## 29 Mar 2018

### Priest (1.4) One. ‘The Bradley 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.1

Gluons and Their Wicked Ways

1.4

Brief summary:

(1.4.1) We will now discuss why the gluon cannot be an object on account of a vicious regress. (1.4.2) In the Bradley regress, a binding factor is posited as being a member of the unity it binds. But that only leaves us to find yet another binding factor that would bind the first into the whole. There can be no end so long as the binding factors are consider as object/parts. In terms of gluons, if we make the gluon be an object/part, then we will always need yet another gluon to explain how the prior gluon is bound into the whole. (1.4.3) We cannot simply account for unity by saying it is gluons all the way down. For, no such gluon is sufficient to explain the unity. All of them require something in addition. So, simply saying the unity is found in yet another part never tells us in what the unity consists. (1.4.4) In conclusion, on account of the Bradley Regress, we cannot explain the unity of objects as being another object.

1.4.1

[Preview: The Gluon as a Non-Object on Account of a Vicious Regress]

1.4.2

1.4.3

[Unity is Not Gluons All the Way Down]

1.4.4

[Unity is Not Another Object]

Bibliography

Summary

1.4.1

[Preview: The Gluon as a Non-Object on Account of a Vicious Regress]

[We will now discuss why the gluon cannot be an object on account of a vicious regress.]

[Recall from section 1.3.4 the notion of the gluon. From the brief summary of that section: “The unifying factor in a thing is called its gluon. It both is and is not an object/part. It is an object insofar as we name it and conceive it. But it is not an object insofar as it is what constitutes the organizing and unifying factor of the thing, because as such it needs to be over and beyond any of the parts rather than simply being another part.” And here is the paragraph in full, where the gluon is given a slightly more formal account:

Here, then, is our problem of unity. Let me lay it out in abstract terms. Take any thing, object, entity, with parts, p1, .. , pn. (Suppose that there is a finite number of these; nothing hangs on this.) A thing is not merely a plurality of parts: it is a unity. There must, therefore, be something9 which constitutes them as a single thing, a unity. Let us call it, neutrally (and with a nod in the direction of particle physics), the gluon of the object, g.10 Now what of this gluon? Ask whether it itself is a thing, object, entity? It both is and is not. It is, since we have just talked about it, referred to it, thought about it. But it is not, since, if it is, p1, .. , pn, g, would appear to form a congeries, a plurality, just as much as the original one. If its behaviour is to provide an explanation of unity, it cannot simply be an object.

(p.9, section 1.3.4)

9. Or some things; but it will turn out that there is only one.

10. The name was coined, with essentially this meaning, in the Conclusion to Priest (1995a).

(p.9, section 1.3.4)

Although it may seem like the gluon could be an object, as it in some sense is a nameable something with regard to a thing’s unity, Priest now will explain why it cannot be an object.]

It will pay to become clearer about why a gluon cannot be an object. A vicious regress stands behind this.12

12. This kind of regress argument is very old. In the form of the “third man argument” it is used in Plato’s Parmenides as an argument against the theory of forms. Plato is there concerned with what makes all, for example, red things one (namely, red). Invoking a form of redness produces the regress. Being one by being red is not the same thing as being one by being parts of something, and Plato’s form is not (obviously) a gluon. However, structurally, the situations are similar. We will come to the third man argument itself in Chapter 8.

(9)

[contents]

1.4.2

[In the Bradley regress, a binding factor is posited as being a member of the unity it binds. But that only leaves us to find yet another binding factor that would bind the first into the whole. There can be no end so long as the binding factors are consider as object/parts. In terms of gluons, if we make the gluon be an object/part, then we will always need yet another gluon to explain how the prior gluon is bound into the whole.]

[In section 1.3.3 we discussed Frege’s problem of accounting for the unity of a proposition. The unity is to be found in the relation between a function-part (like a predicate) and an argument part (like a subject to the predicate). That unity comes undone when trying to make statements about concepts themselves, because then something which is not an object is also bestowed that status by means of the propositional structure.] Priest returns now to the problem of unity in a proposition, but this time turning to Russell rather than Frege. [The idea seems to be the Following. Russell wants to account for the unity of the proposition, and he locates it in the copula ‘is’. For, it is what unites the subject and predicate. He furthermore claims that the copula cannot be a constituent of the proposition, and it can only be a “way in which the constituents are put together.” For, suppose that it were a constituent. We would still need to find something else that puts those constituents together (the subject, with the ‘is’, with the predicate). And supposing that binding element to be a component too, we would need yet another such binding factor. Under such a structuring pattern, we would reach no ultimate binding factor, despite that being our very aim.]

Return to the matter of the unity of the proposition again. At one stage in his career, Russell was much concerned with this, and one possibility he considered was that it was the copula, ‘is’, that binds the constituents together. (So, in Fregean terms, there is just one concept, which is the copula.13) He then explains why the copula cannot be on a footing with the other constituents:14

It might be thought that ‘is’, here, is a constant constituent. But this would be a mistake: ‘x is a’ is obtained from ‘Socrates is human’, which is to be regarded as a subject-predicate proposition, and such propositions, we said, have only two constituents [Socrates and humanity]. Thus ‘is’ represents merely the way in which the constituents are put together. This cannot be a new constituent, for if it were there would have to be a new way in which it and the two other constituents are put together, and if we take this way as again a constituent, we find ourselves embarked on an infinite regress.

(10)

13. A discussion of this view, in the context of its regress, is given in Gaskin (1995).

14. Eames and Blackwell (1973), p. 98.

(10)

Priest says that Russell here is using an argument by F.H. Bradley that was also related to the issue of the unity of the proposition and to unity in general.

Russell is using an argument used earlier to great effect by Bradley.15 Again, addressing the problem of the unity of the proposition, Bradley starts by supposing that a proposition has components A and B. What constitutes them into a unity? A natural thought is that it is some relation between them, C. But, he continues:16

[we] have made no progress. The relation C has been admitted different from A and B ... Something, however, seems to be said of this relation C, and said, again, of A and B ... [This] would appear to be another relation, D, in which C, on one side, and, on the other side, A and B, stand. But such a makeshift leads at once to the infinite process ... [W]e must have recourse to a fresh relation, E, which comes between D and whatever we had before. But this must lead to another, F; and so on indefinitely ... [The situation] either demands a new relation, and so on without end, or it leaves us where we were, entangled in difficulties.

And Bradley is, in fact, aware that this is not just a problem concerning the unity of the proposition. It is much more general. Thus, in discussing the unity of the mind, Bradley writes:17

When we ask ‘What is the composition of Mind,’ we break up that state, which comes to us as a whole, into units of feeling. But since it is clear that these units, by themselves, are not all the ‘composition’, we are forced to recognize the existence of the relations ... If units have to exist together, they must stand in relation to one another; and, if these relations are also units, it would seem that the second class must also stand in relation to the first. If A and B are feelings, and if C their relation is another feeling, you must either suppose | that component parts can exist without standing in relation to one another, or else that there is a fresh relation between C and AB. Let this be D, and once more we are launched off on the infinite process of finding a relation between D and C–AB; and so on forever. If relations are facts that exist between facts, then what comes between the relations and the other facts? (10-11)

15. In fact, it had been used some 600 years earlier by Jean Buridan in his Questiones in Metaphysicam Aristotelis (Bk V, q. 8). (See Normore (1985), p. 197f.) It should therefore be called the Buridan/Bradley regress.

16. Allard and Stock (1994), p. 120.

17. Allard and Stock (1994), pp. 78–9. (10)

Priest then reformulates this in terms of gluons. Suppose that the gluon is a member among the other parts. We would then need another gluon to bind it with them. And that gluon would need yet another, and so on without end.

We can state the regress problem generally in terms of gluons. Suppose that we have a unity comprising the parts, a, b, c, d, for example. There must be something which, metaphysically speaking, binds them together.This is the object’s gluon, g. But then there must be something which binds g and a, b, c, d together, a hyper-gluon, g′. There must, then, be something which binds g′, g, and a, b, c, d together, a hyper-hyper-gluon, g′′. Obviously we are off on an infinite regress. Moreover, it is a vicious one.

(11)

[contents]

1.4.3

[Unity is Not Gluons All the Way Down]

[We cannot simply account for unity by saying it is gluons all the way down. For, no such gluon is sufficient to explain the unity. All of them require something in addition. So, simply saying the unity is found in yet another part never tells us in what the unity consists.]

Priest next explains why we cannot simply say that it is gluons all the way down, or in other words, that there is an infinity of gluons. [I may not capture his insight here. My best attempt for now is the following. What we want is an explanation for unity. Suppose we say it is gluons all the way down. This fails, because at no point in the going down is there a structuring part that unifies the whole. For, given any gluon in the infinite chain, none is sufficient to account for the unity. And to say that it is always to be found in yet another part only makes this problem unsolvable, because it makes it impossible to ever identify the ultimate unifying component. Let me quote, as I am probably not putting that in the best way.]

Perhaps it is not immediately obvious that this is so. Could there not just be a whole lot’a gluin’ goin’ on? To understand why this is not a valid response, we must come back to what is at issue here. Our original problem was how a unity of parts is possible. We need an explanation. Given a bunch of parts, simply invoking another object does not do this. We still have the original problem of how a unity of parts is possible. Thus is a new step triggered, and so on indefinitely. Even invoking an infinite regress of objects does not solve the problem. We still have no explanation of how a unity is constituted. If one is asked how to join two links of a chain together, it helps not one iota to say that one inserts an intervening link. (And adding that one might need an infinite number of such links merely makes the matter worse.) In vicious regresses of this kind (I do not think it is the only kind) the infinity has, in fact, precious little to do with matters. The point is that something has already gone wrong at the first step: a failure of explanation.18

(11)

18. ‘[I]t is the first step in the regress that counts, for we at once, in taking it, draw attention to the fact that the alleged explanation or justification has failed to advance matters; that if there was any difficulty in the original situation, it breaks out in exactly the same form in the alleged explanation. If this is so, the regress at once develops . . . ’ Passmore (1961), p. 31.

(11)

[contents]

1.4.4

[Unity is Not Another Object]

[In conclusion, on account of the Bradley Regress, we cannot explain the unity of objects as being another object.]

Thus: “As Frege realized, if something is to perform the role of explaining how it is that a unity of objects is achieved, it cannot just be another object” (11).

[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.

Or if otherwise cited:

Allard, J. W., and Stock, G. (eds.) (1994), F. H. Bradley: Writings on Logic and Metaphysics, Oxford: Oxford University Press.

Eames, E., and Blackwell, K. (eds.) (1973), Collected Papers of Bertrand Russell, vol. 7: Theory of Knowledge, London: Allen and Unwin.

Gaskin R. (1995), ‘Bradley’s Regress, the Copula and the Unity of the Proposition’, Philosophical Quarterly 45: 161–80. E

Normore, C. (1985), ‘Buridan’s Ontology’, pp. 189–203 of J. Bogen and E. McGuire (eds.), How Things Are, Dordrecht: Reidel Publishing Company.

Passmore, J. (1961), ‘The Infinite Regress’, ch. 2 of Philosophical Reasoning, London: Duckworth.

.

## 28 Mar 2018

### Priest (1.6) An Introduction to Non-Classical Logic, ‘Conditionals’, 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

1.

Classical Logic and the Material Conditional

1.6

Conditionals

Brief summary:

(1.6.1) We now will examine conditionality in classical propositional logic. (1.6.2) A conditional contains two propositions. One is the consequent, which depends in some sense on the other proposition, called the antecedent. In English they are often formed using “if” or similar constructions. (1.6.3) When writing the antecedent or consequent by themselves, we often need to make changes to the verb tense or mood of the sentence, especially when formulating inferences. (1.6.4) Not all English “if” constructions are conditionals. We can test them by seeing if they can be expressed under the form ‘that A implies B’.

Contents

1.6.1

[Preview: Conditionals]

1.6.2

[The Structure of Conditionals]

1.6.3

[Grammar and Isolated Conditional Parts]

1.6.4

[Testing English Constructions for Conditionality]

Summary

1.6.1

[Preview: Conditionals]

[We now will examine conditionality in classical propositional logic.]

[In section 1.3 we learned the semantics for classical propositional logic.] For the rest of chapter 1, Priest will discuss the nature of conditionality that the semantics we have seen so far will give us. We will also evaluate the inadequacies of this notion of the conditional. We begin now with the question, what is a conditional? (11)

[contents]

1.6.2

[The Structure of Conditionals]

[A conditional contains two propositions. One is the consequent, which depends in some sense on the other proposition, called the antecedent. In English they are often formed using “if” or similar constructions.]

Priest now describes the general structure of a conditional. It has two propositions, with one depending on the other in some sense. The one that doing the depending is the consequent, and the one that is being depended upon is the antecedent. In English, we express conditionality using “if” or equivalent constructions.

Conditionals relate some proposition (the consequent) to some other proposition (the antecedent) on which, in some sense, it depends. They are expressed in English by ‘if’ or cognate constructions:

If the bough breaks (then) the cradle will fall.

The cradle will fall if the bough breaks.

The bough breaks only if the cradle falls. |

If the bough were to break the cradle would fall.

Were the bough to break the cradle would fall.

(11-12)

[contents]

1.6.3

[Grammar and Isolated Conditional Parts]

[When writing the antecedent or consequent by themselves, we often need to make changes to the verb tense or mood of the sentence, especially when formulating inferences.]

Priest’s next point clarifies something about how certain grammatical structurings are needed when the antecedent or consequent are written by themselves, especially in the context of making inferences.

Note that the grammar of conditionals imposes certain requirements on the tense (past, present, future) and mood (indicative, subjunctive) of the sentences expressing the antecedent and consequent within it. These may be different when the antecedent and consequent stand alone. To see this, just consider the following applications of modus ponens (if A then B; A; hence B):

If he takes a plane he will get there quicker.

He will take a plane.

Hence, he will get there quicker.

If he had come in the window there would have been foot-marks.

He did come in the window.

So, there are foot-marks.

(12)

[contents]

1.6.4

[Testing English Constructions for Conditionality]

[Not all English “if” constructions are conditionals. We can test them by seeing if they can be expressed under the form ‘that A implies B’.]

Some English constructions use “if,” but they do not form conditional structures. One example Priest gives is: “If I may say so, you have a nice ear-ring” (12). Priest then provides a basic test for “if” constructions to determine whether they are conditionals or not. We see if they can be “rewritten equivalently as ‘that A implies that B’” (12).

Note, also, that not all sentences using ‘if’ are conditionals; consider, for example, ‘If I may say so, you have a nice ear-ring’, ‘(Even) if he was plump, he could still run fast’, or ‘If you want a banana, there is one in the kitchen.’ A rough and ready test for ‘if A, B’ to be a conditional is that it can be rewritten equivalently as ‘that A implies that B’.

(12)

[contents]

From:

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

.

### Priest (4.1) An Introduction to Non-Classical Logic, ‘Introduction [to ch.4, “Non-Normal Modal Logics; Strict Conditionals”],’ 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

4.1

Introduction

Brief summary:

(4.1.1) In the following sections of this chapter, we will examine non-normal modal logics. They involve non-normal worlds, which are ones with different truth conditions for the modal operators. (4.1.2) Following that in the chapter is an examination of the strict conditional.

Contents

4.1.1

[Non-Normal Worlds and Non-Normal Modal Logics]

4.1.2

[The Strict Conditional]

Summary

4.1.1

[Non-Normal Worlds and Non-Normal Modal Logics]

[In the following sections of this chapter, we will examine non-normal modal logics. They involve non-normal worlds, which are ones with different truth conditions for the modal operators.]

[Recall from section 3.1 that we have been working with “normal” modal logics, given the name K. The varieties of K logics are made by applying restrictions to the accessibility relation R. Here were some restrictions:

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 3.2.7)

We will now examine non-normal modal logics. Priest writes here that in non-normal worlds, the truth-conditions for the modal operators are different. We learn the details of that in section 4.2. The other notion we should mention here is that these non-normal modal logics are “weaker” than K. I do not know what the terms “stronger” and “weaker” mean yet. Later at section 4.4.4, he writes:

Note that Kρστ(Kυ) is the strongest of all the logics we have looked at: every normal system that we looked at is contained in Kρστ (3.2.9), and every non-normal system that we looked at is contained in the corresponding normal system (4.4.1, 4.4.2). N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system.

(p.68, section 4.4.4)

If we take into consideration what we said in section 3.2.8, and combine it with what is said here, perhaps we can assess the meaning of these terms, but this is all my guesswork (sorry). What is said above suggests that when a certain logic is contained within another, then it is weaker than that other one. But I am not sure what containment is. It could be having a set of interpretations that is a subset of that of another logic, or it could be having a set of valid inferences that is a subset of that other logic (again, see section 3.2.8). But from what is written above, it would seem that Kρσ is contained within Kρστ, and thus Kρσ is weaker. With regard to subsets, I think the set of valid inferences of Kρσ is a subset of that of Kρστ. So my guess is the following. To be a weaker system means that it has fewer valid inferences. Non-normal modal logics are weaker than normal logics, thus they would have fewer valid inferences. Sorry for not knowing for sure; please consult the quotation below.]

In this chapter we look at some systems of modal logic weaker than K (and so non-normal). These involve so-called non-normal worlds. Nonnormal worlds are worlds where the truth conditions of modal operators are different.

(64)

[contents]

4.1.2

[The Strict Conditional]

[Following that in the chapter is an examination of the strict conditional.]

After that, we will examine strict conditionals.

We are then in a position to return to the issue of the conditional, and have a look at an account of a modal conditional called the strict conditional.

(64)

[contents]

From:

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

.

### Priest (3.3) An Introduction to Non-Classical Logic, ‘Tableaux for Normal Modal Logics’, 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

3.3

Tableaux for Normal Modal Logics

Brief summary:

(3.3.1) To make tableaux for other normal modal logics, we will add rules regarding the R accessibility relation. (3.3.2) The tableaux for the different normal modal logics take rules reflecting the properties of the accessibility relations that characterize them.

 Tableaux Rules for Kρ, Kσ, and Kτ ρ . ↓ iri . . “ρrD” σ irj ↓ jri . . “σrD” τ irj jrk ↓ .irk . “τrD”

(3.3.3) In the first tableau example for normal modal logics, we learn that p p is valid in Kρ but not in K; thus Kρ is a proper extension of K. (3.3.4) In Priest’s second example, we learn that p ⊃ □◊p is not valid in K but it is valid in Kσ, thus Kσ is a proper extension of K. (3.3.5) In the third of Priest’s examples, we learn that □p ⊃ □□p is not valid in K but it is valid in Kτ, thus Kτ is a proper extension of K. (3.3.6) For compound systems, we must apply the rules for each restriction. When making the tableau, we should apply the ◊-rule first. Then secondly we compute and add all the needed new facts about r that then arise. Lastly we should backtrack whenever necessary to apply the □-rule in cases of r where it is required. (3.3.7) We make counter-models by assigning worlds in accordance with the i numbers on an open branch, r relations in accordance with the irj formulations, p,i formulations as  vwi(p) = 1, ¬p,i formulations as vwi(p) = 0, and if neither of those two cases show for some p, we can assign it any value we want. (3.3.8) These tableaux are both sound and complete.

Contents

3.3.1

[Adding Accessibility Rules for Other Normal Modal Logic Tableaux]

3.3.2

[Rules for Kρ, Kσ, and Kτ]

3.3.3

[Tableau example 1: Kρ (Reflexive)]

3.3.4

[Tableau example 2: Kσ (Symmetrical)]

3.3.5

[Tableau example 3: Kτ  (Transitive)]

3.3.6

[Tableau example 4: Kστ (Symmetrical and Transitive)]

3.3.7

[Counter-Models]

3.3.8

[Soundness and Completeness of the Normal Modal Tableaux]

Summary

3.3.1

[Adding Accessibility Rules for Other Normal Modal Logic Tableaux]

[To make tableaux for other normal modal logics, we will add rules regarding the R accessibility relation.]

[In section 2.4 we examined the tableaux rules for modal logic. And in section 3.2 we learned the semantics for normal modal logics, with many of them being fashioned by adding constraints to the accessibility relation of the interpretations. Now we will extend the tableaux rules so that they work for these other normal modal logics, and this will primarily involve rules involving the R accessibility relation.]

The tableau rules for K can be extended to work for other normal systems as well. Essentially, this is done by adding rules which introduce further information about r on branches. Since this information comes into play when the rule for □ is applied, the effect of this is to increase the number of applications of that rule.

(38)

[contents]

3.3.2

[Rules for Kρ, Kσ, and Kτ]

[The tableaux for the different normal modal logics take rules reflecting the properties of the accessibility relations that characterize them.]

Priest now gives the rules for certain normal modal logics. [Recall the constraints from section 3.2.3:

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)

Priest will give the rules for the first three, and we see the fourth in the next section.

 Tableaux Rules for Kρ, Kσ, and Kτ ρ . ↓ iri . . “ρrD” σ irj ↓ jri . . “σrD” τ irj jrk ↓ .irk . “τrD”

| (We come to the rule for η in the next section.) The rule for ρ means that if i is any integer on the tableau, we introduce iri. It can therefore be applied to world 0 after the initial list, and, thereafter, after the introduction of any new integer. The other two rules are self-explanatory. Note that if the application of a rule would result in just repeating lines already on the branch, it is not applied. Thus, for example, if we apply the σ-rule to irj to get jri, we do not then apply it again to jri to get irj. The following three subsections give examples of tableaux for Kρ, Kσ and Kτ , respectively.

[contents]

3.3.3

[Tableau example 1: Kρ (Reflexive)]

[In the first tableau example for normal modal logics, we learn that p p is valid in Kρ but not in K; thus Kρ is a proper extension of K.]

Here is Priest’s first tableau example (recall the other rules from section 2.4.4):

Kρ □p ⊃ p

 ⊢Kρ □p ⊃ p 1. . 2. . 3. . 4. . 5. ¬(□p ⊃ p),0 ↓ 0r0 ↓ □p,0 ↓ ¬p,0 ↓ p,0 × P . 1ρrD . 1¬⊃D . 1¬⊃D . 2,3□rD (4)

(39, with naming and enumeration added)

The last line is obtained from □p, 0, since 0r0. Since □p p is not valid in K (2.12, problem 2(o)), this shows that Kρ is a proper extension of K. (That is, Kρ is not exactly the same as K.)

(39, boldface mine)

[contents]

3.3.4

[Tableau example 2: Kσ (Symmetrical)]

[In Priest’s second example, we learn that p ⊃ □◊p is not valid in K but it is valid in Kσ, thus Kσ is a proper extension of K.]

Here is Priest’s second tableau example (again, we refer to the other rules from section 2.4.4):

Kσ p ⊃ □◊p

 ⊢Kσ p ⊃ □◊p 1. . 2. . 3. . 4. . 5a. 5b. . 6. . 7. . 8. . . ¬(p ⊃ □◊p),0 ↓ p,0 ↓ ¬□◊p,0 ↓ ◊¬◊p,0 ↓ 0r1 ¬◊p,1 ↓ 1r0 ↓ □¬p,1 ↓ ¬p,0 × P . 1¬⊃D . 1¬⊃D . 3¬□D . 4◊rD 4◊rD . 5aσrD . 5b¬◊D . 6,7□rD (2)

(39, with naming and enumeration added.)

The last line follows from the fact that □¬p, 1, since 1r0. Since p ⊃ □◊p is not valid in K (2.12, problem 2(t)), this shows that Kσ is a proper extension of K.

(39, boldface mine)

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3.3.5

[Tableau example 3: Kτ  (Transitive)]

[In the third of Priest’s examples, we learn that □p ⊃ □□p is not valid in K but it is valid in Kτ, thus Kτ is a proper extension of K.]

Here is Priest’s third tableau example (again, refer to the other rules in section 2.4.4):

Kτ □p ⊃ □□p

 ⊢Kτ □p ⊃ □□p 1. . 2. . 3. . 4. . 5a. 5b. . 6. . 7a. 7b. . 8. . 9. . . ¬(□p ⊃ □□p),0 ↓ □p,0 ↓ ¬□□p,0 ↓ ◊¬□p,0 ↓ 0r1 ¬□p,1 ↓ ◊¬p,1 ↓ 1r2 ¬p,2 ↓ 0r2 ↓ p,2 × P . 1¬⊃D . 1¬⊃D . 3¬□D . 4◊rD 4◊rD . 5b¬◊D . 6◊rD 6◊rD . 5a,5bτrD . 2,8□rD (7b)

(40, with naming and enumeration added.)

When we add 1r2 to the tableau because of the ◊-rule, we already have 0r1; hence, we add 0r2. Since □p holds at 0, an application of the rule for □ immediately closes the tableau. Since □p ⊃ □□p is not valid in K (2.12, problem 2(r)), this shows that Kτ is a proper extension of K.

(40, boldface mine)

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3.3.6

[Tableau example 4: Kστ (Symmetrical and Transitive)]

[For compound systems, we must apply the rules for each restriction. When making the tableau, we should apply the ◊-rule first. Then secondly we compute and add all the needed new facts about r that then arise. Lastly we should backtrack whenever necessary to apply the □-rule in cases of r where it is required.]

Priest now explains how to do tableaux for normal modal logics with compound restrictions. (I will quote in the entirety as it is well explained in the original):

For ‘compound’ systems, all the relevant rules must be applied. There may be some interplay between them. To keep track of this, adopt the following procedure. New worlds are normally introduced by the ◊-rule. Apply this first. Then compute all the new facts about r that need to be added, and add them. Finally, backtrack if necessary and apply the □-rule wherever the new r facts require it. The procedure is illustrated in the following tableau, demonstrating that Kστp ⊃ □◊p. For brevity’s sake, we write more than one piece of information about r on the same line.

 ⊢Kστ ◊p ⊃ □◊p 1. . 2. . 3. . 4a. 4b. . 5. . 6. . 7a. 7b. . 8. . 9. . 10. . 11. . 12. . . ¬(◊p ⊃ □◊p),0 ↓ ◊p,0 ↓ ¬□◊p,0 ↓ 0r1 p,1 ↓ 1r0,1r1,0r0 ↓ ◊¬◊p,0 ↓ 0r2 ¬◊p,2 ↓ 2r0, 2r2, 1r2, 2r1 ↓ □¬p,2 ↓ ¬p,2 ↓ ¬p,1 ↓ ¬p,0 × P . 1¬⊃D . 1¬⊃D . 2◊rD 2◊rD . 4a,5στD . 3¬□D . 6◊rD 6◊rD . 5,7a,8στrD . 7b¬◊D . 8,9□rD . 8,9□rD . 8,9□rD (11×4b)

The line ◊¬◊p, 0 requires the construction of a new world, 2, with an application of the ◊-rule. This is done on the next two lines. We then add all the new information about r that the creation of world 2 requires. 2r0 is added because of symmetry; 2r2 is added because of transitivity and the fact that we have 2r0 and 0r2; 1r2 is added because of transitivity and the fact that we have 1r0 and 0r2; similarly, 2r1 is added because of transitivity. Symmetry and transitivity require no other facts about r. In constructing a tableau, it may help to keep track of things if one draws a diagram of the world structure, as it emerges.

(40-41, with naming and enumeration added to the tableau. Page break comes in the middle of the tableau)

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3.3.7

[Counter-Models]

[We make counter-models by assigning worlds in accordance with the i numbers on an open branch, r relations in accordance with the irj formulations, p,i formulations as  vwi(p) = 1, ¬p,i formulations as vwi(p) = 0, and if neither of those two cases show for some p, we can assign it any value we want.]

[Recall from section 2.4.7 how counter-models are formed:

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)

Priest now explains how to do it with the inclusion of the extra r information in our alternate normal modal logics.]

Counter-models read off from an open branch of a tableau incorporate the information about r in the obvious way. Thus, consider the following tableau, which shows that ⊬Kρσ p ⊃ □□p.

 ⊬Kρσ □p ⊃ □□p 1. . 2. . 3. . 4. . 5. . 6. . 7a. 7b. . 8. . 9. . 10. . 11a. 11b. . 12. . . ¬(□p ⊃ □□p),0 ↓ 0r0 ↓ □p,0 ↓ ¬□□p,0 ↓ p,0 ↓ ◊¬□p,0 ↓ 0r1 ¬□p,1 ↓ 1r1, 1r0 ↓ p,1 ↓ ◊¬p,1 ↓ 1r2 ¬p,2 ↓ 2r2, 2r1 P . 1ρrD . 1¬⊃D . 1¬⊃D . 2,3□rD . 4¬□D . 6◊rD 6◊rD . 7ρσrD . 5,7a□rD . 7b¬□D . 10◊rD 10◊rD . 11,ρσrD (open)

The counter-model is ⟨W, R, v⟩, where W = {w0,w1,w2}, R is such that w0Rw0, w1Rw1, w2Rw2, w0Rw1, w1Rw0, w1Rw2 and w2Rw1, and v is such that | vw0(p)  = vw1(p) = 1, vw2(p) = 0. In pictures:

xxxxxxxx

woxxw1xxw2

pxxxxpxxxx¬p

(41-42, with naming and enumeration added to the tableau)

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3.3.8

[Soundness and Completeness of the Normal Modal Tableaux]

[These tableaux are both sound and complete.]

Priest notes lastly that: “The tableau systems above are all sound and complete with respect to their respective semantics. The proof of this can be found in 3.7” (42).

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From:

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

.