23 Apr 2018

Priest (2.1) One, “How Gluons Glue”, summary

 

by Corry Shores

 

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[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

 

2.1

How Gluons Glue

 

 

 

Brief summary:

(2.1.1) The gluon is the factor that binds parts into a unity. It has the contradictory properties of both being and not being an object. We now will see how gluons bind parts into unities, which involves breaking the Bradley regress. (2.1.2) The binding action of gluons involves non-transitive identity.

 

 

 


 

Contents:

 

2.1.1

[How Gluons Will Glue]

 

2.1.2

[Gluons’ Non-Transitive Identity]

 

Bibliography

 

 

 

 

 

Summary

 

2.1.1

[How Gluons Will Glue]

 

[The gluon is the factor that binds parts into a unity. It has the contradictory properties of both being and not being an object. We now will see how gluons bind parts into unities, which involves breaking the Bradley regress.]

 

[In section 1.3.1 we noted how things have parts that together compose the unity they belong to by means of a factor that binds the parts together in a unifying way. This unificatory binding factor is called the gluon. We discussed the problems with seeing it just as an object-part and with seeing it just as not being an object-part (as for example being instead a relation) (see section 1.4, section 1.5, and section 1.6). As a result, in section 1.6.6, we proposed that it is best to hold the dialetheic position that gluons both are and are not objects. In sections 1.3.4 and 1.6.6, we noted that the gluon is a contradictory entity, having the contradictory properties both of being an object and of not being an object (and see section 1.3.5). But we still need to explain how the gluon binds the parts into a unity. (On this importance of the how question when accounting for something philosophically, see section 1.5.5.) This is what we now endeavor, which involves us seeing how we will break the Bradley regress (see section 1.4 especially).]

In the previous chapter we saw that there must be something which accounts for a unity composed of parts where one exists – a gluon; and we saw that a gluon may be expected to have contradictory properties. But we have not yet faced the question of how the gluon does its job: how does it bind the parts (including itself) into a whole? Its having contradictory properties does not immediately address this question (though, one might suspect, it is going to play an important role). In this chapter, we look at the answer. The key is breaking the Bradley regress. We will start by seeing how.

(16)

[contents]

 

 

 

 

2.1.2

[Gluons’ Non-Transitive Identity]

 

[The binding action of gluons involves non-transitive identity.]

 

[In order to explain how gluons glue, we will also need to change the logical properties that we normally ascribe to identity, namely, we need to conceive it as being non-transitive. First Priest will give an informal account of gluonic non-transitive identity, and at the end, he provides a more technical account.]

This will immediately launch us into a discussion of identity. Identity cannot work in the way that orthodoxy takes it to if gluons are to do their job. In particular, it must be non-transitive. How so? The rest of the chapter explains, and articulates the nature of gluons more precisely in this theoretical context. The ideas are spelled out informally. Full technical details can be found in the technical appendix to the chapter, Section 2.10, which can be skipped without loss of continuity by those with no taste for such things.

(16)

[contents]

 

 

 

 


 

Bibliography:

 

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.

 

 

 


.

22 Apr 2018

Priest (4.5) An Introduction to Non-Classical Logic, ‘Strict Conditionals,’ summary.

 

by Corry Shores

 

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

 

4.5

Strict Conditionals

 

 

 

 

Brief summary:

(4.5.1) We now will examine the conditional in the context of modal logic. (4.5.2) There are contingently true material conditionals that we would not want to say are true on account of their contingency, like, “The sun is shining ⊃ Canberra is the federal capital of Australia.” For, “Things could have been quite otherwise, in which case the material conditional would have been false” (72). To remedy this, we could defined the conditional as: “‘if A then B’ as □(AB), where □ expresses an appropriate notion of necessity” (72). (4.5.3) This definition of the conditional using modal logic is called the strict conditional, symbolized as AB and defined as □(AB). (4.5.4) The strict conditional does not validate the problematic counter-examples (that we have seen) for the material conditional.

 

 

 

 

Contents

 

4.5.1

[The Conditional in Modal Logic]

 

4.5.2

[Problems with the Conditional When It Is Contingently True]

 

4.5.3

[The Strict Conditional Defined]

 

4.5.4

[The Strict Conditional as Intuitively Valid, Unlike the Material Conditional]

 

 

 

 

 

 

Summary

 

4.5.1

[The Conditional in Modal Logic]

 

[We now will examine the conditional in the context of modal logic.]

 

Priest says that we have learned the basics of modal logic, and we can now turn to matters regarding the conditional (72).

[contents]

 

 

 

 

4.5.2

[Problems with the Conditional When It Is Contingently True]

 

[There are contingently true material conditionals that we would not want to say are true on account of their contingency, like, “The sun is shining ⊃ Canberra is the federal capital of Australia.” For, “Things could have been quite otherwise, in which case the material conditional would have been false” (72). To remedy this, we could defined the conditional as: “‘if A then B’ as □(AB), where □ expresses an appropriate notion of necessity” (72).]

 

[Priest notes that there are ways that a material conditional can be technically true but nonetheless we would not want to say it is true. (The problem here does not seem to be relevance but rather contingency. He gives the example of: The sun is shining ⊃ Canberra is the federal capital of Australia. The problem it seems is that although this can be a true conditional, we might be bothered by the fact that the sun could very well not be shining, and maybe even that some other city could very easily have become the capital. It is not entirely clear to me yet how to understand this problem, but let us look at the solution. It would be to understand structures of the form ‘if A then B’ as □(AB). Perhaps the idea is that we can easily think of a world much like ours where “The sun is shining ⊃ Canberra is the federal capital of Australia” is not true ((for, perhaps, in this world on a certain day the sun is shining, but there is some other capital of Australia)). Instead, perhaps, we would want something that would be true even under very strong variations of our world, as for example, “The sun is shining ⊃ it is day” perhaps, or something like that, where it is inconceivable how in any world it could be false. For, that is one sense we give to the conditional, it seems. We might think of it as saying, given any occurrence of the antecedent, under whatever possible circumstance, you will have the consequent too. For otherwise, it may only designate a coincidence.)]

Consider a true material conditional, such as ‘The sun is shining ⊃ Canberra is the federal capital of Australia’. One is inclined to reject this as a true conditional just because the truth of the material conditional is too contingent an affair. Things could have been quite otherwise, in which case the material conditional would have been false. This suggests defining the conditional, ‘if A then B’ as □(AB), where □ expresses an appropriate notion of necessity.

(72)

[contents]

 

 

 

 

4.5.3

[The Strict Conditional Defined]

 

[This definition of the conditional using modal logic is called the strict conditional, symbolized as AB and defined as □(AB).]

 

Priest says that Lewis created modern modal logic out of his dissatisfaction with the material conditional. He favored what is called the strict conditional, symbolized as ⥽ and defined like we saw above in section 4.5.2, and so AB is defined as □(AB).

When Lewis created modern modal logic, he was not, in fact, concerned with modality as such. He was dissatisfied with the material conditional. He defined AB as □(AB), and suggested this as a correct account of the conditional. ⥽ is usually called the strict conditional.

(72)

[contents]

 

 

 

 

4.5.4

[The Strict Conditional as Intuitively Valid, Unlike the Material Conditional]

 

[The strict conditional does not validate the problematic counter-examples (that we have seen) for the material conditional.]

 

[Priest will now show how this definition of the strict conditional does not validate any of the problematic counter-examples for the material conditional that we saw in section 1.7, section 1.8, and section 1.9. One is

¬(AB) ⊨ A

Priest says that it is a variation on

¬(A B) ⊢ A

which we examined in section 1.9. Let us look more closely at it, since we made a possible tableau proof for it, showing it to be valid under the semantics of the material conditional. Here was the possible proof for it:

 

(This table in not in the text and probably is mistaken. Please trust your own proofs over my attempt below.)

¬(A ⊃ B) ⊢ A

1.

.

2.

.

3.

.

4.

¬(A ⊃ B)

.

¬A

¬B

A

 ×

P

.

P

.

1¬⊃D

.

1¬⊃D

(4×2)

Valid

(not in Priest’s text, but see section 1.9.1)

 

We will try to see that under the strict conditional, rendered now as

¬(AB) ⊨ A

this is not valid in Kρστ. To do this, we first translate it as:

¬□(AB) ⊨ A

And we make our tableau using the rules from section 2.4 and 3.3.2. Supposing this to give us open branches, we then use the rules for creating a counter-models from section 2.4.7. So

 

¬□(AB) ⊨ A

(tested as the following, with counter-modeling, to determine the invalidity of the above:)

¬□(AB) ⊢ A

 

(This table is not in the text and probably is mistaken. Please trust your own proofs over my attempt below.)

¬□(A ⊃ B) ⊢ A

1.

.

2.

.

3.

.

4.

.

5a.

5b.

.

6.

.

7.

.

8.

.

9.

.

.

¬□(A ⊃ B),0

.

¬A,0

0r0

◊¬(A ⊃ B),0

0r1

¬(A ⊃ B),1

1r0

1r1

A,1

¬B,1

 

P

.

P

.

rD

.

1¬□D

.

4rD

4rD

.

5aσrD

.

5bρrD

.

5b¬⊃D

.

5b¬⊃D

(open)

 

(not in Priest’s text)

 

For the counter model, I would guess it is the following, although this is based on the above tableau, which is likely incorrect:

W = {w1, w2}

w1Rw2, w1Rw1, w2Rw2, w2Rw1

vw1(A) = 0, vw1(B) = 0

vw2(A) = 1, vw2(B) = 0

So let us look again at the formulation in question:

¬□(AB) ⊨ A

Recall how we define validity, from section 2.3.11:

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 vw(B) = 1 for all B ∈ Σ, then vw(A) = 1.

A iff φA, i.e., for all interpretations ⟨W, R, v⟩ and all w W, vw(A) = 1.

(Priest p.23, section 2.3.11)

So in our counter-model, we should have a case where ¬□(AB) is true but A is false. In our counter-model, we have set A as false in world 1. Let us see then if ¬□(AB) is true in world 1. The semantic evaluation for negation is:

vwA) = 1 if vw(A) = 0, and 0 otherwise.

So we next need to know the value of

□(AB)

in world 1. The semantic evaluation for necessity is:

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

So □(AB) is true in world 1 if in world 1 and 2 (AB) is true, and it is false if (AB) is false in either world 1 or world 2. Now, as far as I can tell, the semantic evaluation of the conditional in modal logic was not given yet, but somehow I recall it. I just cannot find it at the moment. John Nolt, in his Logics section 11.2.1, defined it as:

v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;

v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.

(Nolt 315, section 11.2.1)

So let us render that into Priest’s notation:

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

(not in Priest that I know of or where, yet)

Recall:

vw1(A) = 0, vw1(B) = 0

vw2(A) = 1, vw2(B) = 0

So following the proposed semantics for ⊃ above, we have:

vw1(A B) = 1

vw2(A B) = 0

Let us now work backwards. Although vw1(A B) = 1, still

vw1□(A B) = 0

For, vw2(A B) = 0, and thus it is not necessarily true in world 1. And since vw1□(A B) = 0, that means

vw1¬□(A B) = 1

Our original formulation was

¬□(AB) ⊨ A

We see now that it is invalid, because our counter-model gives a premise that is evaluated as true with the conclusion being evaluated as false. Thus:

¬□(AB) ⊭ A

and by translation:

¬(AB) ⊭ A

In all, this shows how the strict conditional gives us a sense of the conditional that seems to correspond with the English conditional.]

It is easy enough to check that all the following are false in Kρστ, and so in all the normal and non-normal logics we have looked at.

BAB

¬AAB

(AB)⥽C ⊨ (AC) ∨ (BC)

(AB) ∧ (CD) ⊨ (AD) ∨ (CB)

¬(AB) ⊨ A

But these inferences are the basis of all the objections to the material account of the conditional that we looked at in 1.7–1.9. Hence, the strict conditional is not subject to any of the objections to which the material conditional is.

(72, referencing section 1.7, section 1.8, and section 1.9)

[contents]

 

 

 

 

 

From:

 

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

 

 

Or if otherwise noted:

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

 

 

 

.

Priest (1.9) An Introduction to Non-Classical Logic, ‘More Counter-Examples’, summary

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[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

 

1.

Classical Logic and the Material Conditional

 

1.9

More Counter-Examples

 

 

 

Brief summary:

(1.9.1) There are three other counter-examples to the material conditional, and they present damning objections to the claim that the English conditional is material. {1} (AB) ⊃ C (A C) ∨ (B C); for example, “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on.” {2} (A B) ∧ (C D) ⊢ (A D) ∨ (C B); for example, “If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.” And {3} ¬(A B) ⊢ A; for example, “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (14-15). (1.9.2) We cannot dismiss these counter-examples on grammatical grounds, because they are all in the indicative mood. And we cannot dismiss them on conversational implicature grounds, because none break the rule of assert the strongest. (1.9.3) We cannot object that in fact the above counter-examples really are valid, provided we stipulate that the English conditional is material in those cases. For, by making that stipulation, we are admitting that naturally the English conditional is not material and is only artificially so. But the whole point of these objections is to show that the English conditional is naturally material.

 

 

 

 

 

Contents

 

1.9.1

[Three Potent Counter-Examples to the Claim that the English Conditional is Material]

 

1.9.2

[The Counter-Examples Evade Objections over Grammatical Mood or Conversational Implicature]

 

1.9.3

[The Ineffectuality of Objecting that the Counter-Examples are Material Under an Additional Stipulation]

 

 

 

 

 

 

Summary

 

1.9.1

[Three Potent Counter-Examples to the Claim that the English Conditional is Material]

 

[There are three other counter-examples to the material conditional, and they present damning objections to the claim that the English conditional is material. {1} (AB) ⊃ C (A C) ∨ (B C); for example, “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on.” {2} (A B) ∧ (C D) ⊢ (A D) ∨ (C B); for example, “If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.” And {3} ¬(A B) ⊢ A; for example, “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (14-15).]

 

[Previously in section 1.7 and section 1.8, Priest showed how the material conditional is not perfectly suited for understanding the logic of the conditional in the English language. He now discusses even more arguments for this. (They are not the simplest structures, so let us go through them slowly starting sometimes with an intuitive rendition and working to the more formal ones. First, we imagine an electrical circuit in which there are two switches in a series where both need to be closed in order for the circuit to close and the light to go on. Think of the first switch as A and the second switch as B. And think of the light as C. Since both switches need to be turned on for the light to go on, and since that is sufficient to bring about the light to turn on, we can think of the logic here as:

(AB) ⊃ C

In other words, if switch A and B are both closed, then light C will be on. Now, intuitively, we would say that it is not enough for just one switch to be on for the light to be on. But under the logic of the material conditional, it is valid. So the following formula is valid, and I will try to see if we can make a tableau to show that.

 

(AB) ⊃ C (A C) ∨ (B C)

 

[This table is not in the text and probably is mistaken. Please trust your own proofs over my attempt below.]

(A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C)

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

9.

.

(A ∧ B) ⊃ C

.

¬((A ⊃ C) ∨ (B ⊃ C))

¬(A ⊃ C)

¬(B ⊃ C)

A

¬C

B

↙                 ↘

¬(A ∧ B)       C    

↙                 ↘                    ×    

¬A       ¬B             

 ×         ×              

 

P

.

P

.

∨D

.

∨D

.

3¬⊃D

.

3/4¬⊃D

.

⊃D

.

1⊃D

(8×6)

∧D

(9×5)

(9×7)

Valid

(not in Priest’s text)

 

In other words, in our electrical circuit example where the light turning on relies conditionally on both the switches being closed, were that conditionally material, we would validly conclude that only one of the two is needed for the light to go on, which is in real fact not what would happen. For the next example, we set up a conjunction of two distinct conditionals. But on account of the semantics for the material conditional, it allows us to derive a disjunction of two conditionals with the same antecedents from before but with switched consequents. The problem with this becomes evident with Priest’s illustration, where the conjunction of conditionals is true, but the derived disjunction of conditionals is not, even though it is a proof-theoretic consequence. Here is Priest’s example:

If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.

(15)

So here is the formulation again, which I will try to prove as best I can with my limited time and skills:

 

(A B) ∧ (C D) ⊢ (A D) ∨ (C B)

 

[This table is not in the text and probably is mistaken. Please trust

your own proofs over my attempt below.]

(A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B)

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

9.

.

10.

.

11.

.

.

.

(A ⊃ B) ∧ (C ⊃ D)

.

¬((A ⊃ D) ∨ (C ⊃ B))

(A ⊃ B)

(C ⊃ D)

¬(A ⊃ D)

¬(C ⊃ B)

A

¬D

C

¬B

↙            ↘

¬A       B

 ×        ×

 

P

.

P

.

1∧D

.

1∧D

.

∨D

.

∨D

.

⊃D

.

⊃D

.

⊃D

.

⊃D

.

3⊃D

(11×7)

(11×10)

Valid

(not in Priest’s text)

 

The next one asserts a negated conditional and then derives the antecedent. It seems simple but is not so intuitive. The example is:

It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god.

(15)

(I am not sure how to grasp the problem with this, but I wonder if it has something to do with the fact that this antecedent asserts the existence of something, which would not seem to be derivable from the negated conditional it is the antecedent in. At any rate, the formula and its possible proof are the following:)

 

¬(A B) ⊢ A

[This table is not in the text and probably is mistaken. Please trust

your own proofs over my attempt below.]

¬(A ⊃ B) ⊢ A

1.

.

2.

.

3.

.

4.

¬(A ⊃ B)

.

¬A

¬B

A

 ×

P

.

P

.

1¬⊃D

.

1¬⊃D

(4×2)

Valid

(not in Priest’s text)

 

Here is Priest’s text:]

There are more fundamental objections against the claim that the indicative English conditional (even if it is distinct from the subjunctive) is material. It is easy to check that the following inferences are valid.

(AB) ⊃ C (A C) ∨ (B C)

(A B) ∧ (C D) ⊢ (A D) ∨ (C B)

¬(A B) ⊢ A

If the English indicative conditional were material, the following inferences would, respectively, be instances of the above, and therefore valid, which they are clearly not.

(1) If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. (Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open.) |

(2) If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.

(3) It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god.

(14-15)

[contents]

 

 

 

 

1.9.2

[The Counter-Examples Evade Objections over Grammatical Mood or Conversational Implicature]

 

[We cannot dismiss these counter-examples on grammatical grounds, because they are all in the indicative mood. And we cannot dismiss them on conversational implicature grounds, because none break the rule of assert the strongest.]

 

[Recall from section 1.8 how objections to some counter-examples for the claim that the English conditional is material involve making a distinction between subjunctive uses of the conditional clause in English from indicative uses, and to argue that indicative uses are material. The examples above in section 1.9.1 are all indicative, so that objection will not apply. Also recall from section 1.7.2 and section 1.7.3 how one might also distinguish certain odd but technically valid cases of the English conditional understood as material as resulting from incorrect uses of conversational rules, namely, to assert the strongest information. (Consider the example, “If New York is in New Zealand then 2 + 2 = 4.” Here, the rule of conversation that can apply is that we should always assert the strongest information. We know the consequent to be true but the antecedent false. So the strongest information is simply that “2 + 2 = 4,” and thus to follow the rule of assert the strongest, we would normally just assert that and leave out “If New York is in New Zealand,” which weakens the assertion with information that we know to be false.) But consider the first problematic counter-example from section 1.9.1 above.

(AB) ⊃ C (A C) ∨ (B C)

(1) If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. (Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open.)

(p.14, section 1.9.1)

Here, Priest notes, we cannot say that either of the disjuncts in the conclusion should be asserted instead of the other, because they both appear to be false.]

Notice that all these conditionals are indicative. Note, also, that appealing to conversational rules cannot explain why the conclusions appear odd, as in 1.7.3. For example, in the first, it is not the case that we already know which disjunct of the conclusion is true: both appear to be false.

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[contents]

 

 

 

 

1.9.3

[The Ineffectuality of Objecting that the Counter-Examples are Material Under an Additional Stipulation]

 

[We cannot object that in fact the above counter-examples really are valid, provided we stipulate that the English conditional is material in those cases. For, by making that stipulation, we are admitting that naturally the English conditional is not material and is only artificially so. But the whole point of these objections is to show that the English conditional is naturally material.]

 

[I do not entirely grasp the next point, but is seems to be the following. There is another objection, which is that someone might say that normally the above counter-examples will seem invalid, but they will have to be valid when we understand the English conditional as material. I am not sure why one would argue that, because it establishes the fact that it can only be valid by an artificial stipulation rather than by an analysis of how natural language works. That might be Priest’s point, because he says that by making this claim, we are admitting that it is not naturally the case that the English conditional is material.]

It might be pointed out that the above arguments are valid if ‘if’ is understood as ⊃. However, this just concedes the point: ‘if’ in English is not understood as ⊃.

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[contents]

 

 

 

 

 

From:

 

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

 

 

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18 Apr 2018

Priest (1.6) One, “The Aporia”, summary

 

by Corry Shores

 

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[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

 

1.6

The Aporia

 

 

 

Brief summary:

(1.6.1) None of our available options of explaining unity in terms of the factor that binds parts together into the whole (which is called the “gluon”) are viable. (1.6.2) We might say there are no gluons, thereby claiming that there are parts in the world but no wholes. This cannot be so, because in thought there are unified mental entities. (1.6.3) We also cannot argue that there are no gluons on account of the world being one whole without containing any parts. For, even in this case we do in practical life think of wholes with parts, meaning that the mental entities involved in those conceptions are wholes with parts and thus have gluons. (1.6.4) Gluons can be referred to, so we cannot claim they are not objects. (1.6.5) We cannot say that the gluon is an object, because that takes unity for granted rather than explain it. (1.6.6) Our best option for understanding gluons is with the dialetheic claim that they both are and are not objects.

 

 

 

 

 


 

Contents:

 

1.6.1

[The Lack of Available, Viable Options to Explain Gluonic Unification]

 

1.6.2

[The Inadequacy of Rejecting the Existence of Gluons and Positing a Wholeless World]

 

1.6.3

[The Inadequacy of Rejecting the Existence of Gluons and Positing a Partless World]

 

1.6.4

[Gluons as Necessarily Objects on Account of Being Referable]

 

1.6.5

[Gluon as not Being an Object]

 

1.6.6

[Gluons as Dialetheias]

 

Bibliography

 

 

 

 

 

Summary


 

1.6.1

[The Lack of Available, Viable Options to Explain Gluonic Unification]

 

[None of our available options of explaining unity in terms of the factor that binds parts together into the whole (which is called the “gluon”) are viable.]

 

[In section 1.5, we discussed the problem of explaining unity. We noted all of our available options, and we saw how none are satisfactory. The unifying factor we want to give an account of is called the gluon (see section 1.3.4). Priest then notes a point he makes in section 1.3.4 and section 1.3.5, namely, that gluons are contradictory objects, because in order to be what they are, they must both be entities while also not being entities. They are entities in that we are talking about them (and anything you can talk about is an entity, for what else would it be?), and yet they are not entities on account of the Bradly regress (see section 1.4, especially 1.4.2), and so by thinking of a gluon simply as an entity makes it a part of the problem of explaining unity rather than a part of the solution. We are thus at an impasse or aporia. Priest says we have three options:

{1} We can say that there are no gluons.

(I suppose for this option, we are claiming that there is no binding, unifying factor in things. But then we are giving up the question altogether it seems. And that is not what we want, because the question is of great philosophical importance.)

{2} We can reject the claim that a gluon is an object.

(Here we would solve the problem of the Bradley Regress, but then we would seem to be unable to talk about it, which is very unhelpful for trying to account for it.)

{3} We can reject the claim that it is not an object.

(This will allow us to talk about it, but then we encounter the Bradley Regress, which prevents us from accounting for unity or the gluon.) (Note, Priest gives his own reasoning in the following sections.) Given that all these options are highly problematic, we seem not to have any good way to proceed.]

We have, then, an aporia.Whatever it is that constitutes the unity of an entity must itself both be and not be an entity. It is an entity since we are talking about it; it is not an entity since it is then part of the problem of a unity, not its solution. ‘Aporia’ is often glossed as ‘puzzle’ or ‘uncertainly’, but it literally means something like ‘impasse’. An aporia is a source of puzzlement and uncertainty precisely because it seems to leave no way to go forward. In the present case, if we wish to go back, there are only three options:

1. We can say that there are no gluons.

2. We can reject the claim that a gluon is an object.

3. We can reject the claim that it is not an object.

Prospects look bleak.

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[contents]

 

 


 

1.6.2

[The Inadequacy of Rejecting the Existence of Gluons and Positing a Wholeless World]

 

[We might say there are no gluons, thereby claiming that there are parts in the world but no wholes. This cannot be so, because in thought there are unified mental entities.]

 

[Recall the first option from section 1.6.1 above.

{1} We can say that there are no gluons.

Because gluons are the binding factor that unifies parts into wholes, if we deny they exist, we also deny that there can be a difference between a unity that has parts and a simple plurality of those parts. (For, without this factor, there would be nothing to make a plurality of parts unified into a whole, and thus it would be no different from an unified plurality of parts.) One way to work around this could be to say that there are just parts but no wholes, and thus “the world is just a congeries of congeries” (14). But this cannot be so. For, we have unities in thought, which although being mental entities, still qualify as unities and thus their gluonic unification still needs to be accounted for.]

Consider the first case. If there are no gluons, then we are bereft of an explanation as to the difference between a unity with parts and the plurality of the parts, which there certainly is. We could avoid this by supposing that there are no unities: the world is just a congeries of congeries. All parts, no unities. But this does not seem to help either. If there are no unities, there certainly appear to be; that is, there are unities in thought. This means that the mind constitutes unities— as, perhaps, for Kant. But in this case, there are gluons.These are mental entities, but they fall foul of the aporia in the usual way.22

(14)

22. The view that there are no material wholes, only simples, is defended in Unger (1979). There are no tables: only atoms ‘arranged table-wise’ (as van Inwagen puts it (1990), p. 72ff). Sider (1993) points out that this commits the view to the (counterintuitive) necessity of the existence of physical simples (partless wholes). (Gluon theory is not so committed.) And Uzquiano (2004) argues that | attempts to paraphrase away talk of unities in the way suggested is problematic. In any case, the view hardly seems credible for abstract objects. A proposition is a single thing: one can believe it, express it. You can not do this to a plurality of meanings arranged proposition-wise, whatever that might be supposed to mean. (14-15)

[contents]

 

 


 

1.6.3

[The Inadequacy of Rejecting the Existence of Gluons and Positing a Partless World]

 

[We also cannot argue that there are no gluons on account of the world being one whole without containing any parts. For, even in this case we do in practical life think of wholes with parts, meaning that the mental entities involved in those conceptions are wholes with parts and thus have gluons.]

 

[Recall yet again the first option from section 1.6.1 above.

{1} We can say that there are no gluons.

Priest now notes another way this could be so. Suppose the world is one whole unity without any parts. It in this sense would also not have any gluons; for there would be no parts to be bound together into wholes. But this goes against common sense and practical life, where for example our car certainly has parts that would render the car inoperable if they were missing. But someone might say that the car is not really such a unity of parts (I am not sure what else it would be, I suppose we only have the one world, and the car is not some unity within it, but I am not sure), and we only mistakenly think that the car is a whole. But even in that case, we are admitting that we conceive of it as a whole containing parts, which means the mental entity of that conception would still involve gluons.]

At the other extreme, one might suppose that there are unities, but that they have no parts, and hence that there are no gluons. All unities, no parts. A very extreme form of this position is to the effect, not only that there are only unities, but there is only one of them. All else is appearance. The view is to be found in Parmenides and Bradley. Supposing that there are only unities with no parts is a desperate move. It flies in the face of common sense: if someone steals a wheel of my car then it is missing an essential part. And before one says that the car is not really a whole, but we only think of it in that way, recall that this means that there is a unity in intention, and we are back with intentional gluons.

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[contents]

 

 

 


 

1.6.4

[Gluons as Necessarily Objects on Account of Being Referable]

 

[Gluons can be referred to, so we cannot claim they are not objects.]

 

[Now recall the second option from section 1.6.1 above.

{2} We can reject the claim that a gluon is an object.
But “we can refer to it, quantify over it, talk about it.” Priest says that there is little other sense to what would qualify something as being an object.]

In the second case, we must insist that the gluon is simply not an object. But this seems even more desperate: we can refer to it, quantify over it, talk about it. If this does not make something an object, I am at a loss to know what could. Anything we can think about is an object, a unity, a single thing (whether or not it exists). There seems little scope here.

(15)

[contents]

 

 


 

1.6.5

[Gluon as not Being an Object]

 

[We cannot say that the gluon is an object, because that takes unity for granted rather than explain it.]

 

[Now finally recall the third option from section 1.6.1 above.

{3} We can reject the claim that it is not an object.

We have seen from section 1.4 (see especially 1.4.2) that this leads to the Bradley Regress. (But Priest’s point this time seems different. I do not quite grasp it, so consult the quotation below. He seems to be saying the following. Let us suppose the gluon is an object. But what we are trying to explain are unified objects. He next says that the only way an object can constitute the unity of another object is by taking unity for granted. That is the part I do not follow so well. Are we talking about taking the unity of the gluon for granted, so that it does not lead to a regress? Or are we taking the whole thing’s unity for granted? At any rate, this somehow involves simply thinking that the unity of things are obvious or unquestionable. But examples of unities very often involve combinations of parts, and we do not explain their compositional bonding by claiming that gluons are objects ((or by otherwise taking unity for granted)). But I am guessing here.]

Finally, in the third case, we may suppose that the gluon is simply an object. But we have seen that this just leaves us bereft of an explanation of the unity of an entity. How could we even have had the impression that any object could constitute the unity of another bunch of objects? Only because of taking the unity for granted. Thus, we write ‘Socrates is a person’ and the rest is obvious. But putting ‘Socrates’ and ‘is a person’ next to each other does not do the job; it just produces a plurality of two things. When we think of the two as cooperating, the magic has already occurred.

(15)

[contents]

 

 


 

1.6.6

[Gluons as Dialetheias]

 

[Our best option for understanding gluons is with the dialetheic claim that they both are and are not objects.]

 

So we cannot use any of the available options, and we should instead say that gluons both are and are not objects. What prevents us is the Principle of Non-Contradiction. But since it is not a well-founded logical principle and since also it is best not obeyed in all cases, we will take the dialetheic position that Gluons have contradictory properties (again, they both are and are not objects.)

If we cannot go back, then we must go forward. What stands in the way? Evidently, the Principle of Non-Contradiction. If we accept that gluons both are and are not objects, then some contradictions are true. Whilst it must be agreed that horror contradictionis is orthodox in Western philosophy, at least since Aristotle’s canonical –but fundamentally flawed – defence, the friends of consistency have done little as yet to establish that there is anything rational in this.23 So let us go forward. Gluons are dialetheic: they have contradictory properties. Of course, if this were all there were to matters, the situation would not be particularly interesting. Going on means crossing the bridge of inconsistency;24 and what is important is what lies on the other side.

(15)

23. See Priest (2006).

24. Not that there are no other good reasons to do so. See Priest (1987) and (1995a).

[contents]

 

 

 

 

 

 


 

Bibliography:

 

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:

 

Priest, G. (1987), In Contradiction, Dordrecht: Martinus Nijhoff; second (extended) edn., Oxford: Oxford University Press, 2006.

 

Priest, G. (1995a), Beyond the Limits of Thought, Cambridge: Cambridge University Press; second (extended) edn., Oxford: Oxford University Press, 2002.

 

Priest, G. (2006), Doubt Truth to be a Liar, Oxford: Oxford University Press.

 

Sider, T. (1993a), ‘Parthood’, Philosophical Review 116: 51–91.

 

Sider, T. (1993b), ‘Van Inwagen and the Possibility of Gunk’, Analysis 53: 285–9.

 

Unger, P. (1979), ‘There are no Ordinary Things’, Synthese 41: 117--54.

 

Uzquiano, G. (2004), ‘Plurals and Simples’, Monist 87: 429–51.

 

 

 

 


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