## 30 Apr 2018

### Priest (4.4a) An Introduction to Non-Classical Logic, ‘S0.5,’ summary

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

[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

4.

Non-Normal Modal Logics; Strict Conditionals

4.4a

S0.5

Brief summary:

(4.4a.1) L is a type of non-normal modal logic. Modal formulas are “sentences of the form □A and ◊A.” And in L, “modal formulas are assigned arbitrary truth values at non-normal worlds” (69). (4.4a.2) In L, the evaluation function v “assigns each modal formula a truth value at every non-normal world” (69). (4.4a.3) The tableau rules for L are the same as for N, except “there are no rules applying to modal formulas or their negations at worlds other than 0. That is, the rules of 2.4.4 apply at world 0 and world 0 only” (69). (4.4a.4) Priest then gives an example of how to create the tableau for a valid formula and another for an invalid one. (4.4a.5) We fashion counter-models from open branches in tableaux in L in the following way. Worlds in W are assigned according to i numbers: “For each number, i, that occurs on the branch, there is a world, wi” (p.27, section 2.4.7). World 0 is the only normal world, and all others are non-normal: “N = {w0}” (70). Accessibility relations follow the irj formulations: “wiRwj iff irj occurs on the branch” (p.27, section 2.4.7). Atomic propositional formulas are true for their indicated world, and negated atomic formulas are false for their world: “for every propositional parameter, p, if p,i occurs on the branch, vwi(p) = 1, if ¬p,i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes” (p.27, section 2.4.7). All necessity and possibility operated formulas, no matter how complex, in non-normal worlds are assigned truth values in the same way: “if i > 0, and □A,i, is on the branch, vwi (□A) = 1; if ¬□A,i, is on the branch, vwi (□A) = 0; similarly for ◊A” (70). (4.4a.6) Priest then provides an example of a counter-model. (4.4a.7) We obtain extensions of L by applying constraints on the accessibility relation, as for example ρ (reflexivity), σ (symmetry), and τ (transitivity). (4.4a.8) “The tableaux for L and its extensions are sound and complete with respect to their semantics” (70). (4.4a.9) On account of the historical development of modal logics, S0.50 and S0.5 are L and Lρ respectively, only without the possibility operator used in them. (4.4a.10) We cannot define the possibility operator for L in the same way as for K and N, i.e., ‘◊A’ as ‘¬□¬A’, because these formulas do not necessarily have the same truth value in all worlds for L. (4.4a.11) “If we wish to make ◊ behave in L as it does when it is defined, we have to add an extra constraint: for every world, w, vw(◊A) = vw(¬□¬A) (that is, vw(◊A) = 1 − vw(□¬A)” (71). (4.4a.12) But this equivalence breaks down in non-normal worlds in L. (4.4a.13) N is a proper extension of L, and L is the weakest modal logic we have seen so far. (4.4a.14) Since L has non-normal worlds, the Rule of Necessitation fails in L. Thus, “that ‘logic need not hold’ at non-normal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world” (71). (4.4a.15) “□A is valid in L (and Lρ) iff A is a truth-functional tautology, or, more accurately, is valid in virtue of its truth-functional structure” (71).

Contents

4.4a.1

[Non-Normal Modal Logic L]

4.4a.2

[Modal Formula Evaluation in L]

4.4a.3

[Tableau Rules for L]

4.4a.4

[Two Tableau Examples]

4.4a.5

[Counter-Models]

4.4a.6

[A Counter-Model Example]

4.4a.7

[Extensions of L]

4.4a.8

[The Soundness and Completeness of L]

4.4a.9

[S0.50 and S0.5 as Historically Without the Possibility Operator]

4.4a.10

[The Failure of the Standard Definition for Possibility in L]

4.4a.11

[A Worldhood Constraint for Defining ◊ in L]

4.4a.12

[The Non-Viability of the Worldhood Constraint Definition]

4.4a.13

[N as a Proper Extension of L]

4.4a.14

[The Failure of the Rule of Necessitation in L]

4.4a.15

[The Conditions of Validity for □A in L]

Summary

4.4a.1

[Non-Normal Modal Logic L]

[L is a type of non-normal modal logic. Modal formulas are “sentences of the form □A and ◊A.” And in L, “modal formulas are assigned arbitrary truth values at non-normal worlds” (69).]

[In this chapter, we have been examining non-normal modal logics (see especially section 4.2 and section 4.4.) Priest will now examine a type of such logics called L. He defines modal formulas as “sentences of the form □A and ◊A.” And in L, “modal formulas are assigned arbitrary truth values at non-normal worlds”.]

Before we leave the topic of non-normal modal logics, there is one further (very small) family of such logics that is worth noting. I will call the basic system of this family L (after Lemmon). Let us call sentences of the form □A and ◊A modal formulas. In interpretations for L, modal formulas are assigned arbitrary truth values at non-normal worlds.

(69)

[contents]

4.4a.2

[Modal Formula Evaluation in L]

[In L, the evaluation function v “assigns each modal formula a truth value at every non-normal world” (69).]

[Recall from section 4.2.2 that interpretations of non-normal modal logics, which are named N, take the structure ⟨W, N, R, v⟩, where 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. The interpretations in our non-normal modal logic L will be the same as for N, with just one exception. (I may not get this right. Recall from section 4.2.3 that in N, 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. Now Priest says that in L, “v also assigns each modal formula a truth value at every non-normal world, as well.” I am not certain, but maybe the idea is that unlike in N, not all necessary propositions will have to be false and not all possible ones will have to be true. I am just guessing at the moment.)]

Thus, interpretations for L are exactly the same as those for N, with one modification. In any interpretation for L, the evaluation function, v, assigns each propositional parameter, p, a truth value at every world, as usual. But v also assigns each modal formula a truth value at every non-normal world, as well.

(69)

[contents]

4.4a.3

[Tableau Rules for L]

[The tableau rules for L are the same as for N, except “there are no rules applying to modal formulas or their negations at worlds other than 0. That is, the rules of 2.4.4 apply at world 0 and world 0 only” (69).]

[(I will guess that the idea is the following. There were four tableau rules in section 2.4.4 that involved modal formulas or their negations. It seems now that they apply only to world 0, and there are no rules at all for modal formulas or their negations at every other world. As such, I am guessing that we would reformulate those rules as follows.

(The following are my guesses and are not in the text.)

 L / S0.5 Negated Necessity Development (¬□D) ¬□A,0 ↓ ◊¬A,0

 L / S0.5 Negated Possibility Development (¬◊D) ¬◊A,0 ↓ □¬A,0

 L / S0.5 Relative Necessity Development (□rD) □A,0 0r0 ↓ A,0

 L / S0.5 Relative Possibility Development (◊rD) ◊A,0 ↓ 0r1 A,1

(modified from p.24, section 2.4.4, but are not in the text and probably are mistaken)

]

Tableaux for L are the same as those for N, except that there are no rules applying to modal formulas or their negations at worlds other than 0. That is, the rules of 2.4.4 apply at world 0 and world 0 only.

(69)

[contents]

4.4a.4

[Two Tableau Examples]

[Priest then gives an example of how to create the tableau for a valid formula and another for an invalid one.]

[Priest then gives some examples.]

Here are tableaux to show that ⊢L □(□A ∨ ¬□A) and ⊬L □(□(pp) ∨ ◊(q ∧ ¬q)):

 ⊢L □(□A ∨ ¬□A) 1. . 2. . 3a. 3b. . 4. . 5. . ¬□(□A ∨ ¬□A),0 ↓ ◊¬(□A ∨ ¬□A),0 ↓ 0r1 ¬(□A ∨ ¬□A),1 ↓ ¬□A,1 ↓ ¬¬□A,1  × P . 1¬□D . 2◊rD 2◊rD . 3b¬∨D . 3b¬∨D (5×4) Valid

(enumeration and step accounting are my own and are probably mistaken)

 ⊬L □(□(p ⊃ p) ∨ ◊(q ∧ ¬q)) 1. . 2. . 3a. 3b. . 4. . 5. . ¬□(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),0 ↓ ◊¬(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),0 ↓ 0r1 ¬(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),1 ↓ ¬□(p ⊃ p),1 ↓ ¬◊(q ∧ ¬q),1 P . 1¬□D . 2◊rD 2◊rD . 3b¬∨D . 3b¬∨D open

(enumeration and step accounting are my own and are probably mistaken)

| The second tableau is now finished, since no modal rules are applicable at world 1.

(69-70)

[contents]

4.4a.5

[Counter-Models]

[We fashion counter-models from open branches in tableaux in L in the following way. Worlds in W are assigned according to i numbers: “For each number, i, that occurs on the branch, there is a world, wi” (p.27, section 2.4.7). World 0 is the only normal world, and all others are non-normal: “N = {w0}” (70). Accessibility relations follow the irj formulations: “wiRwj iff irj occurs on the branch” (p.27, section 2.4.7). Atomic propositional formulas are true for their indicated world, and negated atomic formulas are false for their world: “for every propositional parameter, p, if p,i occurs on the branch, vwi(p) = 1, if ¬p,i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes” (p.27, section 2.4.7). All necessity and possibility operated formulas, no matter how complex, in non-normal worlds are assigned truth values in the same way: “if i > 0, and □A,i, is on the branch, vwi (□A) = 1; if ¬□A,i, is on the branch, vwi (□A) = 0; similarly for ◊A” (70).]

[Recall from section 2.4.7 how in general we fashion counter-models from open branches:

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, wi; wiRwj iff irj occurs on the branch; for every propositional parameter, p, if p,i occurs on the branch, vwi(p) = 1, if ¬p,i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes).

(p.27, section 2.4.7)

And recall from section 4.3.1 the notion of being □-inhabited: “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” (p.65, section 4.3.1). And finally, recall from section 4.3.5 how we make counter-models in non-normal logics:

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.

(p.67, section 4.3.5)

It seems that we make counter-models in the following way. All the numbers after commas are worlds in W. World 0 is normal and is thus included in N. I am not sure if the other worlds can be included in N too, when they are □-inhabited. I cannot tell from the example in the next section, because world 1 is not necessity inhabited. But since it says, “the worlds and accessibility relation are read off as usual,” so that makes me think that necessity inhabited worlds are normal. However, later in the text it seems that worlds greater than 0 are non-normal. So I am very unsure. There are more indicators in this paragraph that all worlds higher than 0 are non-normal, so I am going to side with that interpretation for now. World accessibilities are listed in accordance with the irj formulations. Atomic formulas for a world are assigned true in that world, and negated atomic formulas are assigned false for their respected world. It seems we will also assign values to modally operated formulas no matter how complex, but only for worlds other than 0, which we do in the same way as above only this time we keep the operator.]

To read off a counter-model from an open branch of a tableau, the worlds and accessibility relation are read off as usual, N = {w0}, the truth values of propositional parameters are read off in the usual way, and the truth values of modal formulas at non-normal worlds are read off in exactly the same way. Thus, if i > 0, and □A,i, is on the branch, vwi (□A) = 1; if ¬□A,i, is on the branch, vwi (□A) = 0; similarly for ◊A.

(70)

[contents]

4.4a.6

[A Counter-Model Example]

[Priest then provides an example of a counter-model.]

[Priest now fashions a counter-model from the open tableau in section 4.4a.4 above, which was:

 ⊬L □(□(p ⊃ p) ∨ ◊(q ∧ ¬q)) 1. . 2. . 3a. 3b. . 4. . 5. . ¬□(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),0 ↓ ◊¬(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),0 ↓ 0r1 ¬(□(p ⊃ p) ∨ ◊(q ∧ ¬q)),1 ↓ ¬□(p ⊃ p),1 ↓ ¬◊(q ∧ ¬q),1 P . 1¬□D . 2◊rD 2◊rD . 3b¬∨D . 3b¬∨D open

We see here that there are two worlds, 0 and 1. 0 accesses 1. We have no propositional parameters in 0. But we have negated necessity and possibility formulas in world 1, so we set the unnegated form to false in world 1. Intuitively speaking, we see that the original formula we are testing is invalid, because we can make a counter-model where there is a related world where neither disjunct is true, thus the whole disjunct is false, thus it is not necessary in the original world in question.]

Hence, the counter-model given by the open tableau of 4.4a.4, is such that W = {w0, w1}; N = {w0}; w0Rw1, vw1(□(p p)) = 0, vw1(◊(q ∧¬q)) = 0 (all other values of v being irrelevant). In a diagram:

→xxxx

xxxxw0xxxx→xxxx_____

xxxxw0xxxxxxxx|xw1x|xxx¬□(p p)

xxxxw0xxxx→xxxx|____|xxx¬◊(q ∧¬q)

xxxx

(70)

[contents]

4.4a.7

[Extensions of L]

[We obtain extensions of L by applying constraints on the accessibility relation, as for example ρ (reflexivity), σ (symmetry), and τ (transitivity).]

[Recall from section 3.2.3 the constraints on the accessibility relation that generate variations of a modal logic:

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)

In section 3.2.4, we saw how when applied to the normal modal logic system K, this produces such varieties as

Kρ = T

Kη = D

Kρσ = B

Kρτ = S4

Kρστ = S5

(p.37, section 3.2.4)

And in section 4.2.6, we also saw such extensions of N for non-normal systems. Priest says now that we obtain extensions of L in this same way, and when we do, we add to the systems the appropriate tableau rules for those constraints (see section 3.3.2). Priest notes also some other properties of the extensions.]

Extensions of L are obtained by adding constraints on the accessibility relation in the usual fashion, and adding the corresponding tableau rules. This gives the systems Lρ, Lστ, etc. L is sometimes called S0.50. Lρ is often called S0.5, and is stronger than S0.50, since ⊢Lρ A A. Though it is not immediately obvious, the addition of each of σ and τ has no effect on validity. Jointly, they have an effect on L, but not Lρ. (See 4.10.6 and 4.13, problem 9.)

(70)

[contents]

4.4a.8

[The Soundness and Completeness of L]

[“The tableaux for L and its extensions are sound and complete with respect to their semantics” (70).]

[Priest then notes that:]

The tableaux for L and its extensions are sound and complete with respect to their semantics. This is proved in 4.10.5.

(70)

[contents]

4.4a.9

[S0.50 and S0.5 as Historically Without the Possibility Operator]

[On account of the historical development of modal logics, S0.50 and S0.5 are L and Lρ respectively, only without the possibility operator used in them.]

[Priest’s next point is not so easy for me to follow. It is a historical point. It seems to be that what is now called S0.50 and S0.5 are L and Lρ respectively, only without the possibility operator used in them. The were originally formulated as such, because it seems that modal languages in the beginning only had the necessity operator. But check the quotation to follow:]

One further wrinkle should be noted here. In the earlier years of modal logic, it was common to take a modal language to contain only one modal operator, normally □. The other was then defined. The historical S0.50 and S0.5 are actually the ◊-free fragments of L and Lρ respectively.

(70)

[contents]

4.4a.10

[The Failure of the Standard Definition for Possibility in L]

[We cannot define the possibility operator for L in the same way as for K and N, i.e., ‘◊A’ as ‘¬□¬A’, because these formulas do not necessarily have the same truth value in all worlds for L.]

[The idea in this paragraph seems to be the following. We cannot define the possibility operator for L in the same way as for K and N. Normally we define ‘◊A’ as ‘¬□¬A’, which works in K and N but not in L. I do not grasp why this is so really, so please consult the quotation below.]

The standard definition for ‘◊A’ is ‘¬□¬A’. To take ◊ to be defined in this way, instead of as primitive, has no effect on its behaviour in logics in the K family and N family. This is because ◊A and ¬□¬A have the same truth value at every world (normal or non-normal). But this is not the case in the L family. Given the way in which I set things up, ⊢LA ≡ ¬□¬A (and ⊢L A ≡ ¬◊¬A). However, it does not follow that the formulas on each side of the biconditional have the same truth value in all | worlds, since at a non-normal world ◊A and □¬A can be assigned the same truth value.

(70-71)

[contents]

4.4a.11

[A Worldhood Constraint for Defining ◊ in L]

[“If we wish to make ◊ behave in L as it does when it is defined, we have to add an extra constraint: for every world, w, vw(◊A) = vw(¬□¬A) (that is, vw(◊A) = 1 − vw(□¬A)” (71).]

[I will not be able to summarize the next point, so please skip to the quotation below. Here we are speaking about ◊ being defined. I do not know if we mean defined as we saw above: ‘◊A’ is ‘¬□¬A’, or if we mean giving some or another definition rather than none at all. My guess is that it means to be defined in that particular way, and my next guess is that to make it operate in L in the way the definition normally works requires a constraint. I do not understand what makes it different except that it stipulates that the equivalence be in the same world. I do not know why, but my guess is that it is because in L the values of the formulas in the equivalence can vary from world to world (especially for non-normal worlds), so we need to designate in which world we are making the equivalence. Sorry that I cannot summarize this part:]

Because of this, defining ◊ does affect the inferences that involve it. For example, it is not difficult to check that:

(*) ¬◊(◊p ∧ □¬p)

is not valid in L, but ¬◊(¬□¬p ∧ □¬p) is. Hence, (*) is valid with a defined ◊. If we wish to make ◊ behave in L as it does when it is defined, we have to add an extra constraint: for every world, w, vw(◊A) = vw(¬□¬A) (that is, vw(◊A) = 1 − vw(□¬A). Clearly, this makes for a stronger system.

(71)

[contents]

4.4a.12

[The Non-Viability of the Worldhood Constraint Definition]

[But this equivalence breaks down in non-normal worlds in L.]

[And again I cannot summarize this next part. The idea seems to be that the above constraint is just an option, and it is not perfect anyway. So recall the rule again from section 4.4a.11 above.

for every world, w, vw(◊A) = vw(¬□¬A) (that is, vw(◊A) = 1 − vw(□¬A)

Priest now says that “□A and □¬¬A (and so ¬¬□¬¬A) may have different truth values at a non-normal world.” I am going to make some wild guesses. In the same world, ¬¬□¬¬A = ¬◊¬A = □¬¬A = □A. So we would think that □A and □¬¬A (and so ¬¬□¬¬A) should have the same values. But Priest says that in non-normal worlds in L they might not have the same value. I do not know how all this works, so check the quotation below.]

It should be noted, though, that even this constraint does not ensure that □A and ¬◊¬A have the same truth value at every world, since □A and □¬¬A (and so ¬¬□¬¬A) may have different truth values at a non-normal world. Neither, for essentially the same reason, is ¬◊(□p ∧ ◊¬p) logically valid, as is easy to check. This shows a displeasing lack of symmetry. It is clearly better to treat □ and ◊ even-handedly, as I have done.

(71)

[contents]

4.4a.13

[N as a Proper Extension of L]

[N is a proper extension of L, and L is the weakest modal logic we have seen so far.]

[I will not be able to follow the technicalities here, so consult the quotation below. The main ideas we need to gather from this part are: N is a proper extension of L (all valid formulas in L are valid in N, but not all formulas in N are valid in L.) (Now, since the constraints increase the number of formulas that are valid, that means L is the weakest modal logic we have seen so far, as it has the fewest valid formulas. Sorry that I may have this wrong, so see the quotation:)]

Any N interpretation is an L interpretation (where v makes □ and ◊ behave in the appropriate fashion). Hence, N is an extension of L. It is a proper extension. We have just noted that ¬◊(□p ∧ ◊¬p) is not valid in L. It is not difficult to check that it is valid in N. Similar comments apply to extensions of L and N formed by adding constraints on the accessibility relation. L is, thus, the weakest modal logic we have come across.

(71)

[contents]

4.4a.14

[The Failure of the Rule of Necessitation in L]

[Since L has non-normal worlds, the Rule of Necessitation fails in L. Thus, “that ‘logic need not hold’ at non-normal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world” (71).]

[Recall from section 4.4.6 the Rule of Necessitation:

Let us finally, now, return to the question of the meaning of non-normal worlds. For any normal system, ℒ, if ⊨ A then ⊨ □A. (This is sometimes called the Rule of Necessitation.) For if ⊨ A then A is true at all worlds of all ℒ-interpretations. Hence, if w is any such world, A is true at all worlds accessible from w. Hence, □A is true at w. Thus, ⊨ □A.

(p.68, section 4.4.6)

In section 4.4.7, we noted why the Rule of Necessitation fails in non-normal systems:

The Rule of Necessitation fails in every non-normal logic, ℒ, however. Consider, for example, A∨¬A. This holds at all worlds, normal or non-normal. Hence, □(A∨¬A) holds at all normal worlds, i.e., ⊨ □(A∨¬A). But at any non-normal world, □(A∨¬A) is false. Now consider an interpretation where there is a normal world that accesses such a world. Then □□(A∨¬A) is false at that world. So, ⊭ □□(A∨¬A).

(p.68, section 4.4.7)

It seems that Priest’s next point is that since L also has non-normal worlds, the Rule of Necessitation fails in L. Thus, “that ‘logic need not hold’ at non-normal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world” (71).

]

The rule of Necessitation fails in L for essentially the same reason that it fails in N and its extensions (4.4.7). Indeed, that ‘logic need not hold’ at non-normal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world.

(71)

[contents]

4.4a.15

[The Conditions of Validity for □A in L]

[“□A is valid in L (and Lρ) iff A is a truth-functional tautology, or, more accurately, is valid in virtue of its truth-functional structure” (71).]

[The final point says that “□A is valid in L (and Lρ) iff A is a truth-functional tautology, or, more accurately, is valid in virtue of its truth-functional structure” and that this is proven later. But I am not sure what it means entirely.]

It is worth noting one final fact: □A is valid in L (and Lρ) iff A is a truth-functional tautology, or, more accurately, is valid in virtue of its truth-functional structure.4 The proof is in 4.10.7.

(71)

4. □A ∨ ¬□A is not, strictly speaking, a truth-functional tautology since it contains a □, but it is valid in virtue of its truth- functional structure.

(71)

[contents]

From:

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

.

## 24 Apr 2018

### Priest (2.2) One, “Breaking the Regress”, summary

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[Priest, One, entry directory]

[The following is summary. You will find typos and other distracting mistakes, because I have not finished proofreading. Bracketed commentary is my own. Please consult the original text, as my summaries could be wrong.]

Summary of

Graham Priest

One:

Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness

Ch.2

Identity and Gluons

2.2

Breaking the Regress

Brief summary:

(2.2.1) To explain how gluons bind parts into a unified whole, we need to break the Bradley regress, which prevents gluons from simply being object-parts. (2.2.2) We might name the parts of a unified object with letters, as for example, a, b, c, and d. The gluon, symbolized 中, is what binds all the other parts into the unified whole. If the gluon were distinct from the other parts (in the sense of not being identical to them), there would always be room for another gluon to intervene between the first gluon and the given parts, which leads to the Bradley regress. To avoid it, we say that the gluon is identical to each of the parts, thereby closing those “gaps”. (2.2.3) The gluon is non-transitively identical with each and every part. That means that although each part is identical to the gluon, they are not thereby identical to one another. And, parts can themselves be composed of parts by means of another internal gluon.

x

xxxxb

xxxx||

ax=xx=xc

xxxx||

xxxxd

x

(2.2.4) Gluonic unity involves non-transitive identity, meaning that a = 中 and 中 = c, but not thereby a = c.

2.2.1

[Gluons as Needing to Break the Bradley Regress]

2.2.2

[The Gluon as Identical to the Parts]

2.2.3

[The Non-Transitive Identity of Parts to the Gluon, Part 1]

2.2.4

[The Non-Transitive Identity of Parts to the Gluon, Part 2]

Bibliography

Summary

2.2.1

[Gluons as Needing to Break the Bradley Regress]

[To explain how gluons bind parts into a unified whole, we need to break the Bradley regress, which prevents gluons from simply being object-parts.]

[Recall from section 1.4.2 that we cannot think of the gluon as an object-part, because as such, it would require yet another gluon to explain its binding among the parts it binds, and that newer gluon will need its own to bind it into the unity, and there can be no end to this regress. Priest reminds us that to explain how gluons bind parts into the unity, we need to break this regress.]

The problem of unity is to explain how it is that gluons glue. What stands in the way of an explanation is the Bradley regress. As we saw in Section 1.4, this is vicious, and so it must be broken. But how?

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[contents]

2.2.2

[The Gluon as Identical to the Parts]

[We might name the parts of a unified object with letters, as for example, a, b, c, and d. The gluon, symbolized 中, is what binds all the other parts into the unified whole. If the gluon were distinct from the other parts (in the sense of not being identical to them), there would always be room for another gluon to intervene between the first gluon and the given parts, which leads to the Bradley regress. To avoid it, we say that the gluon is identical to each of the parts, thereby closing those “gaps”.]

[Priest will symbolize the gluon with the Chinese (and maybe also Japanese) character 中. We think of a thing having a number of parts, which we can give letter names to, like a, b, c, and d. The gluon 中, then, is what binds these parts into the whole. Now, the gluon cannot simply be another part, for then it will cause our account to fall victim to the Bradley regress. Priest notes that the regress happens when we come to think of the gluon as being another object for which yet another gluon could intervene between the first and the other parts. Priest then says that if we make the gluon identical to the parts, that closes the metaphysical “gap” that otherwise would stand between the gluon and the parts. So we will need to make the gluon identical to the parts to avoid the Bradley regress.]

Suppose that an object has parts a, b, c, and d, and that these are held together by a gluon 中.1 The Bradley regress is generated by the thought that 中 is distinct | from each of the other parts. If this is the case, then there is room, as it were, for something to be inserted between 中 and a, and so on. Or to use another metaphor, there is a metaphysical space between 中 and a, and one requires something in the space to make the join. Thus, the regress will be broken if 中 is identical to a. There will then be no space, or need, for anything to be inserted.

(16-17, boldface mine)

1. The character 中 (Chinese: zhong; Japanese: chu) means centre, which seems like a pretty good symbol for a gluon. (By coincidence, it is also sometimes used as part of one of the Chinese names for Madhyamaka Buddhism: zhong dao zong.) As the amount of logic increases, it also seems a good time for Western logicians to move to some less familiar languages in search of symbols. Unfortunately, | I will use the character in this section only, due to the current difficulty of typesetting Chinese characters in heavily symbolic contexts.

(16-17)

[contents]

2.2.3

[The Non-Transitive Identity of Parts to the Gluon, Part 1]

[The gluon is non-transitively identical with each and every part. That means that although each part is identical to the gluon, they are not thereby identical to one another. And, parts can themselves be composed of parts by means of another internal gluon.]

[The gluon is non-transitively identical with all of the parts, including itself. We can depict it by having all of the parts equal the gluon, but none of the parts equaling one another.

x

xxxxb

xxxx||

ax=xx=xc

xxxx||

xxxxd

x

Priest then gives an analogy to explain why the identity here is not transitive. We think of how the mortar between bricks binds the bricks without making them one solid brick. Likewise, the gluon binds the parts by being identical to them, without making those parts be identical with each other. The footnote here is important, but I did not quite grasp it all. The basic idea seems to be that each part can be consider as being made of parts, bound with their own subgluon of sorts (not his term). These subparts still form parts of the whole, but how all this works I did not quite follow. Let me just go slowly through that footnote, line by line:

Of course, the parts of an object can themselves have parts.

This we noted.

Thus, it could be the case that, for example, c has parts m and n.

That simply sets up a naming convention.

These will be joined by a gluon, 中′.

That would seem to be what I called the subgluon, namely, the gluon that unifies the subpartitions of a thing’s main parts.

So we will have m = 中′ = n.

This simply says that the subgluon bears the same non-transitive identity relation with respect to the subpartitions it unifies.

If one takes the parthood relation to be transitive, m, 中′, and n, are also parts of the original object. So we will have 中 = m, 中 = 中′, and so on.

Here is where I get lost. As far as I can tell, such main parts as a, b, c, d, 中 are bound by non-transitive identity. And I would assume also that for m and n we would also not want them to be identical. So when Priest says, “If one takes the parthood relation to be transitive,” I am guessing by “parthood relation” he refers to the subpartition’s relation to the main part. But how does that work? Let us try to make a figure for it:

x

xxxxb

xxxx||

ax=xx=xc {m = 中′ = n}

xxxx||

xxxxd

xSomething we have not established here, which would show the

In order to get enough levels of parthood for a transitive relation, let us call the whole object Ω:

x

Ω {xxxxb

xxxxxx||

xxax=xx=xc {{m = 中′ = n}}

xxxxxx||

xxxxxxdxx}

x

My best guess at the moment is that when Priest says, “If one takes the parthood relation to be transitive” he does not mean that the parthood relation is one of identity. I am guessing he simply means that if m is a part of c, and if c is a part of Ω, then m is a part of Ω. But then, I do not understand the line using equations: “So we will have 中 = m, 中 = 中′, and so on.” Because if 中 = m and 中 = n, and this is a transitive identity relation, then we have n = m. In the two given formulas, 中 = m, 中 = 中′, there is in both cases an equation of a higher scale part with a lower scale part. So the only way I can think of this all working is if we think instead of the parthood relation as a non-transitive identity relation. And maybe specifically it is the non-transitive identity relation between the subpartitions specifically with the main gluon. This would bind each subpartition into the whole without equating them. I apologize; please read the text below to see what is really meant here.]

Of course, 中 must be identical with b, c, d, for exactly the same reason. Thus, 中 is able to combine the parts into a unity by being identical with each one (including itself). The situation may be depicted thus:

x

xxxxb

xxxx||

ax=xx=xc

xxxx||

xxxxd

x

The explanation of how it is that the gluon manages to unite the disparate bunch is, then, that it is identical with each of them.2 Consider, if it helps, an analogy. Suppose that one wants to join two physical bricks together with physical glue. The glue is inserted between the bricks. It bonds to each one, and so joins them. It does not make the two bricks one, but the molecules of the glue and each brick become physically indissoluble. In the metaphysical case, the parts of an object do not become identical either, but the gluon bonds with each part in the most intimate way, by being identical with it.

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2. Of course, the parts of an object can themselves have parts. Thus, it could be the case that, for example, c has parts m and n. These will be joined by a gluon, 中′. So we will have m = 中′ = n. If one takes the parthood relation to be transitive, m, 中′, and n, are also parts of the original object. So we will have 中 = m, 中 = 中′, and so on.

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[contents]

2.2.4

[The Non-Transitive Identity of Parts to the Gluon, Part 2]

[Gluonic unity involves non-transitive identity, meaning that a = 中 and 中 = c, but not thereby a = c.]

[Priest now explains how the transitivity of identity fails under this conception of gluonic unity. For, “We have a = 中 and 中 = c, but we will not have a = c.” Priest will now provide a more precise theory of non-transitive identity to show how it is a coherent notion.]

It should be immediately obvious that the relation of identity invoked here will not behave in the way that identity is often supposed to behave. In particular, the transitivity of identity will fail. We have a = 中 and 中 = c, but we will not have a = c. Two bricks of a house are not identical. It might be doubted that there is any such coherent notion, or that, if there is, it is really one of identity. These concerns cannot be set aside lightly, and the only way to assuage them is to provide a precise theory of identity which delivers what is required. Let us turn to this.

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[contents]

Priest, Graham. 2014. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University.

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### Priest (3.5) An Introduction to Non-Classical Logic, ‘S5’, summary

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

[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

3.

Normal Modal Logics

3.5

S5

Brief summary:

(3.5.1) The normal modal logic S5 has the universal or υ (upsilon) constraint, meaning that every world relates to every other world: for all w1 and w2, w1Rw2. (3.5.2) Given that under an υ-interpretation, all worlds access all others, we need not be concerned with the parts of our semantic evaluation rules that mention the R relation. As such, we evaluate necessity and possibility operators in the following way:

vw(□A) = 1 iff for all w′ ∈ W, vw(A) = 1

vw(◊A) = 1 iff for some w′W , vw′(A) = 1

(3.5.3) We make our tableaux for S5 using the tableau rules for modal logic, but eliminating the r designations; and: “Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j” (45).

 S5 Relative Necessity Development (□rD) □A,i ↓ A,j (for every j)

 S5 Relative Possibility Development (◊rD) ◊A,i ↓ A,j (j must be new: it cannot occur anywhere above on the branch)

(modified from p.24, section 2.4.4)

(3.5.4) Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other. (3.5.5) S5 stands for both Kυ and Kρστ, on account of their logical equivalence. (3.5.6) S numbering indicates the system’s relative strength.

Contents

3.5.1

[S5 and the υ (Upsilon) Constraint]

3.5.2

[The Lack of a Need for the R Relation in υ-Interpretations]

3.5.3

[Tableau Rules for S5]

3.5.4

[The Logical Equivalence of Kρστ and Kυ]

3.5.5

[Kυ and Kρστ as Both S5]

3.5.6

[S Numbering as Strength Indicator]

Summary

3.5.1

[S5 and the υ (Upsilon) Constraint]

[The normal modal logic S5 has the universal or υ (upsilon) constraint, meaning that every world relates to every other world: for all w1 and w2, w1Rw2.]

[Priest will now show us the normal logic S5. Recall first from section 3.2.3 the various constraints we saw for the accessibility relation R that generate different types of modal logics:

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)

It seems, but I am not sure, that S5 is defined primarily by a new constraint, namely upsilon, υ, meaning “universal.” Here, every world relates to every other world.]

The system S5 is special. To see how, let an υ-interpretation – ‘υ’ (upsilon) for universal – be an interpretation in which R satisfies the following condition: for all w1 and w2, w1Rw2 – everything relates to everything.

(45)

[contents]

3.5.2

[The Lack of a Need for the R Relation in υ-Interpretations]

[Given that under an υ-interpretation, all worlds access all others, we need not be concerned with the parts of our semantic evaluation rules that mention the R relation. As such, we evaluate necessity and possibility operators in the following way: vw(□A) = 1 iff for all w′ ∈ W, vw(A) = 1; vw(◊A) = 1 iff for some w′W , vw′(A) = 1.]

[Recall from section 2.3.5 the semantic evaluation for necessity and possibility:

For any world wW:

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

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

(Priest p.22, section 2.3.5)

Priest’s next point is that under our new “υ-interpretation, R drops out of the picture altogether, in effect” (45). I am not certain, but the idea might be that under this constraint, we can be sure that all worlds access all others, and thus it makes no difference really whether or not we keep the stipulation above, “... such that wRw′”. It would seem rather that we would formulate the two in the following way, but please trust the quotation below. (Note we also use “iff” instead of “if...otherwise”.)

vw(□A) = 1 iff for all w′ ∈ W, vw(A) = 1

vw(◊A) = 1 iff for some w′W , vw′(A) = 1

]

In an υ-interpretation, R drops out of the picture altogether, in effect. We can just as well define an υ-interpretation to be a pair ⟨W, v⟩, where the truth conditions for □ are simply: vw(□A) = 1 iff for all w′ ∈ W, vw(A) = 1; and similarly for ◊.

(45)

[contents]

3.5.3

[Tableau Rules for S5]

[We make our tableaux for S5 using the tableau rules for modal logic, but eliminating the r designations; and: “Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j” (45).]

[Recall from section 2.4 the tableau rules for modal logic. We use the same rules for the truth functional operators. Our rules for the modal operators will be slightly different, because they normally specify world relativities, which we no longer need to do. So let us modify those rules from section 2.4.4.

 S5 Relative Necessity Development (□rD) □A,i ↓ A,j (for every j)

 S5 Relative Possibility Development (◊rD) ◊A,i ↓ A,j (j must be new: it cannot occur anywhere above on the branch)

(modified from p.24, section 2.4.4)

We of course do not need to use the tableau rules for Kρ, Kσ, and Kτ (see section 3.3.2), because we can assume that all these constraints are already in operation.]

Tableaux for Kυ can also be formulated very simply: r is never mentioned. Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j. For example,

KυA ⊃ □◊A:

 ⊢Kυ ◊A ⊃ □◊A 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . ¬(◊A ⊃ □◊A),0 ↓ ◊A,0 ↓ ¬□◊A,0 ↓ ◊¬◊A,0 ↓ A,1 ↓ ¬◊A,2 ↓ □¬A,2 ↓ ¬A,0 ↓ ¬A,1 ↓ ¬A,2 × P . 1¬⊃D . 1¬⊃D . 3◊¬□D . 2◊rD . 4◊rD . 6¬◊D . 7□rD . 7□rD . 7□rD (9×5)

(45, enumerations and step accounting are my own and are not to be trusted)

[contents]

3.5.4

[The Logical Equivalence of Kρστ and Kυ]

[Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other.]

[Priest now notes that Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other.]

Now, Kρστ and Kυ are, in fact, equivalent, in the sense that Σ ⊨Kρστ A iff Σ ⊨ A. Half of this fact is obvious. It is easy to check that if a relationship satisfies the condition υ it satisfies the conditions ρ, σ and τ. Hence, if truth is preserved at all worlds of all ρστ-interpretations, it is preserved at all worlds of all υ-interpretations. Hence, if Σ ⊨Kρστ A, then Σ ⊨ A. The converse is not so obvious. (A proof can be found in 3.7.5.)

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3.5.5

[Kυ and Kρστ as Both S5]

[S5 stands for both Kυ and Kρστ, on account of their logical equivalence.]

[Priest also notes that often times the name S5 is used for both Kυ and Kρστ, on account of their logical equivalence.]

Because of the equivalence between Kυ and Kρστ, the name S5 tends to be used, indifferently, for either of these systems.

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[contents]

3.5.6

[S Numbering as Strength Indicator]

[S numbering indicates the system’s relative strength.]

[It seems that the number in the S system nomenclature is often based on their relative strengths.]

There are many other normal modal logics. Some of these glorify in names such as S4.2. The number indicates that the system is between S4 and S5 in strength, but otherwise is not to be taken too seriously.

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[contents]

From:

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

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