29 Jun 2018

Dupréel (ED) La consistance et la probabilité constructive, entry directory

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Eugène Dupréel, entry directory]

 

 

 

Entry Directory for

 

Eugène Dupréel

 

La consistance et la probabilité constructive

 

Part 1

“La consistance”

 

1.1

Les contraires

 

1.2

Les consistance des êtres

 

1.3

La similitude

 

1.4

L’amalgamation

 

1.5

Hiérarchie des êtres selon la consistance

 

1.6

Hiérarchie des êtres spatio-temporels

 

1.7

Hiérarchie des notions

 

 

 

 

 

 

 

 

 

 

Dupréel, Eugène. (1961). La consistance et la probabilité constructive. (Classe des lettres et des sciences morales et politiques 55, no.2). Brussels: Académie Royale de Belgique.

PDF at:

http://www.academieroyale.be/fr/les-publications-memoires-detail/oeuvres-2/la-consistance-et-la-probabilite-constructive/.\

 

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Eugène Dupréel, entry directory

 

by Corry Shores

 

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

 

[Central Entry Directory]

 

 

 

Entry Directory for

 

Eugène Dupréel

 

eugene-dupreel

(Source: SBP)

 

 

 

 

La consistance et la probabilité constructive

 

Dupréel. La consistance et la probabilité constructive, entry directory

 

 

 

 

 

 

Image Source:

Société Belge de Philosophie (SBP)

http://www.sobelphi.be/histo.htm

 

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27 Jun 2018

Priest (9.7) Introduction to Non-Classical Logic, ‘Impossible Worlds and Relevant Logic,’ summary

 

by Corry Shores

 

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[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

 

9.

Logics with Gaps, Gluts and Worlds

 

9.7

Impossible Worlds and Relevant Logic

 

 

 

 

Brief summary:

(9.7.1) We will now discuss philosophical matters regarding K4 , N4 , K , and N. (9.7.2) We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different. (9.7.3) Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds. (9.7.4) We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.” (9.7.5) Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling. (9.7.6) In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to. (9.7.7) Logically impossible worlds can also in fact be ones where laws of logic indeed are broken. (9.7.8) Relevant propositional logics are ones where whenever “A B is logically valid, A and B have a propositional parameter in common” (172). (9.7.9) But N4 is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds. (9.7.10) In a similar way, N is also a relevant logic. (9.7.11) Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters. (9.7.12) There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173). (9.7.13) Relevance in our systems here however is not conditions added on top of classical validity. (9.7.14) If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.

 

 

 

 

Contents

 

9.7.1

[Moving to a Discussion on K4 , N4 , K , and N]

 

9.7.2

[Logically Impossible Worlds]

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

9.7.8

[Relevant Logics and the Conditional]

 

9.7.9

[N4 as Relevant]

 

9.7.10

[N as Relevant]

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

9.7.12

[Filter Logics]

 

9.7.13

[More Than Classical Relevance]

 

9.7.14

[Preserving Conventional Truth in This System]

 

 

 

 

 

 

 

 

Summary

 

9.7.1

[Moving to a Discussion on K4 , N4 , K , and N]

 

[We will now discuss philosophical matters regarding K4 , N4 , K , and N.]

 

[Recall the semantics and tableau constructions from previous sections: K4 (9.2 and 9.3); N4 (9.4 and 9.5) and K and N (9.6). Now Priest will discuss philosophical matters regarding these constructions.]

We are now in a position to make some comments on the import of the previous constructions.

(171)

[contents]

 

 

 

 

9.7.2

[Logically Impossible Worlds]

 

[We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different.]

 

[Recall from section 4.2.3 that for modal logics, non-normal worlds are ones where nothing is necessary and all is possible; for, at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. In section 9.2, we discussed a possible worlds First Degree Entailment (and thus four value-situationed) system called K4. In section 9.4, we noted that K4 has the following problematic valid formula: ⊨ p → (qq). This was problematic, because we want to be able to say things about what would follow if certain laws of logic were suspended. So the following would be valid, even though we would want it to be invalid: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (p.167, section 9.4.3 ). But “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (p.167). We thus incorporated non-normal worlds into K4 in order to get N4. But we note here that in non-normal worlds, the normal laws of logic are different, and we will call such non-normal worlds “logically impossible worlds.”]

As we saw (9.4.4 9.4.6), non-normal worlds of the kind we have employed in this chapter are worlds where the laws of logic are different. Let us call these ‘logically impossible worlds’.

(171)

[contents]

 

 

 

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

[Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds.]

 

[In section 3.6.5 (not yet summarized), Priest discussed physically impossible worlds: “Something is physically necessary if it is determined by the laws of nature, and physically possible if it is compatible with the laws of nature. Thus, it is physically impossible for me to jump thirty metres into the air (though this is not a logical impossibility)” (46). Priest says now that there is no reason why there cannot likewise be logically impossible worlds.]

There seems to be no reason why there should not be logically impossible worlds, in whatever sense there are possible worlds. Physically impossible worlds, where the laws of physics are different, are entirely routine (see 3.6.5). And just as there are worlds where the laws of physics are different, there must be worlds where the laws of logic are different.

(171)

[contents]

 

 

 

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

[We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.”]

 

[In fact, when we discuss non-classical logics, we seem to suppose such “logically impossible” worlds (even though in fact the real world might be one where the laws of classical logic do indeed fail). It is implied for example when we say that “if intuitionist logic were correct, the law of double negation would fail.”]

After all, we seem to envisage just such worlds when we evaluate conditionals such as ‘if intuitionist logic were correct, the law of double negation would fail’ (true), ‘if intuitionist logic were correct, the law of | identity would fail’ (false). Even if one is a modal realist (2.6), why should there not be such worlds?

(171-172)

[contents]

 

 

 

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

[Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling.]

 

[Priest next deals with some objections to the idea that there can be logically impossible worlds. {1} Objection: Someone might say that logical laws should always hold at possible worlds, by definition. Reply: we are not dealing with possible worlds but rather impossible ones. {2} Objection: One might say that some proposed logically law for a possible world that breaks one of our normal logical laws cannot be the case, simply because it is breaking a normal logical law. For example, suppose someone claims that there is a world where it is a logical law that A → (B ∧ ¬B) holds and so does A. Then, by modus ponens we can infer that B ∧ ¬B. The objector can say that this is a contradiction and it cannot be the case. Reply: {2a} Some might have philosophical reasons to say that the normal laws can be broken. For example, a dialetheist would say that the law of non-contradiction is breakable. {2b} The objection assumes that modus ponens holds in this world. But as an impossible world, it may not.]

One might suggest that there can be no worlds at which logical laws fail: by definition, logical laws hold at all possible worlds. Maybe so. But it is precisely impossible worlds that we are dealing with here. Or one might say: take a world in which it is a logical law that A → (B ∧ ¬B) and in which A is also true. It would follow that B ∧ ¬B is true at that world, which cannot be the case. This argument is hardly likely to persuade someone who accepts the possibility of truth-value gluts. But in any case, it is fallacious. For who says that modus ponens holds at that world? In the semantics we have looked at, it is entirely possible to have both A and A C holding at a non-normal world, without C holding there.

(172)

[contents]

 

 

 

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

[In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to.]

 

[Priest next notes that logically impossible worlds do not necessarily have cases of broken laws of logic, just like how physically impossible worlds may allow for certain alternate physical situations without them ever obtaining. However, one could specifically define logically impossible worlds as ones where the laws of logic are in fact broken.]

Note that one might take ‘logically impossible world’ to mean something other than ‘world where the laws of logic are different’. One might equally take it to mean ‘world where the logically impossible happens’. This need not be the same thing. If this is not clear, just consider physically impossible worlds. The fact that the laws of physics are different does not necessarily mean that physically impossible things happen there (though the converse is true). For example, even if the laws of physics were to permit things to accelerate past the speed of light, it does not follow that anything actually would. Things at that world might be accelerating very slowly, and the world might not last long enough for any of them to reach super-luminal speeds.

(172)

[contents]

 

 

 

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

[Logically impossible worlds can also in fact be ones where laws of logic indeed are broken.]

 

[I might be mistaken about this next point. It might be that we know there are logically impossible worlds where the laws of logic are broken, because we have already seen that there is a world where A and A C are true, but C is not. In the footnote Priest mentions some inferences that do not hold in any impossible world. However, these are instances without conditionals, and it is conditionals that express the laws of logic (but I do not myself know why that is.)]

But logically impossible worlds, in the sense that these occur in the semantics we have been looking at, may be logically impossible in the second sense as well. For example, there are, as has just been noted, worlds where A and A C are true, but C is not.6

(172)

6. There are no worlds at which AB is true, but A is not, or at which ¬¬A is true, but A is not. But it is conditionals that express the laws of logic, not conjunctions or negations. That is why it is their behaviour (and only theirs) that changes at non-normal worlds.

(172)

[contents]

 

 

 

 

9.7.8

[Relevant Logics and the Conditional]

 

[Relevant propositional logics are ones where whenever “A B is logically valid, A and B have a propositional parameter in common” (172). ]

 

[Priest now defines relevant logic: “A propositional logic is relevant iff whenever A B is logically valid, A and B have a propositional parameter in common” (172). This may seem odd, because A and B would seem to be propositional parameters, and surely we are not saying that conditions need to be of the form A A to be relevant. So recall from section 1.2.3: “I use capital Roman letters, A, B, C, ..., to represent arbitrary formulas of the object language. Lower-case Roman letters, p, q, r, ..., represent arbitrary, | but distinct, propositional parameters” (4-5). So maybe we would need to look at the lower-case sorts of formulations, when looking for relevance. We will see some examples. Or maybe A B is shorthand for more complex formulations, like we will see below, as with ⊨ A ⥽ (B ∨ ¬B). He says that conditionals that suffer from the paradoxes of implication, including the strict conditional, are not relevant. Let us look at the paradoxes of strict implication. In section 4.6.3 we saw that the following  are valid for the strict conditional.

A ⥽ (B ∨ ¬B)

⊨ (A ∧ ¬A) ⥽ B

We might fill them out with propositional parameters I am going to guess in the following way.

p ⥽ (q ∨ ¬q)

⊨ (p ∧ ¬p) ⥽ q

But I am not sure about much here yet. Yet we can see that the strict conditional is not part of a relevant logic. And these same forms are not valid in K4.

K4 pq ∨ ¬q

K4 (p ∧ ¬p) → q

Nonetheless, Priest says that neither K4 and K are relevant. To see why, we first recall from section 9.4.2 that in K4, ⊨ p → (qq) is valid, and in section 9.6.6 we saw that it if valid in K too.]

A propositional logic is relevant iff whenever A B is logically valid, A and B have a propositional parameter in common. Obviously, any conditional that suffers from paradoxes of implication (material implication, | strict implication, the intuitionist conditional) is not relevant. Neither are K4 and K relevant, as we have seen (9.4.2 and 9.6.6).

(172-173)

[contents]

 

 

 

 

9.7.9

[N4 as Relevant]

 

[But N4 is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds.]

 

[Priest will now show that N4 is a relevant logic. It gets very technical, and I am the wrong person to summarize this, so please skip to the quotation below. I will try to say some things still, but they probably will not help you. Recall that in N4, there are non-normal worlds. And recall from section 4.2.5 and 9.4.9 that inferences are valid only if they preserve truth in all interpretations at all normal worlds. I might have this wrong, but I think that means it cannot be that the premises are at least true and the conclusion not at least true. If I am following even a little here (and probably not), Priest is going to do the following. Let me first note that I am not certain if we are dealing with an inference, like in his cited problem, or a simple conditional, like mentioned in the last line of this paragraph. I am also not sure if it makes a difference. Let us for now say that we are dealing with the conditional A B, and we want to know if it would be valid/true in N4 whenever there is no relevance of the antecedent A to the consequent B. Priest will make a model where it is false, even though in K4 presumably it would be true. It seems that the way this will work will have to do with the fact that in normal worlds we evaluate conditionals on the basis of all other worlds, whether normal or not. But as we saw in section 9.4.6, we do not evaluate the conditionals in non-normal worlds compositionally in terms of the component terms’ values but rather we assign their values arbitrarily however we please. Priest will exploit those two features of N4 in order to make a non-relevant A B be false/invalid in a normal world. So our model will have two worlds, 0 and 1, and world 1 is the non-normal one. It is still unclear to me if we are dealing with an inference from A to B or a conditional, but I am guessing wildly that Priest is covering both options. (Sorry, please read the text). For non-relevance, we suppose that we have antecedent A and consequent B (or premise(s) A and conclusion B), but A and B share no propositional parameters in common. We also consider a propositional parameter (or conditional) called D, which can be included either in antecedent A or consequent B (or in premise(s) A or conclusion B); but it cannot be in both, because as we said, A and B share no parameters in common, thus if D is in one, it cannot be in the other. We will assign our values for the conditional in non-normal world 1 arbitrarily, as that is how it works in non-normal worlds (see section 9.4.6). So we say, if D is among the antecedent (or premises), then we assign it both as true and also as false. Or, if instead it is in the consequent (or conclusion), then we assign it neither true nor false. Now, let us stick with conditionals for a second. Recall from section 9.2.4 that this is how we evaluate conditionals in N4:

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

A Bρw0 iff for some w′ ∈ W, Aρw1 and Bρw0

(p.164, section 9.2.4)

We ask, A Bρw0??? In other words, we want to determine the value of A B in world 0, the normal world. Priest’s way of proving this uses the induction method, which I have not learned yet (see section 0.2). Were we to perform it, we would find that somehow, regardless of whether D is in the antecedent (or premises) or in the consequent (or conclusion), A will be both true and false (and thus at least true) and B will be neither true nor false (and thus not at least true). And hence the formula will be false/invalid in world 0, the normal world. Please read the quotation, as I am not grasping this one very well at all.]

But N4 is a relevant logic. This can be seen by modifying the argument of 8.10, problem 5. Suppose that A and B share no propositional parameters, and consider an interpretation ⟨W, N, ρ⟩, where W = {w0, w1}; N ={w0}; if D is a propositional parameter or a conditional in A, w11 and w10; if D is a propositional parameter or a conditional in B, neither w11 nor w10. (D cannot occur in both, since A and B have no parameters in common.) It is easy to check that w11 and w10, but neither w11 nor = w10.7 In particular, A is true at w1 and B is not. Hence A B is not true at w0.

(173)

7. Proof: For the first, what we show is that every formula made up from the propositional parameters occurring in A – and so, in particular, A – the result holds. Similarly for B. This is proved by induction on the construction of sentences, but an induction slightly different from the normal kind. Note that every formula can be built up from conditionals and parameters using the extensional connectives. Hence, the result may be proved by induction, with parameters and conditionals as the basis case, and induction cases for the extensional connectives. The basis case is true by definition. The induction cases are as in the notes to 8.4.6 and 8.4.9.

(173)

[contents]

 

 

 

 

9.7.10

[N as Relevant]

 

[In a similar way, N is also a relevant logic.]

 

[Priest next shows how in a similar way N is a relevant logic. Please consult the text for the details.]

A similar argument shows that Nis a relevant logic. Take a ∗ interpretation ⟨W, N, ∗, v⟩, where W = {w0, w1, w2}; N = {w0}, w*o  = w0, w*1 = w2, w*2 = w1; for every propositional parameter or conditional, D, in A, vw1(D) = 1 and vw2(D) = 0; for every propositional parameter or conditional, D, in B, vw1(D) = 0 and vw2(D) = 1. One can check that vw1(A) = 1, and vw1(B) = 0. Hence vw0(A B) = 0. Details are left as an exercise.

(172)

[contents]

 

 

 

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

[Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters.]

 

[We have the intuition already that “for a conditional to be true there must be some connection between its antecedent and consequent” (172). But it is not always obvious how to do that in a formalized way. Yet we saw in section 9.7.8 that one way is to require shared parameters.]

It is a natural thought that for a conditional to be true there must be some connection between its antecedent and consequent. It was precisely this idea that led to the development of relevant logic. A sensible notion of connection is not so easy to spell out, however (as we saw, in effect, in 4.9.2). The parameter-sharing condition of 9.7.8 gives some content to the idea.

(172)

[contents]

 

 

 

 

9.7.12

[Filter Logics]

 

[There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173).]

 

[Priest then notes another sort of relevant logic called filter logics. Here “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter.” But it is a different sort of logic than the ones of this book. Priest notes that often times filter logics break the principle of transitivity.]

There are some approaches to relevant logic where a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter. (These are | sometimes called filter logics, since the extra constraint filters out ‘undesirables’.) Characteristically, such approaches give rise to relevant logics of a kind different from those considered in this book. For example, if the parameter-sharing filter is used, (p ∧ (¬pq)) → q is valid, which it is not in the relevant logics of this, and subsequent, chapters. Typically (though not invariably), a feature of filter logics is the failure of the principle of transitivity: if AB and B C then A C (thus breaking the argument of 4.9.2).

(173-174)

[contents]

 

 

 

 

9.7.13

[More Than Classical Relevance]

 

[Relevance in our systems here however is not conditions added on top of classical validity.]

 

[The way that we are dealing with relevance here “is not some extra condition imposed on top of classical validity” (174). Rather, it is something else (but I am not sure I understand what it is and what the distinction is. So see the quote below.)]

In the present approach, relevance is not some extra condition imposed on top of classical validity. Rather, relevance, in the form of parameter sharing, falls out of something more fundamental, namely the taking into account of a suitably wide range of situations.

(174)

[contents]

 

 

 

 

9.7.14

[Preserving Conventional Truth in This System]

 

[If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.]

 

[Priest then notes some additional concerns. We might want within this four value-situationed non-normal worlds logic to reserve certain conventional properties of truth for the “real” or actual world. Priest then explains how this would work. We symbolize @ as the actual world, and we say truth in this restricted sense is truth at @, and validity is truth preservation at @ for all interpretations. Then, we can add constraints in @ to model the properties we think truth should have in the actual real world. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts (see section 8.4.6 and section 8.4.9).]

One final comment: one might hold that truth – real truth, not just truth in some world – has some special properties; that unlike truth in an arbitrary world, truth itself can have no gaps or gluts. To accommodate this view, one could take an interpretation to include a distinguished normal world, @ (for actuality), such that truth (simpliciter) is truth at @. Validity would then be defined as truth preservation at @ in all interpretations.8 The special properties of truth would be reflected in semantic constraints on @. Thus, if it be held that there are no truth value gluts in @, one would impose the constraint that ρ@ satisfy the condition Exclusion of 8.4.6. If it be held that there are no truth-value gaps in @, then one would impose the constraint that ρ@ satisfy the condition Exhaustion of 8.4.9.9 Or in a ∗ interpretation, one might require that @ = @∗, which rules out gaps and gluts. But from the present | perspectives, these conditions would require justification by some novel considerations.

(174-175)

8 One could, in fact, set up all the possible-world semantics that we have had till now in this way. But since these semantics contain nothing to distinguish @ from any other normal world, this would have had no effect on validity.

9 Strictly speaking, these conditions are not sufficient. To rule out truth-value gluts and gaps with formulas containing → s, we need to make another change as well. Specifically, to rule out truth-value gaps, the falsity conditions for A B at @ have to read:

A Bρ@0 iff (for some w′, Aρw1 and Bρw 0) or (it is not the case that A Bρw@1) and to rule out truth-value gluts, they have to read:

A Bρ@0 iff (for some w′, Aρw1 and Bρw 0) and (it is not the case that A Bρw@1).

(174)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

.

 

26 Jun 2018

Priest (9.6) Introduction to Non-Classical Logic, ‘Star Again,’ 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

 

9.

Logics with Gaps, Gluts and Worlds

 

9.6

Star Again

 

 

 

 

Brief summary:

(9.6.1) We can apply the N4 constructions to Routley Star ∗ semantics. (9.6.2) To the Routley semantics that we have seen before, we now add the rule for the conditional →, which gives us K. Here is the formalization:

Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, ∗ is a function from worlds to worlds such that w∗∗ = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:

vw(AB) = 1 if vw(A) = vw (B) = 1, otherwise it is 0. .

vw(AB) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.

vwA) = 1 if vw*(A) = 0, otherwise it is 0.

| Note that vw*A) = 1 iff vw**(A) = 0 iff vw(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.

(p.151-152, section 8.5.3)

Let ⟨W, ∗, v⟩ be any Routley interpretation (8.5.3). This becomes an interpretation for the augmented language when we add the following truth condition for →:

vw(A B) = 1 iff for all w′ ∈ W such that vw (A) = 1, vw(B) = 1

Call the logic that this generates, K.

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(9.6.3) Priest then supplies the tableau rules for K.

 

Conjunction

Development, True (D,+x)

A ∧ B,+x

A,+x

B,+x

 

Conjunction

Development, False (D,−x)

A ∧ B,−x

↙      ↘

A,−x       B,−x

 

 Disjunction

Development, True (∨D,+x)

A ∨ B,+x

↙      ↘

A,+x        B,+x

 

 Disjunction

Development, False (∨D,−x)

A ∨ B,-x

A,-x

B,-x

 

 Negation

Development, True (¬D,+x)

¬A,+x

A,-

 

 Negation

Development, False (¬D,−x)

¬A,-x

A,+

 

 Conditional

Development, True (→D,+x)

A → B,+x

↙      ↘

A,-y      B,+y

.

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch

 

 Conditional

Development, False (D,−x)

A → B,-x

A,+j

B,-j

.

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch. j must be new.

(last two are based on p.169, and those above from p.152, section 8.5.4, with names and bottom text added, possibly mistakenly; please consult the original text)

(9.6.4) Priest then gives an example tableau of an invalid formula: that p ¬q ⊬ ¬(p q). (9.6.5) We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world. “W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0” (170). (9.6.6) In K, we still have the problematic valid formula: ⊨ p → (qq). We can remedy this by adding non-normal worlds to get N. “An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N” (170). (9.6.7) We make our tableau for Nthe same way as for for K, only now the rules for the conditional → only apply for world 0. We generate counter-models the same way too. (9.6.8) The tableaux for K and N are sound and complete. (9.6.9) K4 and N4 are not equivalent to K and N. For example,  K and N validate  contraposition: p q ⊨ ¬q → ¬p, but K4 and N4 do not. (9.6.10) Additionally, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.

 

 

 

 

 

Contents

 

9.6.1

[N4 and Routley ∗ ]

 

9.6.2

[K Semantics]

 

9.6.3

[K Tableau Rules]

 

9.6.4

[An Example Tableau: Invalid]

 

9.6.5

[Counter-Models]

 

9.6.6

[Non-Normal Worlds Routley ∗ :  N]

 

9.6.7

[Tableaux and Counter-Models for N]

 

9.6.8

[The Soundness and Completeness of N]

 

9.6.9

[The Non-Equivalence of K4 / N4 to K / N]

 

9.6.10

[More Non-Equivalence of K4 / N4 to K / N]

 

 

 

 

 

 

 

Summary

 

9.6.1

[N4 and Routley ∗ ]

 

[We can apply the N4 constructions to Routley Star ∗ semantics.]

 

[We should first review the basic ideas of Routley Star ∗ semantics, from section 8.5. The following is our brief summary of that section.

FDE can be given an equivalent, two-valued, possible world semantics in which the negation is an intensional operator, meaning that it is defined by means of related possible worlds. In this case, we use Routley’s star worlds. We have a star function, *, which maps a world to is “star” or “reverse” world (and back again to the first one, if applied yet another time. That bringing back to the first is what defines the function). So for any world w, the * function gives us its companion star world w. We evaluate conjunctions and disjunctions based on values in the given world. But what is notable in Routley Star semantics is that a negated formula in a world w is valued true in that world not on the basis of it unnegated form being false in that world w, but rather on the basis of its unnegated form being false in the star world w (so a negated formula is 1 in w if its unnegated form is 0 in w*.) Validity is defined as truth preservation for all worlds and interpretations. For constructing tableaux in Routley Star logic, we designate not just the truth-value for a formula but also the world in which that formula has that value. This is especially important for negation, where the derived formulas are found in the companion star world. Here is how the tableau rules work for Routley Star logic [quoting Priest, except for the rules tables, where I add my own names and abbreviations, following David Agler]:

Nodes are now of the form A,+x or A,−x, where x is either i or i#, i being a natural number. (In fact, i will always be 0, but we set things up in a slightly more general way for reasons to do with later chapters.) Intuitively, i# represents the star world of i. Closure occurs if we have a pair of the form A,+x and A,−x. The initial list comprises a node B,+0 for every premise, B, and A,−0, where A is the conclusion. The tableau rules are as follows, where x is either i or i#, and whichever of these it is, is the other.

 

Conjunction

Development, True (D,+x)

AB,+x

A,+x

B,+x

 

Conjunction

Development, False (D,−x)

AB,−x

↙      ↘

A,−x       B,−x

 

 Disjunction

Development, True (∨D,+x)

A ∨ B,+x

↙       ↘

A,+x        B,+x

 

 Disjunction

Development, False (∨D,−x)

A ∨ B,-x

A,-x

B,-x

 

 Negation

Development, True (¬D,+x)

¬A,+x

A,-

 

 Negation

Development, False (¬D,−x)

¬A,-x

A,+

(p.152, section 8.5.4, quoting. Note, names and abbreviations are my own and are not in the text. )

 

We test for validity first [as noted above] by setting every premise to true in the non-star world and the conclusion to false in the non-star world. We then apply all the rules possible, and if all the branches are closed [recall from above that closure occurs if we have a pair of the form A,+x and A,−x] then it is valid, and invalid otherwise [so it is invalid if any branches are open]. We then can make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world (that is to say, “ if p,+x occurs on the branch, vwx(p) = 1, and if p,−x occurs on the branch, vwx(p) = 0.”) The equivalence between Routley Star semantics and FDE becomes apparent when we make the following translation: vw(p) = 1 iff 1; vw(p) = 0 iff 0.

(Brief summary for 8.5)

Next we need to recall from section 9.5 how Priest provided us the tableau rules for N4, which is the non-normal worlds variation of K4, which is a possible worlds First Degree Entailment (and thus four value-situationed) system. Priest says now that we can perform the same sorts of constructions for Routley ∗ semantics.

]

Before we move on to consider some of the implications of the preceding, let us pause to note that exactly the same sorts of construction can be performed with respect to the ∗ semantics.

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[contents]

 

 

 

 

 

9.6.2

[K Semantics]

 

[To the Routley semantics that we have seen before, we now add the rule for the conditional →, which gives us K.]

 

[In section 8.5.3, Priest wrote:

Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, ∗ is a function from worlds to worlds such that w∗∗ = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:

vw(AB) = 1 if vw(A) = vw (B) = 1, otherwise it is 0. .

vw(AB) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.

vwA) = 1 if vw*(A) = 0, otherwise it is 0.

| Note that vw*A) = 1 iff vw**(A) = 0 iff vw(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.

(p.151-152, section 8.5.3)

Now Priest will add the rule for the conditional operator →, which will generate in our augmented language: K.]

Let ⟨W, ∗, v⟩ be any Routley interpretation (8.5.3). This becomes an interpretation for the augmented language when we add the following truth condition for →:

vw(A B) = 1 iff for all w′ ∈ W such that vw (A) = 1, vw(B) = 1

Call the logic that this generates, K.

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[contents]

 

 

 

 

 

9.6.3

[K Tableau Rules]

 

[Priest then supplies the tableau rules for K.]

 

[Recall from section 9.6.1 above that we listed from section 8.5.4 the tableau rules for Routley ∗ logic. Priest says that we can obtain the full tableau rules for K by adding to those the rules for →. In the quotation]

Tableaux for K can be obtained by adding to the rules of 8.5.4, these rules for →:

Conjunction

Development, True (D,+x)

A ∧ B,+x

A,+x

B,+x

 

Conjunction

Development, False (D,−x)

A ∧ B,−x

↙      ↘

A,−x       B,−x

 

 Disjunction

Development, True (∨D,+x)

A ∨ B,+x

↙      ↘

A,+x        B,+x

 

 Disjunction

Development, False (∨D,−x)

A ∨ B,-x

A,-x

B,-x

 

 Negation

Development, True (¬D,+x)

¬A,+x

A,-

 

 Negation

Development, False (¬D,−x)

¬A,-x

A,+

 

 Conditional

Development, True (→D,+x)

A → B,+x

↙      ↘

A,-y      B,+y

.

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch

 

 Conditional

Development, False (D,−x)

A → B,-x

A,+j

B,-j

.

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch. j must be new.

(last two are based on p.169, and those above from p.152, section 8.5.4, with names and bottom text added, possibly mistakenly; please consult the original text)

 

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch;4 and in the second rule, j must be new. (Note that we do not need rules for negated →. The ∗ rules take care of that.)

(169)

4. So for a completed tableau, if either j or j# occurs on the branch, the rule needs to be applied to both j and j#.

(169)

[contents]

 

 

 

 

 

9.6.4

[An Example Tableau: Invalid]

 

[Priest then gives an example tableau of an invalid formula: that p ¬q ⊬ ¬(p q).]

 

[Priest then provides an example to show how to do a tableau in K. By the way, note for line 8 that the left branch closes, but the same rule could be still applied there as is applied on the right (because the development rule for true conditionals says it applies to all worlds on the branch), but it is unnecessary to do so.]

Here is a tableau to show that p ¬q ⊬ ¬(p q):

p ∧ ¬q ⊬ ¬(p → q)

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

p ∧ ¬q,+0

¬(p → q),−0

.

p,+0

¬q,+0

q,−0#

p → q,+0#

↙             ↘

p,−0            q,+0

     ×           ↙        ↘

                p,−0#     q,+0#

                         ×

  .

P

.

P

.

1∧+

.

1∧+

. 

4¬−

. 

2¬→−

  .

6→+

(7×4)

6→+ 

(8×5)

open

invalid

[based on p.153. Enumeration and step accounting are my own and are possibly mistaken.]

The splits are caused by applying the rule for true → to the line immediately before the first split. There are two worlds, 0 and 0#, so the rule has to be applied to both of them.

(169-170)

[contents]

 

 

 

 

 

9.6.5

[Counter-Models]

 

[We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world. “W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0” (170).]

 

[Recall from section 8.5.6 how we construct counter-models. In summary we said: “We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world.” And Priest himself writes:

To read off a counter-model from an open branch: W = {w0,w0# } (there are only ever two worlds);  w*0 = w0# and (w0#)* = w0. (W and ∗ are always the same, no matter what the tableau.) v is such that if p,+x occurs on the branch, vwx(p) = 1, and if p,−x occurs on the branch, vwx(p) = 0. Thus, the counter-model defined by the righthand open branch of the second tableau of 8.5.5 has vwo(p) = 1, vwo(r) = 0 and vwo#(q) = 0. It is easy to check directly that this interpretation does the job. Since q is false at w0# , ¬q is true at w0, as, therefore, is q∨¬q; but p is true at w0, hence p ∧ (q ∨ ¬q) is true at w0. But r is false at w0, as required.

(p.153, section 8.5.6)

]

Counter-models are read off as is done without → (8.5.6), except that there may be more than two worlds now. Thus, W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0. Thus, the counter-model from the open branch of the tableau of 9.6.4 may be depicted thus:

x

xxx+pxxx−p

xxx+qxxx−q

xxxw0xxxw*0

x

Since q is not true at w*0, ¬q is true at w0, as, then, is p ∧ ¬q. But at every world where p is true, q is true. Hence, p q is true at w*0, and so ¬(p q) is false (untrue) at w0.

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[contents]

 

 

 

 

 

9.6.6

[Non-Normal Worlds Routley ∗ :  N]

 

[In K, we still have the problematic valid formula: ⊨ p → (qq). We can remedy this by adding non-normal worlds to get N. “An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N” (170).]

 

[Recall the following issue from section 9.4.

(9.4.2) In K4, if ⊨ A then ⊨ B A. That means ⊨ p → (qq) is valid, because ⊨ q q is valid. (9.4.3) Even though ⊨ p → (qq) , which contains the law of identity, is valid, we can think of a paradoxical instance of this that shows how the law of identity can fail: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (167). (9.4.4) As we noted, the conditional should be able to express things that go against the laws of logic, like the law of identity. We should be able to formulate sentences in which the antecedent supposes some law of logic not to hold, and then the consequent would express what sorts of things would follow from that. Non-normal worlds are ones where the normal laws of logic may fail; so we should implement non-normal worlds: “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (167). (9.4.5) We thus need to consider non-normal worlds where the laws of logic fail and, given how the conditionals express those laws, where the conditional takes on values different than it would in normal worlds (K4). (9.4.6) At a non-normal world, A B might be able to take on any sort of value, because the laws of logic may change in that world. (9.4.7) We “take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N W is the set of normal worlds (so that W N is the set of non-normal worlds), and ρ does two things. For every w, ρw is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every non-normal world, w, ρw is a relation between formulas of the form A B and truth values” (167). (9.4.8) The truth conditions for connectives in our non-normal worlds K4 are the same as for K4, except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation.  Here are the truth conditions for normal worlds:

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρw0 or Bρw0

(p.164, section 9.2.3)

Aw1 iff w1 or w1

Aw0 iff w0 and w0

¬w1 iff Aρw0

¬w0 iff w1

(not in the text)

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

A Bρw0 iff for some w′ ∈ W, Aρw1 and Bρw0

(p.164, section 9.2.4)

(9.4.9) Our non-normal worlds FDE system will be called N4, and it will define validity in the same way as for K4, namely, as truth preservation at all normal worlds of all interpretations.

(from the brief summary of section 9.4)

Priest notes now that in K, ⊨ p → (qq). We can fixe that problem like we did in in section 9.4 by adding non-normal worlds, giving us N.]

As in K4, in K, ⊨ p → (qq), as may easily be checked. To change this, we may add non-normal worlds in the same way. An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N.

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[contents]

 

 

 

 

 

9.6.7

[Tableaux and Counter-Models for N]

 

[We make our tableau for Nthe same way as for for K, only now the rules for the conditional → only apply for world 0. We generate counter-models the same way too.]

 

[Priest then notes that we use the same tableau rules for Nas for K, except now the rules for the conditional only apply for world 0 (the only normal world). And counter-models are constructed the same way.]

The tableaux for Nare the same as those for K, except that the rules for → (9.6.3) are applied only at 0. Counter-models are also read off in the same way. Again, only w0 is normal.

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[contents]

 

 

 

 

 

9.6.8

[The Soundness and Completeness of N]

 

[The tableaux for K and N are sound and complete.]

 

[Priest then notes that.]

Soundness and completeness for the tableaux for K and N are proved in 9.8.10–9.8.13.

[contents]

 

 

 

 

 

9.6.9

[The Non-Equivalence of K4 / N4 to K / N]

 

[K4 and N4 are not equivalent to K and N. For example,  K and N validate  contraposition: p q ⊨ ¬q → ¬p, but K4 and N4 do not.]

 

[Recall from section 8.5.8 that Routley Star semantics and FDE are equivalent when we make the translation: vw(p) = 1 iff 1; vw(p) = 0 iff 0. However, we lose this equivalence when we add the conditional →. The relational systems (I am guessing: K4 and N4) do not validate contraposition: p q ⊨ ¬q → ¬p. But the ∗ systems (K and N, I am guessing) do validate it.]

It should be noted that although the relational semantics and the ∗ semantics are equivalent for FDE, as we saw in 8.5.8, this equivalence no longer obtains once we add →. For a start, the ∗ systems (K and N) validate contraposition: p q ⊨ ¬q → ¬p. (Details are left as an exercise.) The relational systems do not. (We saw that this is not valid in K4, and a fortiori N4, in 9.3.6.)5

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5. This may be changed by redefining the truth conditions of → (at normal worlds) in the relational semantics, as:

A Bρw1 iff for all w′ ∈ W (if Aρw1 then Bρw 1, and if Bρw 0 then Aρw 0).

Or, more simply, and equivalently, defining a new conditional AB as (A B) ∧ (¬B ¬A), and working with this.

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[contents]

 

 

 

 

 

9.6.10

[More Non-Equivalence of K4 / N4 to K / N]

 

[Additionally, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.]

 

[Also, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.]

More fundamentally, because of the falsity conditions for →, the relation semantics (normal and non-normal) verify p ∧ ¬q ⊨ ¬(p q). (Details are left as an exercise.) But this inference fails in K (and a fortiori N), as we saw in 9.6.4.

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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.

 

 

.