nothing is necessary and all is possible. (4.2.4) At every world, including non-normal ones, ¬□A and ◊¬A have the same truth value. ¬◊A and □¬A do too. (4.2.5) Inferences are valid only if they preserve truth in all interpretations at all normal worlds. (4.2.6) Non-normal modal logics with the structure ⟨W, N, R, v⟩ in which R is a binary relation on W are called N, with such R constraints as ρ, σ, τ etc. creating extensions of N like Nρ, etc. (So here we have N for non-normal modal logics where we previously had K and its extensions for normal modal logics.) And, “As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.” (4.2.7) Nρ = S2; Nρτ = S3; and Nρστ = S3.5, with the first two S’s being “Lewis systems” and the last one being a “non-Lewis system”. (4.2.8) Although non-normal worlds originally were fashioned solely for technical reasons, in fact they have a philosophical meaning too.

4.2.1

[Preview: Technicalities]

4.2.2

[The Components of Non-Normal Interpretations]

4.2.3

[The Universal Possibilities and the Absence of Necessities in Non-Normal Worlds]

4.2.4

[Equivalences Holding in Non-Normal Worlds Too]

4.2.5

[Validity]

4.2.6

[N Non-Normal Modal Logic and Its Extensions]

4.2.7

[N Logics in Terms of Lewis Systems]

4.2.8

[Origins and Philosophical Potential of Non-Normal Logics]

Summary

4.2.1

[Preview: Technicalities]

[We will first examine the technical elements of non-normality.]

We will now first look “at the technicalities concerning non-normality,” and we later gradually will “discuss what they mean” (64).

4.2.2

[The Components of Non-Normal Interpretations]

[Our interpretations of non-normal modal logics take the structure ⟨W, N, R, v⟩. W is the set of worlds. R is the accessibility relation. v is the valuation function. And N is the set of normal worlds, with all the remaining worlds in W being non-normal ones.]

[Recall the semantics for modal logic that we saw in section 2.3. The following comes from our brief summary there:

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.

v_w(¬A) = 1 if v_w(A) = 0, and 0 otherwise.

v_w(A ∧ B) = 1 if v_w(A) = v_w(B) = 1, and 0 otherwise.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w(B) = 1, and 0 otherwise.

(p.21, section 2.3)

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:

v_w(◊A) = 1 if, for some w′ ∈ W such that wRw′, v_w_′(A) = 1; and 0 otherwise.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w_′(A) = 1; and 0 otherwise.

(p.22, section 2.3)

In non-normal logics, most of this stays the same, with the following exceptions. We designate a subset of worlds that are normal, meaning that all the above semantics holds for them, and all the rest of the worlds are non-normal, meaning that for them all necessity formulations are false and all possibility formulations are true. In other words, in non-normal worlds, nothing is necessarily the case, and all things are possible. (For the set notation towards the end of the quote below, see sections 0.1.6 and 0.1.8.)]

A non-normal interpretation of a modal propositional language is a structure, ⟨W, N, R, v⟩, where W, R and v are as in previous chapters, and N ⊆ W. Worlds in N are called normal. Worlds in W−N (the worlds that are not normal) are called non-normal.

(64)

4.2.3

[The Universal Possibilities and the Absence of Necessities in Non-Normal Worlds]

[The semantics are the same for non-normal worlds, except that at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. For, in non-normal worlds, nothing is necessary and all is possible.]

[In the above section 4.2.2, we mentioned the semantics for the truth functions. Priest says they are the same I think for both normal and non-normal worlds. The truth conditions for necessity and possibility, also given above in section 4.2.2, hold still for normal worlds, but we have different rules for non-normal worlds. If something is necessary, then it is assigned the value false, and if something is possible, it is assigned the value true. This carries the sense that in non-normal worlds, nothing is necessary, and everything is possible.]

The truth conditions for the truth functions, ∧,∨,¬, etc. are the same as before (2.3.4). The truth conditions for □ and ◊ at normal worlds are also as before (2.3.5). But if w is non-normal:

v_w(□A) = 0

v_w(◊A) = 1

In a sense, at non-normal worlds, everything is possible, and nothing is necessary.

(64, boldface mine)

4.2.4

[Equivalences Holding in Non-Normal Worlds Too]

[At every world, including non-normal ones, ¬□A and ◊¬A have the same truth value. ¬◊A and □¬A do too.]

[In section 2.3.9 Priest gives a proof for why ¬◊A at any (normal) world is equivalent to □¬A. I was not able to summarize the reasoning in that proof adequately. It seemed to me at the time that the idea was the following. If it is not that something is necessary, then that means that at least in one related world it is not true. In that world where it is not true, its negated form would be true (under classical assumptions about negation). So for the first world that we started with, in a related world the negation is true, thus the negation is possible in that first world. I am just guessing, sorry. Now his point is that this holds also for non-normal worlds. I am not entirely sure how and why. My best guess is that by assigning all necessities as 0 and all possibilities as 1 does not thereby change the rules governing the way the worlds are related. Thus we will find the same conditions that would cause these equivalence to remain. I quote:]

Note that at every world, w,¬□A and ◊¬A still have the same truth value, as do ¬◊A and □¬A. We saw this to be the case for normal worlds in 2.3.9 and 2.3.10. It is easy to see that this is also true if w is non-normal.

(65)

4.2.5

[Validity]

[Inferences are valid only if they preserve truth in all interpretations at all normal worlds.]

[Now we discuss validity. Recall from section 2.3.11 that 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). And, a logical truth (or tautology) is a formula that is true in all worlds of all interpretations. Here is a quotation from that section, with the formal definition.

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

(p.23, section 2.3.11)

In our non-normal modal logics, we have the added issue of non-normal worlds. Priest says that an inference or formula is valid so long as it preserves truth at all normal worlds (leaving open the possibility that it is not preserved in non-normal worlds).]

Logical validity is defined in terms of truth preservation at normal worlds, thus:

∑ ⊨ A iff for all interpretations ⟨W, N, R, v⟩ and all w ∈ N: if v_w(B) = 1 for all B ∈ ∑ then v_w(A) = 1.

⊨ A iff φ ⊨ A, i.e., iff for all ⟨W, N, R, v⟩ and all w ∈ N, v_w(A) = 1.

(65)

4.2.6

[N Non-Normal Modal Logic and Its Extensions]

[Non-normal modal logics with the structure ⟨W, N, R, v⟩ in which R is a binary relation on W are called N, with such R constraints as ρ, σ, τ etc. creating extensions of N like Nρ, etc. (So here we have N for non-normal modal logics where we previously had K and its extensions for normal modal logics.) And, “As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.”]

[Recall from section 2.1.2 that we call the most basic modal logic K. In section 3.2.2 we learned that K is the most basic “normal” modal logic, and in section 3.2.3 Priest explains how we may fashion extensions of K by placing constraints on the R relation, like reflexivity, etc.:

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.

(p.36, section 3.2.3)

A normal modal logic K under the symmetry and transitivity constraints, for example, will be called Kστ. Priest is saying now that instead of K, when R is a binary relation on W, then our logic is now called N, with the R constraints producing such extensions of N as Nστ, for example. My guess is that for the construction to be an N, the R relation and its constraints would need to apply to all worlds, normal and non-normal. The last notion we should note is this idea of being an extension, such that “Nρτ is an extension of Nρ, which is an extension of N”. The idea as I understand it so far (and I do not quite grasp it fully yet) is that to be an extension means that a logic has all the same valid inferences as the one it is extending, and probably more in addition to that. See the discussion in section 3.2.8 and the examples from sections 3.3.3, 3.3.4, and 3.3.5.]

If the accessibility relation, R, may be any binary relation on W, the logic this construction gives will be called N.¹As with normal modal logics, additional logics can be formed by placing constraints on R, such as reflexivity, transitivity, symmetry, etc. (as in 3.2). In fact, of course, how R behaves at non-normal worlds is irrelevant, since this plays no role in determining truth values. We use Nρ to refer to the non-normal logic determined by the class of all interpretations where R is reflexive; Nστ, to refer to the non-normal logic determined by the class of all interpretations where R is symmetric and transitive, and so on. As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.

(65)

1. The name is not standard, but is sensible enough. Note that N is also used for the normal worlds in an interpretation. Context, however, will disambiguate. (65)

4.2.7

[N Logics in Terms of Lewis Systems]

[Nρ = S2; Nρτ = S3; and Nρστ = S3.5, with the first two S’s being “Lewis systems” and the last one being a “non-Lewis system”.]

[In section 3.2.5, Priest noted that Kρτ = S4 and Kρστ = S5. Now Priest says that Nρ = S2 and Nρτ = S3, with both S’s being “Lewis systems”, and Nρστ = S3.5, with that being a “non-Lewis system.” I am not familiar with the idea of a “Lewis system”, so I will have to quote for now.]

Historically, Nρ and Nρτ are the Lewis systems S2 and S3 respectively. Nρστ is the non-Lewis system S3.5.

(65)

4.2.8

[Origins and Philosophical Potential of Non-Normal Logics]

[Although non-normal worlds originally were fashioned solely for technical reasons, in fact they have a philosophical meaning too.]

[Priest now gives some interesting and philosophically important historical information about non-normal logics. For these ideas we need the notions of stronger and weaker, which I have not quite grasped yet. We discussed these ideas in section 4.1.1, where my guess was that the extensions are stronger because they have more valid formulas. But I really do not know if that is so. I will quote now, as I cannot summarize any better than what is written:]

Non-normal worlds were originally invented purely as a technical device to give a possible-world semantics for the Lewis systems weaker than S4. As we shall see in due course, though, they have a perfectly good philosophical meaning. For the record, Lewis thought that the correct system of modal logic for logical necessity was S2.

(65)