Priest (7.5) An Introduction to Non-Classical Logic, ‘Many-valued Logics and 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 II:

Quantification and Identity

 

7

Many-valued Logics

 

 

7.5

Many-valued Logics and Conditionals

 

 

 

 

Brief summary:

(7.5.1) We will now examine the conditional operator in many-valued logics. (7.5.2) We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table. (In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.)

 

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(7.5.3) Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional. (7.5.4) We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold. (7.5.5) Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated. (7.5.6) But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)

 

 

 

 

 

 

Contents

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

7.5.3

[Evaluating the Many-Valued Logics]

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

7.5.6

[Counter-Examples to This Claim]

 

 

 

 

 

 

Summary

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

[We will now examine the conditional operator in many-valued logics.]

 

[(ditto)]

Further details of the properties of ∧, ∨ and ¬ in the logics we have just met will emerge in the next chapter. For the present, let us concentrate on the conditional.

(125)

[contents]

 

 

 

 

 

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

[We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table.]

 

[The issue of problematic inferences using the conditional is something we have explored quite a bit in previous sections. See sections 1.6, 1.7, 1.8, 1.9, 1.10, 4.5, 4.6, 4.8, 4.9, and 5.2. Priest now summarizes many of these problematic inferences that use the conditional in a table where we can see whether or not they are are valid in K₃, Ł₃, LP₃, and RM₃. Recall that part of this evaluation involves the designated value, which is the truth-preserving value (like 1 in classical logic and i and 1 in LP) and which is not the same in all of these systems. In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.]

In past chapters, we have met a number of problematic inferences concerning conditionals. The following table summarises whether or not they hold in the various logics we have looked at. (A tick means yes; a cross means no.)

|

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(1) and (2) we met in 1.7, and (3)–(5) we met in 1.9, all in connection with the material conditional. (6)–(8) we met in 5.2, in connection with conditional logics. (9) and (10) we met in 4.6, in connection with the strict conditional. The checking of the details is left as a (quite lengthy) exercise. For K₃, a generally good strategy is to start by assuming that the premises take the value 1 (the only designated value), and recall that, in K₃, if a conditional takes the value 1, then either its antecedent takes the value 0 or the consequent takes the value 1. For L₃, it is similar, except that a conditional with value 1 may also have antecedent and consequent with value i. For LP, a generally good strategy is to start by assuming that the conclusion takes the value 0 (the only undesignated value), and recall that, in LP, if a conditional takes the value 0, then the antecedent takes the value 1 and the consequent takes the value 0. For RM₃, it is similar, except that if a conditional has value 0, the antecedent and consequent may also take the values 1 and i, or i and 0, respectively. And recall that classical inputs (1 or 0) always give the classical outputs.

(125-126)

[contents]

 

 

 

 

 

 

7.5.3

[Evaluating the Many-Valued Logics]

 

[Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional.]

 

[If we look at the table, we can see that there are many check-marks, which means that these problematic inferences are commonly valid in the many-valued logics. If if we ignore conditionals with an enthymematic ceteris paribus clause (see section 5.2), all these many-valued logics still “suffer from some of the same problems as the material conditional” (126). Priest then writes: “K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4) ...” (126-127). I am not sure I get that, but maybe the ideas are the following. One of the problems of the strict conditional is the explosion of contradictions, where contradictions entail everything (see especially section 4.8). But, as we can see from row 10, this is not valid in K₃ and Ł₃. Let us consider the evaluation of ‘⊨ (p ∧ ¬p) ⊃ q’ in K₃ and also ‘p ∧ ¬p ⊨ q’ in K₃, because I am guessing that is the distinction here. Recall from section 7.3.2 that the evaluation for negation, conjunction, and the conditional are the following.

 

f¬
1	0
i	i
0	1

 

f∧	1	i	0
1	1	i	0
i	i	i	0
0	0	0	0

 

f⊃	1	i	0
1	1	i	0
i	1	i	i
0	1	1	1

(122)

 

And the designated value is 1. So we need to see if there is any instance in the evaluation where the value of the whole conditional is not 1. If there is such an instance, it is invalid, and if there is no such instance, it is valid.

 

p  q	(p	∧	¬p)	⊃	q
1  1	1	0	0	1	1
1  i	1	0	0	1	i
1  0	1	0	0	1	0
i  1	i	i	i	1	1
i  i	i	i	i	i	i
i  0	i	i	i	i	0
0  1	0	0	1	1	1
0  i	0	0	1	1	i
0  0	0	0	1	1	0

 

As we can see, ‘⊨ (p ∧ ¬p) ⊃ q’ is not valid in K₃, because when p is i and q is either i or 0, then the conditional is i. Now let us evaluate p ∧ ¬p ⊨ q’ in K₃. If there are any cases where the premises are 1 and the conclusion is not 1, then it is valid. But note that if there are no cases where the premises are 1 to begin with, then it will still be valid, although “vacuously valid” (see Priest’s Logic: A Very Short Introduction, section 2.)

 

p  q	(p	∧	¬p)	⊨	q
1  1	1	0	0		1
1  i	1	0	0		i
1  0	1	0	0		0
i  1	i	i	i		1
i  i	i	i	i		i
i  0	i	i	i		0
0  1	0	0	1		1
0  i	0	0	1		i
0  0	0	0	1		0

 

As we can see, there are no cases where the premises are 1, so it is vacuously valid and thus explosion holds in K₃, and it also holds in Ł₃. But, since in LP, i is a designated value, and since when p is i and q is 0 the premises are i but the conclusion 0, that means it is not valid in LP (see section 7.4.4). Yet Priest continues, “but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows.” I am not sure why this formula shows that modus ponens is invalid. I will need to come back to this. But for now I will note that under the exact same evaluation there is the same row as the counter-example for modus ponens, as we saw in section 7.4.5:

 

p  q	p	p ⊃ q	⊨ q
1  1	1	1	1
1  i	1	i	i
1  0	1	0	0
i  1	i	1	1
i  i	i	i	i
i  0	i	i	0
0  1	0	1	1
0  i	0	1	i
0  0	0	1	0

 

In all, the system that has the fewest valid problematic inferences using the conditional is RM₃.]

As can be seen from the number of ticks, the conditionals do not fare very well. If one’s concern is with the ordinary conditional, and not with conditionals with an enthymematic ceteris paribus clause, then one may ignore lines (6)–(8). But all the logics suffer from some of the same problems as the material conditional. K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4), but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows. (Modus ponens is valid for the other logics, as may easily be checked.) About the best of the bunch is RM₃.

(126-127)

[contents]

 

 

 

 

 

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

[We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold.]

[Priest will now explain why the conditional in finitely many-valued logics will probably be problematic. In this section we make the first step, so the full reasoning will not here be given. But we need to note a few important things here. First consider how disjunction works. If we can affirm either just A on the one hand or alternatively just B on the other hand, then we should be able to affirm A ∨ B, because we know one of the disjuncts would be true, which is enough for the whole disjunction to be true. Thus:

(i) if A (or B) is designated, so is A ∨ B

(127)

The next observation is that A ≡ A should be a logical truth, even in a many-valued logic. Why? The reasoning is a little tricky, but it seems to be the following. Suppose A is valued at i. Let us make that concrete with an example. We have a moving object, and for A we have, “the moving object is at point 1 at time 1.” Suppose we think that should be both true and false. Regardless, would we not still think that the following should at least also be both true and false: “If the moving object is at point 1 at time 1, then the moving object is at point 1 at time 1”? So, we are claiming that it is reasonable to say that ‘if A then A’ has designated value whenever A itself has one. Now, since A is A, then that implies the inversion of the ‘if A then A’ also holds, and so A iff A should also hold as well. Now, in this case, A will have to have the same value as itself, so both sides of the biconditional will always have the same value, and regardless, the whole biconditional will have a designated value. The next idea seems to be that we suppose we have B, but we assign it the same value as A. And then we say that the same thing should hold for the biconditional here, namely, that:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

I may have the reasoning wrong there, but this is my guess. Priest ends by saying that this holds in all the finitely many-valued logics we have seen, but for K₃, (ii) fails. ]

But there are quite general reasons as to why the conditional of any finitely many-valued logic is bound to be problematic. For a start, if disjunction is to behave in a natural way, the inference from A (or B) to A ∨ B must be valid. Hence, we must have:

(i) if A (or B) is designated, so is A ∨ B

Also, A ≡ A ought to be a logical truth. (Even if A is neither true nor false, for example, it would still seem to be the case that if A then A, and so, that A iff A.) Hence:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Note that both of these conditions hold for all the logics that we have looked at, with the exception of K₃, for which (ii) fails.

(127)

[contents]

 

 

 

 

 

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

[Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

 

[The argumentation continues to get tricky. I may have the reasoning wrong, so see the quotation below. I am guessing we are doing the following. We are dealing with finitely many-valued logics. So that means regardless of the number of values, they will have a total number of values, that we can use the variable n to signify. So first we will consider any n-valued logic that satisfies

(i) if A (or B) is designated, so is A ∨ B

and

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Now we will consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Recall from section 1.2 that a propositional parameter is something like a formula which in our texts above we are writing as A, B, or C. The point is that we have one more formula than the total number of possible truth-values for that system. So at least two formulas will need to have the same truth-value. Now, given what (ii) says, that means for these two same-valued formulas, their biconditional must have a designated value. And, since we now have at least one designated biconditional, that would make it that the disjunction of all biconditionals will have to be designated, by rule (i). So in sum, suppose we have a many-valued logic with one more formula than there are truth-values, that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

Now, take any n-valued logic that satisfies (i) and (ii), and consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Since there are only n truth values, in any interpretation, two of these must receive the same value. Hence, by (ii), for some j and k, p_j ≡ p_k must be designated. But then the disjunction of all biconditionals of this form must also be designated, by (i). Hence, this disjunction is logically valid.

[contents]

 

 

 

 

 

 

7.5.6

[Counter-Examples to This Claim]

 

[But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)]

 

[So as we saw in section 7.5.5, if we accept the following two intuitive notions:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Then we should conclude that for any finitely many-valued logical system with n truth-values and n+1 propositional parameters, that the disjunction of all its biconditionals will necessarily be true. Priest then gives a counter-example. “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’.” Priest says that any biconditional of these formulas would seem to be false, and thus the disjunction of all of them will not have any true one in it. I am not following this well, and I also do not see very how this shows us why the conditional for any finitely many-valued logic is bound to be problematic. Let me go through this as best I can. The idea originally was that in such a set of propositions, at least two would have to have the same value. So here, I would think that there would be two with the value false, and thus at least one biconditional that is true. Suppose we are using a three-valued logic with the values 0, 1, and i. And suppose John has 1 hair on his head. That means we have the following four propositions:

(1) John has 1 hair on his head.

(2) John has 2 hairs on his head.

(3) John has 3 hairs on his head.

(4) John has 4 hairs on his head.

Now, 3 and 4 are false, so their biconditional is true. So I do not see yet why “Any biconditional relating a pair of these would appear to be false.” But this is my failing. I am just not sure where I go wrong in the above explication. My best guess at the moment is that the problems come from the intuitive sense of these biconditional formulations: One of them will be, “John has 1 hair on his head if and only if John has 2 hairs on his head.” Now, as we know, this is senseless. John does in fact have 1 hair on his head, but this cannot be biconditional with him having 2 hairs on his head. By extension, even though technically “John has 3 hairs on his head if and only if John has 4 hairs on his head” is true, it is also for the same reason senseless. In other words, John’s not having some number of hairs on his head should not be biconditional on him having some other false number of hairs on his head. In other words, because every biconditional will biconditionally equate statements saying John has a different number of hairs on his head would seem on the level of its sense to necessarily always be false, even though technically they might evaluate as true whenever both sides of the biconditional are false. So on the level of sense, John’s having x number of hairs cannot be conditioned on him having x+1 number of hairs, and vice versa. That is my best guess at the moment. Still, even supposing this to be the case, I am not entirely sure why that would show there to be something problematic with conditionals in finitely many-valued logics. Is it because the biconditional is composed of conditionals, so if there is a paradox with the biconditionals there is a problem with conditionals in general? In other words, since the biconditionals imply that we will have  conditionals of the form “John has x number of hairs if John has x+1 number of hairs” where at least one biconditional combination of them will have to be true, even though we know that cannot be so in any case, given the intuitive sense of the constituent conditional sentences, that we will always have this problem with conditionals in finitely many-valued logics. Let me quote, as I am guessing very wildly here.]

But this seems entirely wrong. Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true.

[contents]

 

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

Dupréel (6.3.3) Essais pluralistes, ch.6: Théorie de la consolidation, sect 6.3.3, ‘Probabilité de la Consolidation’, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

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[The following is summary and not translation. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, so typos are present, including in the quotations. Please consult the original text to be sure about the contents. Also, I welcome corrections to my interpretations, because I am not especially good with French.]

 

 

 

Summary of

 

Eugène Dupréel

 

Essais pluralistes

 

Ch.6

Théorie de la consolidation.

Esquisse d’une théorie de la vie d’inspiration sociologique.

 

6.3

[Purpose/Finality in Sociology]

 

6.3.3

Probabilité de la Consolidation

 

 

 

 

Brief summary:

(6.3.3.1) Whether or not parts consolidate is often a matter of probability. (6.3.3.2) Whether or not two (spatial) coexistents come to be consolidated in the same solid or body is a matter of probability, which increases or decreases depending on whether the spatial interval between them increases or decreases. For example, if two flint pebbles are one centimeter apart in the soft binding material, their chances of consolidating together is much greater than for pebbles set a meter apart. (6.3.3.3) Time is also required for consolidation. In the case of the puddingstone, gravity had to hold the pebbles and sand in place for a very long time. (6.3.3.4) When parts are set up to be consolidated, there could be any of three sorts of conditions with respect to the probability of their co-consolidation: {1} there could be unfavorable conditions, like a torrent of water moving the two pebbles very far apart from each other; {2} there could be indifferent conditions, like a light breeze brushing against the pebbles without moving them; and {3} there could be favorable conditions, like a rain of sand that fixes the pebbles in their place. When the supporting force is weak, then many influences can destroy the parts’ ordering before they can consolidate. But if the supporting force is strong, like two nails being hammered near one another in an oak beam, then they will more likely hold their spatial relations despite disruptive influences. (6.3.3.5) This operation of consolidation can be seen as one of fixation or of crystallization. And it could be that this operation still fixes or crystallizes the parts even while modifying their relations a little in the process.

 

 

 

 

 

 

 

Contents

 

6.3.3.1

[Noting the Probability of Consolidation]

 

6.3.3.2

[The Interval Between Parts and Its Effect on the Probability of Consolidation]

 

6.3.3.3

[The Role of Time in Consolidation]

 

6.3.3.4

[Variables on Consolidation Probability]

 

6.3.3.5

[Crystallization or Fixation as Modifying the Parts’ Relation]

 

 

 

 

 

 

 

 

 

Summary

 

 

6.3.3.1

[Noting the Probability of Consolidation]

 

(p.160: “Probabilité de la Consolidation. – Il semble qu’on peut parler ...”)

 

[Whether or not parts consolidate is often a matter of probability.]

 

[The Probability of Consolidation: It seems that we can speak of the degree of probability of some given operation of consolidation. (Recall from section 6.3.2 what consolidation is. There are two phases in the formation of a physical entity that is made of parts and that sustains that composite composition over time. Firstly, the parts are arranged and held in place by an exterior influence, like gravity and soil holding flint pebbles in a particular arrangement within a still-soft binding material. Secondly, the parts’ arrangement becomes fixed such that it no longer depends on the exterior influence for the arrangement to hold over time, like the binding material around the flint pebbles hardening to from puddingstone (see especially section 6.3.2.4 for this example). Note that two things are transferred from the exterior influence to the interiority of the formed object: {1} the parts’ proper arrangement of mutual relations, and {2} the capacity to hold those relations intact over time, which is called solidity. Whenever there is such a transfer, we call it consolidation (see section 6.3.2.2). Such beings whose consolidation is a matter of physical parts placed into a spatial arrangement are called consolidations of coexistents (consolidés de coexistence) (see section 6.3.2.3).]

Probabilité de la Consolidation. – Il semble qu’on peut parler du degré de probabilité d’une opération de consolidation déterminée.

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6.3.3.2

[The Interval Between Parts and Its Effect on the Probability of Consolidation]

 

(p.160: “Si nos deux cailloux de silex ...”)

 

[Whether or not two (spatial) coexistents come to be consolidated in the same solid or body is a matter of probability, which increases or decreases depending on whether the spatial interval between them increases or decreases. For example, if two flint pebbles are one centimeter apart in the soft binding material, their chances of consolidating together is much greater than for pebbles set a meter apart.]

 

[(In rough form: If our two flint pebbles, instead of being deposited less than a centimeter from one another, had been set more than a meter apart, then the chances of them one day becoming  integrated parts of the same solid independent of the site where it was formed would have been much lower. The phenomenon, in fact, depends, in principle, on a favorable layout of the interval. The larger the interval, the greater the chance that agglutination will be lost in a multitude of opposing possibilities.)]

Si nos deux cailloux de silex, au lieu d’être déposés à moins d’un centimètre l’un de l’autre, l’avaient été à plus d’un mètre, les chances d’être un jour parties intégrantes d’un même solide indépendant du site où il s’est formé, auraient été beaucoup moins grandes. Le phénomène, en effet, dépend, au principal, d’un aménagement favorable de l’intervalle. Plus celui-ci est grand, plus les chances d’agglutination se perdent dans une multitude de possibilités contraires.

(160)

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6.3.3.3

[The Role of Time in Consolidation]

 

(p.160-161: “Il faut, en outre, du temps ...”)

 

[Time is also required for consolidation. In the case of the puddingstone, gravity had to hold the pebbles and sand in place for a very long time.]

 

[(In rough form: Also, it requires time for the decisive condition to present itself. In the case of this puddingstone, gravity alone had to hold the two pebbles and the sand in place probably for a span of many years.]

Il faut, en outre, du temps pour que la condition décisive se | présente. Dans le cas de ce poudingue, la seule pesanteur a dû maintenir en place les deux cailloux et le sable probablement durant des années.

(160-161)

[contents]

 

 

 

 

 

 

6.3.3.4

[Variables on Consolidation Probability]

 

(p.161: “Enfin, les chances augmentent ...”)

 

[When parts are set up to be consolidated, there could be any of three sorts of conditions with respect to the probability of their co-consolidation: {1} there could be unfavorable conditions, like a torrent of water moving the two pebbles very far apart from each other; {2} there could be indifferent conditions, like a light breeze brushing against the pebbles without moving them; and {3} there could be favorable conditions, like a rain of sand that fixes the pebbles in their place. When the supporting force is weak, then many influences can destroy the parts’ ordering before they can consolidate. But if the supporting force is strong, like two nails being hammered near one another in an oak beam, then they will more likely hold their spatial relations despite disruptive influences.]

 

[(In rough form: Lastly, the chances (of consolidation) increase with the force of the exterior support. With our two pebbles being placed close to one another, there are three kinds of situations that can take place in the spatial interval between them. {1} There can be conditions that are unfavorable for maintaining this established order; for example, a torrent of water might disperse the stones (away from each other). {2} There can be indifferent conditions, as for instance a light breeze that brushes against the pebbles without displacing them. And {3} there can be favorable circumstances, as for example a rain of sand that fixes them in their place. If the supporting force is quite weak or null, then, all things being equal, there will be many more probable circumstances that will result in the destruction of the supported relations than will there be favorable interactions. So if two pebbles are held in place by a very strong force, like two nail heads hammered one beside the other into an oak beam, then this (supporting) force will cancel out many dangers of destruction, leaving the field free for favorable conditions for consolidation.)]

Enfin, les chances augmentent en même temps que la force de sustentation extérieure. Nos deux cailloux étant déposés l’un près de l’autre, ce qu’il peut se passer dans l’intervalle qui les sépare se ramène à trois espèces de faits : 1° des faits défavorables au maintien de cet ordre établi, par exemple un torrent d’eau qui disperse les cailloux ; 2° des faits indifférents, tels qu’une légère brise qui les caresse sans les déplacer ; 3° des faits favorables, tels qu’une pluie de sable qui les fixe à leur place. Si la force de sustentation est très faible ou nulle, toutes choses égales d’ailleurs les conjonctures probables dont résultera la destruction du rapport soutenu seront bien plus nombreuses que les rencontres favorrables. Que si les deux cailloux sont maintenus en place par une force très grande, comme le seraient deux têtes de clous enfoncés, l’un près de l’autre, dans une poutre de chêne, cette force annulera nombre de dangers de destruction, laissant le champ libre aux conditions favorables à la consolidation (fig. 3).

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6.3.3.5

[Crystallization or Fixation as Modifying the Parts’ Relation]

 

(p.161: “II est bon de remarquer que si notre opération ...”)

 

[This operation of consolidation can be seen as one of fixation or of crystallization. And it could be that this operation still fixes or crystallizes the parts even while modifying their relations a little in the process.]

 

[(In rough form: It is worth noting that if our operation is the fixation or the crystallization  of a spatial relation between terms, it could be that the operation modifies the spatial relation at the same time that it ensures that the spatial relation endures through time. Slow work can occur in the puddingstone that could cause the two pebbles to move a little closer or a little farther apart from one another. It could be that a particular alteration of the relationship be in effect or even be a condition of its consolidation.)]

II est bon de remarquer que si notre opération est la fixation ou la cristallisation d’un rapport spatial entre des termes, il peut arriver qu’elle le modifie plus ou moins en même temps qu’elle en assure la durée. Un travail lent peut se produire dans le poudingue, qui aura pour effet de rapprocher ou d’éloigner quelque peu l’un de l’autre les deux galets. Il se peut qu’une certaine altération du rapport soit un effet, ou même une condition de sa consolidation.

(161)

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Dupréel, Eugène. (1949). Essais pluralistes. Paris: Presses universitaires de France.

.

.

Dupréel (6.3.2) Essais pluralistes, ch.6: Théorie de la consolidation, sect 6.3.2, ‘Les consolidés de coexistence’, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Eugène Dupréel, entry directory]

[Dupréel, Essais pluralistes, entry directory]

 

[The following is summary and not translation. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, so typos are present, including in the quotations. Please consult the original text to be sure about the contents. Also, I welcome corrections to my interpretations, because I am not especially good with French.]

 

[May I please thank the sources of the puddingstone images:

East Herts Geology Club:

http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/

]

 

 

Summary of

 

Eugène Dupréel

 

Essais pluralistes

 

Ch.6

Théorie de la consolidation.

Esquisse d’une théorie de la vie d’inspiration sociologique.

 

6.3

[Purpose/Finality in Sociology]

 

6.3.2

Les consolidés de coexistence

 

 

 

 

Brief summary:

(6.3.2.1) There are two stages in the manufacture of an object: firstly, the parts are manually given the arrangement they will finally hold on their own, and secondly, these structural relations between the parts are then fixed so that the object stands by itself, without the laborer’s interference. (6.3.2.2) We see these two phases of object construction in the molding process: {1} first the mold places the molding material’s parts together into a certain arrangement and holds them there. {2} Next, the material hardens into that form and keeps it all on its own. Note that two things are transferred from the mold to molded material: {1} the parts’ proper arrangement of mutual relations, and {2} the capacity to hold those relations intact over time, which is called solidity. Whenever there is such a transfer, we call it consolidation. (6.3.2.3) In manufactured things, the ordering of the parts is a spatial one. We call such things consolidations of coexistents (consolidés de coexistence). (6.3.2.4) This process of consolidation that we saw in human industry can also be found in natural processes, as for example in the formation of puddingstone. Here pieces of flint are fixed in place within binding materials by the soil and gravity. As the binding material solidifies, a solid rock is formed which no longer relies on the exterior supporting factors to maintain the compositional arrangement of the pebbles in the hardened binding cement. This is a natural example of consolidated coexistents. (6.3.2.5) Consolidations of coexistents are quite common in nature, as  all bodies with connected parts – be they solids or things with more loosely bound parts – are consolidations of coexistents. They are all formed by this two-step process where the exterior order gives arrangement and support to the parts until they solidify. (6.3.2.6) The world of our sensible perception is a totality of consolidations of coexistents.

 

 

 

 

 

 

 

Contents

 

6.3.2.1

[The Two Phases of Object Manufacture: The Arrangement of the Parts and the Fixing of the Arrangement]

 

6.3.2.2

[Molding as a Great Example of the Two-Phased Process. Solidity as Consistency. Consolidation as Exterior-to-Interior Structuration-Support Transfer]

 

6.3.2.3

[Consolidations of Coexistents]

 

6.3.2.4

[Natural Consolidated Coexistents: Puddingstone]

 

6.3.2.5

[The Prevalence of Consolidations of Coexistents in Nature]

 

6.3.2.6

[Our World of Sensible Perception as Being Composed of Consolidations of Coexistents]

 

 

 

 

 

 

 

Summary

 

 

6.3.2.1

[The Two Phases of Object Manufacture: The Arrangement of the Parts and the Fixing of the Arrangement]

 

(p.158: “Les Consolidés de coexistence. – Dans toute fabrication ...”)

 

[There are two stages in the manufacture of an object: firstly, the parts are manually given the arrangement they will finally hold on their own, and secondly, these structural relations between the parts are then fixed so that the object stands by itself, without the laborer’s interference.]

 

[(In rough form: The Consolidateds/Consolidations of Coexistence. Generally speaking, in any manufacturing process, we may distinguish two well-characterized successive states: In the first state, the parts of the object under construction are collected and arranged in the order that they should continue retaining. But at this phase of the labor, this order is only maintained by external and temporary means. It is only in a second and final state that, through an internal arrangement, the parts all on their own will hold together the parts’ positional relations in the completed object. For example, if it is a matter of making a crate, for a few moments, it is the hands of the worker that hold the boards upon each other, which she will fix together with nails. When they are hammered in, the crate “stands on its own”: it went from the first to the second of the two states whose succession we have just mentioned.) (Main ideas in refined form: In a manufacturing process, there are two stages of the object’s production. In the first stage, the object’s parts are arranged into the order they will finally hold. But at this point, the establishment and the maintenance of that order is made by the laborer. In the second stage of production, those arrangements are given a self-standing, fixed form, so that the object’s arrangement holds on its own. Consider for illustration the production of a wooden crate. In the first stage, the laborer’s hand sets up the proper arrangement of the crate’s parts. (Were the laborer to remove the influence of her hands, the crate would collapse). In the second stage, the crate’s boards are hammered together so that it can hold together even without the support of the laborer’s hands.)]

Les Consolidés de coexistence. – Dans toute fabrication, en général, on peut distinguer deux états successifs bien caractérisés : Dans un premier état, les parties de l’objet à construire sont rassemblées et mises dans l’ordre où elles devront demeurer. Mais à ce moment du travail cet ordre ne se maintient que par des moyens extérieurs et provisoires. Ce n’est qu’à un état second et définitif que, par un aménagement intérieur, les parties garderont d’elles-mêmes les rapports de position que comporte l’objet achevé. S’agit-il de faire une caisse, pendant quelques instants, ce sont les mains de l’ouvrier qui retiennent l’une contre l’autre les planches qu‘il va réunir par des clous. Ceux-ci étant enfoncés, la caisse « tient toute seule » : elle est passée du premier au second des deux états dont nous venons de rappeler la succession.

(158)

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6.3.2.2

[Molding as a Great Example of the Two-Phased Process. Solidity as Consistency. Consolidation as Exterior-to-Interior Structuration-Support Transfer]

 

(p.158-159: “Cela est encore plus apparent ...”)

 

[We see these two phases of object construction in the molding process: {1} first the mold places the molding material’s parts together into a certain arrangement and holds them there. {2} Next, the material hardens into that form and keeps it all on its own. Note that two things are transferred from the mold to molded material: {1} the parts’ proper arrangement of mutual relations, and {2} the capacity to hold those relations intact over time, which is called solidity. Whenever there is such a transfer, we call it consolidation.]

 

[(In rough form: This is even more apparent in the process of molding; the duality of the operation’s times is marked by that of the mold itself and of the molded object. Before “the setting” of the cement, the parts of the object are already placed in the proper order, but the force that maintains this order is external to them, and it is the solidity of the mold. The mold can only be removed when it is no longer needed to provide its sustaining role, as the parts in the molding now stand on their own. The order of the parts of the molded object was first supported or determined by the order of the parts of the mold, or its form; the accomplished operation consists in a consolidation of this order, which was initially precarious and inconsistent. Something has been transported from the mold to the molded object, namely, solidity, (which may be understood as a property shared by a number of terms (here, the parts of the object) by which they maintain themselves in a certain mutual relationship and keep this ordering). We will call consolidation any operation where we discern a transporting (of constituent structural relations) of this kind, that is to say, where an order, maintained initially by its dependence on an external order, comes to be supported by an internal capacity, such that the sustaining role of the external order, having become superfluous, can be eliminated). (Main ideas in refined form: This two-step process of first arranging the parts of an object under construction by external means and secondly for that arrangement to reify is seen quite clearly in the molding process. Here, the structural relations that the object’s parts will finally take-on are built into the mold’s form, and the mold itself, under this form, is the external factor that initially gives the molded material’s parts its proper arrangement and continued support in that formation. We then see the second stage when the material hardens, and the mold’s supporting ability becomes superfluous as the material now holds its internal structure without need of additional supporting aid. This process of the relations of the object’s parts becoming self-standing is called consolidation. This happens when solidity is transferred from the mold to the molded object (or more generally from the formational process or structure to the formed object.) Solidity is the property of the parts by which they can  maintain their mutual relations and conserve this ordering over time. Thus, consolidation is the operation by which there is a transfer both of constituent, structural relations and as well the capacity to maintain them. This process starts from an external influence that the constructed object initially depends upon; it then moves into the object itself, thereby endowing it with the internal capacity to maintain its constitution without the need for external support, on account of it obtaining its own solidity.) (Commentary: Here, solidity seems to be similar to what Dupréel calls “consistency” in “La consistance et la probabilité constructive” (see especially section 1.2). And “consolidation” seems to be similar to the amalgamation of similars and their formation of solids (see section 1.4).)]

Cela est encore plus apparent dans l’opération du moulage ; la dualité des temps de l’opération y apparaît marquée par celle du moule et de l’objet moulé. Avant « la prise » du ciment, les parties de l’objet sont déjà placées dans l’ordre qui convient, mais la force qui maintient cet ordre leur est extérieure, c’est la solidité du moule. Celui-ci ne peut être ôté que lorsque son rôle sustentateur est devenu inutile, les parties du moulage se tenant désormais d’elles-mêmes. L’ordre des parties de l’objet moulé était d’abord soutenu ou déterminé par l’ordre des parties du moule, ou sa forme ; l’opération accomplie consiste dans une consolidation de cet ordre, d’abord précaire et inconsistant. Quelque chose s’est transporté du moule vers l’objet moulé, c’est la solidité, ou cette | propriété pour un certain nombre de termes (ici, les parties de l’objet), de se maintenir dans un certain rapport mutuel, de conserver leur ordre. Nous appellerons consolidation toute opération où l’on discerne un transport de cette sorte, où un ordre, maintenu d’abord par sa dépendance à l’égard d’un ordre extérieur, arrive à se soutenir par une capacité interne, de telle sorte que le rôle sustentateur de l’ordre extérieur, devenu superflu, peut s’abolir.

(158-159)

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6.3.2.3

[Consolidations of Coexistents]

 

(p.159: “Dans le cas du moulage ... ”)

 

[In manufactured things, the ordering of the parts is a spatial one. We call such things consolidations of coexistents (consolidés de coexistence).]

 

[(In rough form: In the case of molding, and in general in all manufactured materials, the consolidated order is a spatial order; it is the relation of two or more extended parts that exist simultaneously or that endure together. We call such objects consolidations of coexistents (consolidés de coexistence). All our manufactured materials are consolidations of coexistents, from a pin to a bible or even a railway network.) (Note: consolidations of coexistents is not a very literal translation, but I do not know how better to render a natural translation. I invite your suggestions.)]

Dans le cas du moulage, et en général dans tout fabricat matériel, l’ordre consolidé est un ordre spatial, c’est le rapport de position de deux ou de plusieurs parties étendues, existant simultanément ou durant ensemble : Nous dirons de tels objets que ce sont des consolidés de coexistence. Tous nos fabricats matériels sont des consolidés de coexistence, depuis une épingle jusqu’a une bible ou un réseau de chemins de fer.

(159)

 

[contents]

 

 

 

 

 

 

6.3.2.4

[Natural Consolidated Coexistents: Puddingstone]

 

(p.159: “Nous venons de dégager cette notion ...”)

 

[This process of consolidation that we saw in human industry can also be found in natural processes, as for example in the formation of puddingstone. Here pieces of flint are fixed in place within binding materials by the soil and gravity. As the binding material solidifies, a solid rock is formed which no longer relies on the exterior supporting factors to maintain the compositional arrangement of the pebbles in the hardened binding cement. This is a natural example of consolidated coexistents.]

 

[(In rough form: We have just formulated this notion by considering human industry, which is an eminently finalistic activity. Now, let us note that consolidated coexistents are as much a fact of nature as they are a fact of humans. Consider a piece of puddingstone that  contains two flint pebbles embedded in ferruginous cement. (Here are some images of puddingstone, from the East Herts Geology Club website:

(These beautiful images come from the East Herts Geology Club: http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/)

There is no doubt about the way this composite and solid body was formed. The two pebbles, rolled by the water, stopped, and then landed flat in each other’s neighborhood. Sand deposited around them and filled the gap between them; then, with the help of moisture, the sand hardened, making it all one same solid. Before this agglomeration, the order constituting the two stones considered as terms was maintained by the underlying soil combined with the attraction of the Earth (gravity) (fig. 1).

This supported ordering was therefore exterior to what would constitute our conglomerate. The change in the consistency of the sand made this support – which was once exterior – an internal support, if not to the two pebbles, at least to the object they constitute with the cement that binds them. The soil and gravity are displaced from their sustaining role; I was able to take away this piece of stone, and I can turn it in a hundred ways without altering it; it is a consolidation of coexistents (fig. 2).

)]

Nous venons de dégager cette notion en considérant l’industrie humaine, c’est-à-dire une activité éminemment finaliste. Constatons maintenant que des consolidés de coexistence sont aussi bien le fait de la seule nature que le fait des hommes. Voici un morceau de poudingue. Il contient deux cailloux de silex enrobés dans un ciment ferrugineux. La manière dont ce corps composite et solide s’est formé ne fait pas de doute. Les deux cailloux, roulés par les eaux, se sont arrêtés, posés à plat, dans le voisinage l’un de l’autre. Du sable s’est déposé autour d’eux et a comblé l’intervalle qui les séparait ; puis, l’humidité aidant, ce sable s’est durci, faisant du tout un même solide. Avant cette agglomération, l’ordre que constituaient les deux cailloux considérés comme termes, était maintenu par le sol sous-jacent combiné avec l’attraction de la Terre (fig. 1).

Cet ordre de sustentation était donc à l’extérieur de ce qui allait constituer notre conglomérat. Le changement de consistance du sable a fait que cette sustentation, d’extérieure qu‘elle était, est devenue intérieure, sinon aux deux cailloux, du moins à l’objet qu’ils constituent avec le ciment qui les lie. Le sol et la pesanteur sont évincés de leur rôle sustentateur ; j’ai pu enlever ce morceau de pierre et je peux le tourner de cent façons sans l’altérer ; c’est un consolidé de coexistence (fig. 2).

(159)

(160)

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6.3.2.5

[The Prevalence of Consolidations of Coexistents in Nature]

 

(p.160: “Loin que ce processus soit rare ou exceptionnel ...”)

 

[Consolidations of coexistents are quite common in nature, as all bodies with connected parts – be they solids or things with more loosely bound parts – are consolidations of coexistents. They are all formed by this two-step process where the exterior order gives arrangement and support to the parts until they solidify.]

 

[(In rough form: Far from this process being rare or exceptional, there is nothing more common in nature than such consolidations of coexistents: they are everywhere. All solid bodies, all consistent beings that are formed from parts that are fused together or that are simply attached to one another, have come into existence by means of the operation we have been discussing. They all share the same history/story. A time always passes, with a duration that is sometimes quite short but often very long, during which the parts are only maintained in the relations that constitute the whole only by an exterior order of support which is followed by it obtaining its own consistency when the thing is supported by itself and from within. There are cases where the order of primitive and external support does not disappear even though it became superfluous. This would have been the case for our puddingstone had it remained, with the thousands of similar fragments, “in situ”; the sand would have maintained the two pebbles in the same position where gravity sufficed to hold them. Sometimes, on the contrary, the object at this point has become so detached from the circumstances of its formation that it is no longer possible to reconstitute it. Nevertheless, in this case, like in any other, something subsists in the object which we cannot explain by the nature of its parts, something irreducible to their properties, something which is compatible, on the contrary, with other elements, and which comes from this exterior order that is currently being eliminated.) (Main ideas in refined form: The consolidations of coexistents is quite common in nature, as it is what forms all groupings where parts come together, like in solid bodies where parts are fused or like in more loosely constituted bodies. In all cases, they follow the same course of formation: first their parts and their arrangements are supported by an external order, and following that they obtain their own consistency when they find structural support from within. Sometimes the external support remains even if it becomes superfluous to the structural supporting of the parts. For example, suppose the puddingstone remained in the place of its formation. The gravity would still be acting on the pebbles in the same supporting way, even though the hardened binding is now sufficient to keep the pebbles in their places. (I cannot grasp the following ideas well, so please see the quotation below.) Sometimes, however, the object becomes so detached from the circumstances of its formation that it can no longer be reconstituted. (I am guessing that for instances the binding of the puddingstone does not harden. But I have no clue what the idea is here.) But even in this case, there is something about the parts which is not intrinsic to them but rather is the result of external relations coming from the exterior supporting order. (So perhaps, to continue guessing, even if the binding material of the puddingstone does not harden, the arrangement of the pebbles has its ordering on account of the exterior factors like the gravity and soil structure below it.)]

Loin que ce processus soit rare ou exceptionnel, il n’est rien de plus commun dans la nature que de tels consolidés de coexistence : ils sont partout. Tous les corps solides, tous les êtres consistants formés de parties soudées ou seulement rattachées les unes aux autres, sont venus à l’existence en passant par notre opération. Ils ont tous la même histoire. Un temps, parfois très court, souvent très long, s’est toujours passé pendant lequel les parties n’étaient maintenues dans le rapport qui constitue le tout, que par un ordre de sustentation extérieur, puis est venue la consistance propre, la chose s’est soutenue d’elle-même et par le dedans. Il y a des cas où l’ordre de sustentation primitif et extérieur ne disparaît pas, quoique devenu superflu. Tel aurait été le cas pour notre morceau de poudingue s’il était demeuré, avec des milliers de fragments analogues, « in situ » ; le sable aurait maintenu les deux cailloux dans la même position où la pesanteur suffisait à les conserver. Parfois, au contraire, l’objet s’est à ce point détaché des circonstances de son élaboration qu’il n’est plus possible de les reconstituer. Il n’empêche que, dans ce cas comme dans tout autre, quelque chose subsiste, dans l’objet, qu’on ne saurait expliquer par la nature de ses parties, quelque chose d’irréductible aux propriétés de celles-ci, de compatible, au contraire, avec d’autres éléments, et qui vient de cet ordre extérieur, actuellement aboli.

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6.3.2.6

[Our World of Sensible Perception as Being Composed of Consolidations of Coexistents]

 

(p.160: “Le monde de notre perception sensible …”)

 

[The world of our sensible perception is a totality of consolidations of coexistents.]

 

[(ditto)]

Le monde de notre perception sensible est un ensemble de consolidés de coexistence.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dupréel, Eugène. (1949). Essais pluralistes. Paris: Presses universitaires de France.

 

 

Image Source:

East Herts Geology Club. “Puddingstone Use”.

http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/

Thank you very much for the beautiful pictures.

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