## 19 Jul 2018

### Williams (5.2) Gilles Deleuze’s Philosophy of Time, ‘Eternal return and death,’ summary

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[The following is summary of Williams, with boldface and bracketed commentary being my own additions. Note, I drafted this six years ago so it may seem discontinuous with current entries. For example, I do not in brackets here try to clear up what is said or note what I failed to understand.]

Summary of

James Williams

Gilles Deleuze’s Philosophy of Time:

A Critical Introduction and Guide

Chapter 5

Time and eternal return

5.2

Eternal return and death

Brief Summary:

The eternal return is not a recycling of identical things, because “then there would be no sense of return in them due to the lack of any way of telling there had been a return.” (119b) The eternal return also opens the third synthesis of time, which is thus somehow both circular while also having a before and after and an asymmetrical series. This is conceivable if we view it as the cycling of pure difference, which combines the features of return with non-identity. As conscious beings, we are to live time’s passage in such a way that we participate in the eternal return of difference, and if we fail in that, we die. Deleuze critiques the Freudian notion of the inorganic death drive, because the eternal return is not cycles of life and death. Rather, it is “the passing away of that which is inanimate in sameness and identity and the eternal return of multiple forms of difference.” (123bc) Humans in fact do not dread the inanimate state of death but instead that which makes all beings who are fixed in their identity pass away in time. We should distinguish this sort of death resulting from remaining self-same while time moves past us with the vital death involved in evolving, growth, and change where we affirm continuous differentiation and thereby participate in the eternal return of pure difference. Thus the third synthesis of time is tied both to death and eternal return.

Summary

The eternal return in its purest form also involves death, violence and “the most difficult tests put to living beings” (118c), and

Williams then gives an excellent formulation for why the eternal return is not of something similar or identical. “No two cycles of return could be internally identical, for then there would be no sense of return in them due to the lack of any way of telling there had been a return.” (119b) If we hold this view, we might suffer from two sorts of anguish, the anguish we feel as dear things pass-away, and the anguish about the return of things we do not want to see again. (119c.d) Anguish is what humans feel, but the eternal return is a process applying more broadly than just human experience. (120c) Deleuze writes that we must conceive eternal return as the selecting thought. (120d) But this does not mean it depends on our thinking to operate. (121a) It is the highest test to conceive eternal return as purely differential [perhaps because preexisting concepts will be inadequate?]. (121c)

Williams wonders, how can the third synthesis be both circular but also have a before and after and an asymmetrical series? “The answer is that the third synthesis is both an irreversible series and a cyclical return.” (122a) The third synthesis orders things that remain the same, but for pure difference it is the cycling of pure difference. As conscious beings, how are we to live time’s passage so that we participate in the eternal return of difference? (122b) Deleuze says that if we do not pass this test, we die, as would any being that did not pass the test. (122c)

Williams writes that there are different kinds of processes at work, those working towards an identity and those working in becoming and change, novelty, and transformation. (123a) So a process resisting change is doomed to fall out of existence.

Deleuze critiques Freud’s work on death and also he critiques “any definition of death as a return of the living beings to undifferentiated and inanimate matter.” (123b) For Deleuze, the eternal return is not cycles of life and death, but rather it is “the passing away of that which is inanimate in sameness and identity and the eternal return of multiple forms of difference.” (123bc) So death is not the inanimate state humans dread. It is rather what makes any and every being that remains fixed in its identity pass away in time. (123cd)

But death is not just about the passing of identities but also the survival of difference. Thus evolution and change are forms of death, because they involve the passing of certain beings but also “the moving towards new processes through difference in itself.” (124a) Thus there are two deaths, our personal one concerning the I and self, and the impersonal one that causes things to persist. (124b) “death as subject and death as difference affirming process”. (124b) Because the future is “the deployment and explication of the multiple” (Deleuze qtd 124bc) and because the eternal return promotes the death of all fixed things, we can see how the third synthesis is tied both to death and eternal return. (124bc)

Williams, James. Gilles Deleuze’s Philosophy of Time: A Critical Introduction and Guide. Edinburgh: Edinburgh University Press, 2011.

.

## 18 Jul 2018

### Priest (11a.4) An Introduction to Non-Classical Logic, ‘Modal FDE,’ summary

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

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

Part I:

Propositional Logic

11a

Appendix: Many-valued Modal Logics

11a.4

Modal FDE

Brief summary:

(11a.4.1) In our many-valued modal FDE logic, we have just the connectives ∧, ∨ and ¬, because A B is defined as ¬A B. (11a.4.2) FDE can be formulated as a four-valued logic, and the connectives can be evaluated with the following diamond lattice (Hasse diagram):

 1 ↗               ↖ b                                 n ↖                   ↗ 0

f is the greatest lower bound;  f is the least upper bound; and f¬ maps 1 to 0, vice versa, and each of b and n to itself. (11a.4.3) Our modal FDE logic is called KFDE. “If we ignore the value n in the non-modal case (that is, we insist that formulas take one of the values in {1, b, 0}) we get the logic LP. In the modal case, we get KLP. If we ignore the value b in the non-modal case, we get the logic K3. In the modal case, we get KK3” (245). (11a.4.4) The equivalence between four-valued FDE truth-values and the four value-situations of the relational semantics are: v(A) = 1 iff Aρ1 and it is not the case that Aρ0 ; v(A) = b iff Aρ1 and Aρ0 ; v(A) = n iff it is not the case that Aρ1 and it is not the case that Aρ0 ; and v(A) = 0 iff it is not the case that Aρ1 and Aρ0 . The truth-falsity conditions for the connectives in the relational semantics are: A Bρ1 iff Aρ1 and Bρ1 ; A Bρ0 iff Aρ0 or Bρ0 ; A Bρ1 iff Aρ1 or Bρ1 ; A Bρ0 iff Aρ0 and Bρ0 ; ¬Aρ1 iff Aρ0 ; and ¬Aρ0 iff Aρ1 . “Validity is defined in terms of the preservation of relating to 1” (245). (11a.4.5) KFDE has the same relational semantics but with world designations and with the following rules for the necessity and possibility operators:

vw(A) = 1 iff Aρw1 and it is not the case that Aρw0

vw(A) = b iff Aρw1 and Aρw0

vw(A) = n iff it is not the case that Aρw1 and it is not the case that Aρw0

vw(A) = 0 iff it is not the case that Aρw1 and Aρw0

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρ0 or Bρw0

A Bρw1 iff Aρw1 or Bρw1

A Bρw0 iff Aρw0 and Bρw0

¬Aρw1 iff Aρw0

¬Aρw0 iff Aρw1

Aρw1 iff for all w′ such that wRw′, Aρw1

Aρw0 iff for some w′ such that wRw′, Aρw0

Aρw1 iff for some w′ such that wRw′, Aρw1

Aρw0 iff for all w′ such that wRw′, Aρw0

(based on with quotation from p.245-246)

(11a.4.6) Next Priest provides the argumentation for why the truth/falsity conditions are formulated the way they are. (11a.4.7) By adding a possible worlds exhaustion constraint to KFDE we can get KLP. (11a.4.8) By adding a possible worlds exclusion constraint to KFDE we can get KK3. (11a.4.9) By adding both the exhaustion and exclusion constraints to KFDE, we get the classical modal logic K. (11a.4.10) We can apply the world accessibility relation constraints (like ρ, σ, τ, etc) to our specific many-valued modal logics to get such extensions as: KFDEρ, KLPρτ, KK3σ and so forth. (11a.4.11) There are no logical truths (tautologies) in KFDE and KK3, because to be a logical truth means that the formula is true under all interpretations. But in KFDE and KK3, under any interpretation, every formula can be valued neither true nor false, and thus no formulas can be logical truths. (11a.4.12) The interpretations for the logics of the family we are considering are monotonic. (11a.4.13) A corollary of this is “that ⊨K A iff ⊨KLPA (and similarly for Kρ and KLPρ, etc.)” (247).

Contents

11a.4.1

[The Connectives in Our Many-Valued Modal FDE]

11a.4.2

[FDE as a Four-Valued Logic]

11a.4.3

[KFDE, KL, and KK3]

11a.4.4

[The Relational Semantics of FDE]

11a.4.5

[The Relational Semantics of KFDE]

11a.4.6

[The Rationale for the Modal Operator Truth/Falsity Conditions]

11a.4.7

[Obtaining KLP]

11a.4.8

[Obtaining KK3]

11a.4.9

[Obtaining K]

11a.4.10

[Extensions of Our Many-Valued Modal Logics]

11a.4.11

[The Lack of Logical Truths in KFDE and KK3.]

11a.4.12

[The Monotonicity of These Logics]

11a.4.13

[Corollary]

Summary

11a.4.1

[The Connectives in Our Many-Valued Modal FDE]

[In our many-valued modal FDE logic, we have just the connectives ∧, ∨ and ¬, because A B is defined as ¬A B.]

[Recall from section 11a.2 that we are formulating many-valued modal logics. We will now do a many-valued modal FDE logic. (See ch.8). Recall from section 8.2.1 that we have just the connectives ∧, ∨ and ¬, because A B is defined as ¬A B.]

Let us now look at one many-valued modal logic in more detail. The many-valued logic in question is FDE. The language for this has three connectives: ∧, ∨ and ¬. (Recall that A B is defined as ¬A B.)

(244)

[contents]

11a.4.2

[FDE as a Four-Valued Logic]

[FDE can be formulated as a four-valued logic, and the connectives can be evaluated with the following diamond lattice (Hasse diagram):

 1 ↗               ↖ b                                 n ↖                   ↗ 0

f is the greatest lower bound;  f is the least upper bound; and f¬ maps 1 to 0, vice versa, and each of b and n to itself.]

[Recall from section 8.4 that we can formulate FDE as a four-valued logic (rather than as a two-valued logic with four truth-value situations enabled by a relational truth-value semantics; see section 8.2. In the four-valued version, the four values are: 1 (for just true), 0 (for just false), b (for both), and n (for neither), and the designated values are 1 and b. Next we should recall from section 8.4.3 how the diamond lattice works. Priest’s version is:

 1 ↗               ↖ b                                 n ↖                   ↗ 0

I proposed something with more reminders in it for how it is read.

 1 ↙↗ ↑ ↖↘ ⟳ b | n ⟲ ↘↖ ↓ ↗↙ 0

Negation toggles 0 and 1, and it maps n to itself and b to itself:

Negation and

 1 ↙↗ ↑ ↖↘ ⟳ b | n ⟲ ↘↖ ↓ ↗↙ 0

Priest writes here that: “f¬ maps 1 to 0, vice versa, and each of b and n to itself”. For conjunction, we take the greatest lower bound for both of the conjunct values, that is to say, when moving upward, we find the highest place we can start from in order to arrive at both of the two conjunct values (we can also start and arrive at the same place, if we begin at our destination).

Conjunction >

 1 ↙↗ ↑ ↖↘ ⟳ b | n ⟲ ↘↖ ↓ ↗↙ 0

Priest writes: “f is the meet on this lattice”. (On this wiki page, the greatest lower bound is called the “meet”.) For disjunction, we look for the least upper bound: when moving downward, we seek the lowest place we can start from to arrive at both values.

Disjunction ↓ <

 1 ↙↗ ↑ ↖↘ ⟳ b | n ⟲ ↘↖ ↓ ↗↙ 0

Priest writes: f is the join. (On this wiki page, the least upper bound is called the “join”.) ]

As we saw in chapter 8, FDE can be formulated as a four-valued logic. V = {1, 0, b, n} – true (only), false (only), both and neither. D = {1, b}. The values are ordered as follows:

 1 ↗               ↖ b                                 n ↖                   ↗ 0

f is the meet on this lattice; f is the join; f¬ maps 1 to 0, vice versa, and each of b and n to itself.

(245)

[contents]

11a.4.3

[KFDE, KL, and KK3]

[Our modal FDE logic is called KFDE. “If we ignore the value n in the non-modal case (that is, we insist that formulas take one of the values in {1, b, 0}) we get the logic LP. In the modal case, we get KLP. If we ignore the value b in the non-modal case, we get the logic K3. In the modal case, we get KK3” (245).]

[Recall from section 11a.2.8 that a many-valued modal logic is called KL. Here the L is a many-valued logic, and the K is a modal logic (see section 2.1 and section 2.3). The K part is structured in the following way:

An interpretation for this language is a triple ⟨W, R, v⟩. W is a non-empty set. Formally, W is an arbitrary set of objects. Intuitively, its members are possible worlds. R is a binary relation on W (so that, technically, R W×W). Thus, if u and v are in W, R may or may not relate them to each other. If it does, we will write uRv, and say that v is accessible from u. Intuitively, R is a relation of relative possibility, so that uRv means that, relative to u, situation v is possible. υ is a function that assigns a truth value (1 or 0) to each pair comprising a world, w, and a propositional parameter, p. We write this as vw(p) = 1 (or vw(p) = 0). Intuitively, this is read as ‘at world w, p is true (or false)’.

(p.21, section 2.3.3)

So that gives us the K in KL. The L is a many-valued logic, which has the structure:

a propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c C}⟩, where V is the set of semantic values, D V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation.

(pp.242-243, section 11a.2.1; see section 7.2).

It would seem that for FDE, as a four-valued logic, its ⟨V, D, {fc : c C}⟩ would be filled out in the following way:

V = {1, 0, b, n}

D = {1, b}

C = {¬, ∧, ∨} (with A B as ¬A B)

fc; c C = {f¬, f, f}

 f¬ 1 0 b b n n o 1

 f∧ 1 b n o 1 1 b n 0 b b b 0 0 n n 0 n 0 o 0 0 0 0

 f∨ 1 b n o 1 1 1 1 1 b 1 b 1 b n 1 1 n n o 1 b n 0

So when we combine this FDE four-valued semantics with modal semantics, our KL is more specifically KFDE. Now recall the exhaustion constraint from section 8.4.9 and 8.4.10. When applied to FDE, it disallows there being a neither value, giving us LP. Here we can also thereby obtain KLP. Or, if we apply the exclusion constraint (see section 8.4.6 and section 8.4.7), we can get K3, and thus here in our modal logics, KK3. (Note, the first K is for “Kripke” (our modal logic, see section 2.1.2) and the second one is for “Kleene” (which is  for our many-valued logics; see section 7.3.4)).]

KFDE is obtained by the general construction described. If we ignore the value n in the non-modal case (that is, we insist that formulas take one of the values in {1, b, 0}) we get the logic LP. In the modal case, we get KLP. If we ignore the value b in the non-modal case, we get the logic K3. In the modal case, we get KK3.

(245)

[contents]

11a.4.4

[The Relational Semantics of FDE]

[The equivalence between four-valued FDE truth-values and the four value-situations of the relational semantics are: v(A) = 1 iff Aρ1 and it is not the case that Aρ0 ; v(A) = b iff Aρ1 and Aρ0 ; v(A) = n iff it is not the case that Aρ1 and it is not the case that Aρ0 ; and v(A) = 0 iff it is not the case that Aρ1 and Aρ0 . The truth-falsity conditions for the connectives in the relational semantics are: A Bρ1 iff Aρ1 and Bρ1 ; A Bρ0 iff Aρ0 or Bρ0 ; A Bρ1 iff Aρ1 or Bρ1 ; A Bρ0 iff Aρ0 and Bρ0 ; ¬Aρ1 iff Aρ0 ; and ¬Aρ0 iff Aρ1 . “Validity is defined in terms of the preservation of relating to 1” (245).]

[Recall from section 8.2 that Priest originally formulates FDE using a relational semantics. Here he gives the equivalences between the four-values and the four value-situations along with the truth/falsity conditions for the connectives. Validity is preservation of 1.]

As we also saw in chapter 8, FDE can be formulated equivalently as a logic in which, instead of an evaluation function, v, there is a relation, ρ (not to be confused with the constraint on the accessibility relation), which relates a formula, A, to the values 1 (true) and 0 (false) as follows:

v(A) = 1 iff Aρ1 and it is not the case that Aρ0

v(A) = b iff Aρ1 and Aρ0

v(A) = n iff it is not the case that Aρ1 and it is not the case that Aρ0

v(A) = 0 iff it is not the case that Aρ1 and Aρ0

The appropriate truth/falsity conditions for the connectives are:

A Bρ1 iff Aρ1 and Bρ1

A Bρ0 iff Aρ0 or Bρ0

A Bρ1 iff Aρ1 or Bρ1

A Bρ0 iff Aρ0 and Bρ0

¬Aρ1 iff Aρ0

¬Aρ0 iff Aρ1

Validity is defined in terms of the preservation of relating to 1.

(245)

[contents]

11a.4.5

[The Relational Semantics of KFDE]

[KFDE has the same relational semantics but with world designations and with the following rules for the necessity and possibility operators: □Aρw1 iff for all w′ such that wRw′, Aρw1 ; □Aρw0 iff for some w′ such that wRw′, Aρw0 ; ◊Aρw1 iff for some w′ such that wRw′, Aρw1 ; ◊Aρw0 iff for all w′ such that wRw′, Aρw0 .]

[Priest next says that we can formulate KFDE firstly by using the above formulations from section 11a.4.4 by adding world designations, and secondly by formulating the truth/falsity conditions for the modal operators similarly. Recall from section 11a.2.6 that the modal operators for a many-valued modal logic would be given as:

vw(□A) = Glb{vw(A) : wRw′}

vw(◊A) = Lub{vw(A) : wRw′}

(p.242, section 11a.2.6 )

For necessity, we take the greatest lower bound. So in FDE, if a formula is related both to 1 and 0 in whatever accessible world, the greatest lower bound would be 0. These are the conditions for necessity in our modal version:

Aρw1 iff for all w′ such that wRw′, Aρw1

Aρw0 iff for some w′ such that wRw′, Aρw0

That would seem to fit the lower-bound articulation. Likewise for the possibility operator (see below). As such, I will try to list all the value equivalences and truth/falsity conditions for all the values and operators in KFDE formulated in the relational semantics.

vw(A) = 1 iff Aρw1 and it is not the case that Aρw0

vw(A) = b iff Aρw1 and Aρw0

vw(A) = n iff it is not the case that Aρw1 and it is not the case that Aρw0

vw(A) = 0 iff it is not the case that Aρw1 and Aρw0

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρ0 or Bρw0

A Bρw1 iff Aρw1 or Bρw1

A Bρw0 iff Aρw0 and Bρw0

¬Aρw1 iff Aρw0

¬Aρw0 iff Aρw1

Aρw1 iff for all w′ such that wRw′, Aρw1

Aρw0 iff for some w′ such that wRw′, Aρw0

Aρw1 iff for some w′ such that wRw′, Aρw1

Aρw0 iff for all w′ such that wRw′, Aρw0

(based on with quotation from p.245-246)

]

KFDE can be formulated in the same way. The facts of 11a.4.4 carry over with a subscript w to the vs and ρs. What of the truth/falsity conditions | of the modal operators if FDE is formulated in this way? They may be given, in a very natural way, as follows:

Aρw1 iff for all w′ such that wRw′, Aρw1

Aρw0 iff for some w′ such that wRw′, Aρw0

Aρw1 iff for some w′ such that wRw′, Aρw1

Aρw0 iff for all w′ such that wRw′, Aρw0

(246)

[contents]

11a.4.6

[The Rationale for the Modal Operator Truth/Falsity Conditions]

[Next Priest provides the argumentation for why the truth/falsity conditions are formulated the way they are.]

[(ditto)]

The argument for this is as follows. Consider vw(□A), that is Glb{vw(A) : wRw′}. This has four possible values.

1: In this case, for all w′ such that wRw′ the value of vw(A) is 1. So for all w′ such that wRw′, Aρw1 and it is not the case that Aρw0. In this case, the truth/falsity conditions give that □Aρw1 and it is not the case that □Aρw0, as required.

b: In this case, for all w′ such that wRw′, the value of vw(A) is 1 or b, and at least one is b. That is, for all w′ such that wRw′, Aρw1 and for at least one such w′, Aρw0. In this case, the truth/falsity conditions give that □Aρw1 and □Aρw0, as required.

n: In this case, for all w′ such that wRw′, the value of vw(A) is 1 or n, and at least one is n. That is, for all w′ such that wRw′, it is not the case that Aρw0 and for at least one such w′, it is not the case that Aρw1. In this case, the truth/falsity conditions give that it is not the case that □Aρw1 and it is not the case that □Aρw0, as required.

0: In this case, either there is some w′ such that wRw′ and vw(A) = 0, or there are w′ and w′′, such that wRw′, wRw′′, vw(A) = b and vw′′(A) = n. In the first case, for all w′ such that wRw′, Aρw′0 and it is not the case that Aρw1. In the second case, Aρw′′1 and Aρw′′0, and neither Aρw′′1 nor Aρw′′0. In either case, the truth/falsity conditions give that □Aρw0 and it is not the case that □Aρw1, as required.

The case for ◊ is similar, and is left as an exercise.

(264)

[contents]

11a.4.7

[Obtaining KLP]

[By adding a possible worlds exhaustion constraint to KFDE we can get KLP.]

[Recall the exhaustion constraint from section 8.4.9:

Exhaustion: for all p, either 1 or 0

i.e., every propositional parameter is either true or false – and maybe both. Then it is not difficult to check that, again, the same holds for every sentence, A. That is, nothing takes the value n.

(p.148, section 8.4.9)

In section 8.4.10 we saw that exhaustion-constrained FDE is equivalent to LP (see section 7.4). Priest seems to be saying that if in KFDE we add the world designations to the exhaustion constraint, we can get KLP. But I am not sure if that means it would be:

Exhaustion: for all p, either w1 or w0

or what else it would be.]

In the context of the relational semantics, LP is obtained by requiring that, for all p, either pρ1 or pρ0. (See 8.4.9.) The same is true with the appropriate subscript w on ρ for KLP.

(246)

[contents]

11a.4.8

[Obtaining KK3]

[By adding a possible worlds exclusion constraint to KFDE we can get KK3.]

[Now recall the exclusion constraint from section 8.4.6:

Exclusion: for no p, 1 and 0

| i.e., no propositional parameter is both true and false. Then it is not difficult to check that the same holds for every sentence, A. That is, nothing takes the value b.

(p.147-148, section 8.4.6)

In section 8.4.7 we saw that this gives us a logic equivalent to K3. So similarly to what we did above in section 11a.4.7, we can obtain KK3 by adding the possible worlds exclusion constraint to KFDE. So maybe it is formulated:

Exclusion: for no p, w1 and w0

But I have no idea how it should be written.]

In the context of the relational semantics, K3 is obtained by requiring that, for all p, not both pρ1 and pρ0. (See 8.4.6.) The same is true with the appropriate subscript w on ρ for KK3.

(246)

[contents]

11a.4.9

[Obtaining K]

[By adding both the exhaustion and exclusion constraints to KFDE, we get the classical modal logic K.]

[Next recall from section 8.4.12 that when we add both the exhaustion and exclusion constraints to FDE, we get something equivalent to classical logic. Thus by adding both the exhaustion and exclusion constraints to KFDE, we get the classical modal logic K.]

If we add both conditions in the non-modal case, we get classical logic. In the modal case, we get the classical modal logic K.

(247)

[contents]

11a.4.10

[Extensions of Our Many-Valued Modal Logics]

[We can apply the world accessibility relation constraints (like ρ, σ, τ, etc) to our specific many-valued modal logics to get such extensions as: KFDEρ, KLPρτ, KK3σ and so forth.]

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

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.

τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.

η (eta), extendability: for all w1, there is a w2 such that w1Rw2.

(p.36, section 3.2.3)

We saw in section 11a.2.8 that we can apply them to a many-valued modal logic KL to derive stronger logics, like KLρ, KLσ, KLρτ, and so forth. Thus we can apply them to our specific many-valued modal logics to get such extensions as: KFDEρ, KLPρτ, KK3σ and so forth.]

All the many-valued modal logics may be extended by adding the constraints on the accessibility relation ρ, σ and τ, to give KFDEρ, KLPρτ, KK3σ, etc.

(247)

[contents]

11a.4.11

[The Lack of Logical Truths in KFDE and KK3.]

[There are no logical truths (tautologies) in KFDE and KK3, because to be a logical truth means that the formula is true under all interpretations. But in KFDE and KK3, under any interpretation, every formula can be valued neither true nor false, and thus no formulas can be logical truths.]

[I may not grasp the next idea. Recall from section 1.3.4 that:

A is a logical truth (tautology) (⊨ A) iff it is a semantic consequence of the empty set of premises (φA), that is, every interpretation makes A true.

(p.5, section 1.3.4)

Now, in KFDE and KK3, there is always the option that a formula have neither value. That means that in these logics (and their extensions), they can never be logical truths, because that requires the formula to be true under all interpretations.]

Note that KFDE, KK3, and all their normal extensions have no logical truths. To see this, just consider the interpretation with one world, w, such that wRw, and for all p, neither pρw1 nor pρw0. An easy induction shows the same to be true for all formulas. (Details are left as an exercise.)

(247)

[contents]

11a.4.12

[The Monotonicity of These Logics]

[The interpretations for the logics of the family we are considering are monotonic.]

[Priest’s next point is that all of the interpretations for the logics here are monotonic. But this is not a concept I know about yet or could find explained elsewhere in this text, so I will need to come back to it later.]

Note also that interpretations for any logic in the family we are considering is monotonic, in the following sense. Let 12 iff the two interpretations have the same worlds and accessibility relation, and, in addition, for all propositional parameters, p, and all worlds, w:

if pρ1w1 then pρ2w1

if pρ1w0 then pρ2w0

where ρ1 and ρ2 are the evaluation relations of 1 and 2, respectively. If 12, the displayed conditions obtain for an arbitrary formula, A. The proof is by a simple induction, which is left as an exercise.

(247)

[contents]

11a.4.13

[Corollary]

[A corollary of this is “that ⊨K A iff ⊨KLPA (and similarly for Kρ and KLPρ, etc.)” (247).]

[Priest continues this discussion, but for the same reasons as above, I will need to come back to it later.]

A corollary is that ⊨K A iff ⊨KLPA (and similarly for Kρ and KLPρ, etc.). From right to left, the result is straightforward, since any interpretation of K is an interpretation of KLP. For the converse, suppose that ⊭KLP A. Then there is an interpretation, 2, such that A does not hold at some world, w0, in 2 (i.e., it is not the case that Aρw01). Let 1 be any classical interpretation obtained from 2 simply by resolving contradictory propositional parameters one way or the other. That is, when pρ2w1 and pρ2w0, only one of these holds for ρ1w. Then 12. By monotonicity, A does not hold at w0 in 1; and 1 is an interpretation for K.

(247)

[contents]

From:

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

### Priest (11a.2) An Introduction to Non-Classical Logic, ‘General Structure,’ summary

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

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

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

Part I:

Propositional Logic

11a.

Appendix: Many-valued Modal Logics

11a.2

General Structure

Brief summary:

(11a.2.1) A “propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c C}⟩, where V is the set of semantic values,  D V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (242-243). (11a.2.2) We will assume that the set of truth-values of V are ordered from lesser to greater (or to equal than: ≤) and that “every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering” (242). (11a.2.3) Our many-valued modal logic adds to many-valued logic “the monadic operators, □ and ◊ in the usual way” (242). (11a.2.4) “An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V” (242). (11a.2.5) The value of a connective on formulas in a certain world is determined by the value generated by the corresponding connective function operating recursively on the truth values of those formulas in that particular world: “if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))” (242). (11a.2.6) The truth-conditions for the modal operators are:

vw(□A) = Glb{vw(A) : wRw′}

vw(◊A) = Lub{ vw(A) : wRw′}

(11a.2.7) Validity is defined in the following way: “Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ ” (242). (11a.2.8) Our many-valued modal logic is called KL, and we can apply the accessibility relation constraints on it to derive stronger logics, like KLρ, KLσ, KLρτ, and so forth.

Contents

11a.2.1

[The Structure]

11a.2.2

[Certain Assumptions: V-Ordering and Bounds]

11a.2.3

11a.2.4

[The Structure of Many-Valued Modal Logic]

11a.2.5

[Connective Evaluation]

11a.2.6

[Modal Operator Evaluation]

11a.2.7

[Validity]

11a.2.8

[KL and Its Constraints]

Summary

11a.2.1

[The Structure]

[A “propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c C}⟩, where V is the set of semantic values, D V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (242-243).]

[We will now call to mind the structure for many-valued semantics from section 7.2. It is:

V, D, {fc : c C}⟩

V is the set of assignable truth values. is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic).  C is the set of connectives. c is some particular connective. And fc is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. In a classical bivalent logic,

V = {1, 0}

D = {1}

C = {¬, ∧, ∨, ⊃, ≡} (but we have redefined ≡)

fc; c C = {f¬, f, f, f}

with the connective functions being formulated in the classical way. So for example:

f¬ is a one-place function such that f¬(0) = 1 and f¬(1) = 0;

f is a two-place function such that f(x, y) = 1 if x = y = 1, and f(x, y) = 0 otherwise [...]

 f¬ 1 0 0 1

 f∧ 1 0 1 1 0 0 o o

(pp.120-121, section 7.2.2)

With this structure in mind, we can vary the components, like adding truth-values, adding designated values, and changing the functions for connectives, to generate various other sorts of logics. (See section 7.3 for K3 and Ł3 and section 7.4 for LP and RM3.) We could also do this for continuum valued logics (see section 11.4).]

As we observed in 7.2, semantically, a propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c C}⟩, where V is the set of semantic values, D V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation.

(242-243)

[contents]

11a.2.2

[Certain Assumptions: V-Ordering and Bounds]

[We will assume that the set of truth-values of V are ordered from lesser to greater (or to equal than: ≤) and that “every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering” (242).]

[(ditto)]

It is standard for V to come with an ordering, ≤. We will assume in what follows that this is so. We also assume that every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering.

(242)

[contents]

11a.2.3

[Our many-valued modal logic adds to many-valued logic “the monadic operators, □ and ◊ in the usual way” (242).]

[(ditto)]

The language of a many-valued modal logic is the same as that of the many-valued logic, except that it is augmented by the monadic operators, □ and ◊ in the usual way.

(242)

[contents]

11a.2.4

[The Structure of Many-Valued Modal Logic]

[“An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V” (242).]

[Recall again from section 11a.2.1 the structure of logics: ⟨V, D, {fc : c C}⟩. By having more than the normal two truth-values in V, we obtain a many-valued logic. When the V in this structure produces a many-valued logic, (along with the other changes in the other parts of the structure needed for this purpose), then we can call the whole formulation ⟨V, , {fc : c C}⟩ SL, for “a structure for a many-valued logic, L” (242). Now recall the structure for modal logics from section 2.3.3:

An interpretation for this language is a triple ⟨W, R, v⟩. W is a non-empty set. Formally, W is an arbitrary set of objects. Intuitively, its members are possible worlds. R is a binary relation on W (so that, technically, R W×W). Thus, if u and v are in W, R may or may not relate them to each other. If it does, we will write uRv, and say that v is accessible from u. Intuitively, R is a relation of relative possibility, so that uRv means that, relative to u, situation v is possible. υ is a function that assigns a truth value (1 or 0) to each pair comprising a world, w, and a propositional parameter, p. We write this as vw(p) = 1 (or vw(p) = 0). Intuitively, this is read as ‘at world w, p is true (or false)’.

(p.21, section 2.3.3)

So now we want to combine the modal logic structure ⟨W, R, v⟩ with the many-valued logic structure ⟨V, , {fc : c C}⟩, which is abbreviated as SL, and thus we now have: ⟨W, R, SL, v⟩.]

An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V.

(242)

[contents]

11a.2.5

[Connective Evaluation]

[The value of a connective on formulas in a certain world is determined by the value generated by the corresponding connective function operating recursively on the truth values of those formulas in that particular world: “if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))” (242).]

[Recall from section 11a.2.1 above how we reviewed the semantics for the connective functions. For example, the connective function for conjunction was:

f is a two-place function such that f(x, y) = 1 if x = y = 1, and f(x, y) = 0 otherwise

(p.120, section 7.2.2)

We now do something similar but now including worlds. So, the value of a connective of formulas in a certain world is determined by the value generated by the corresponding connective function operating on the truth values of those formulas in that particular world.]

The truth conditions for the many-valued connectives at a world simply deploy the functions fc. Thus, if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))). (So if c is conjunction, vw(A B) = f(vw(A), vw(B)).)

(242)

[contents]

11a.2.6

[Modal Operator Evaluation]

[The truth-conditions for the modal operators are: vw(□A) = Glb{vw(A) : wRw′} ; vw(◊A) = Lub{ vw(A) : wRw′}.]

[I will probably missummarize the next ideas, so it is best to skip to the quotation. We will formulate the truth-conditions for the modal operators in our many-valued modal logic. Let us consider two cases, one where we have fuzzy values and another where we have FDE values. We want to know the value of □A in world 1. Suppose there is another world 2, and they have access to themselves and each other. In world 1, A has the truth value 0.5, and in world 2 A has the value 0.25. From what I can tell, □A in world 1 would be valued 0.25, but I am not sure. I am guessing that we take all the values for A in all the worlds, and we look for the least value. The real formula is: vw(□A) = Glb{vw(A) : wRw′}, and I am not entirely sure what it means. Suppose instead that we have FDE values, and in world 1 A is 1 and in world 2 A is both 1 and 0. I would guess that □A would then be 0. But I am not sure what to say if in world 2 A has neither value. I would guess that □A would then also have neither value, but I am not sure how it works with the scaling of values outside fuzzy logic. Now let us consider ◊A, but we will keep the above value assignments. In the fuzzy interpretation, it would seem that  ◊A in world 1 would be 0.5, because the truth-condition is: vw(◊A) = Lub{vw(A) : wRw′}. But I am guessing. And for the many-valued interpretation, I would guess that  ◊A in world 1 would be 1. I am sorry that I am simply guessing here. The quote follows.]

The natural generalisation of the two-valued truth conditions for the modal operators is as follows:1

vw(□A) = Glb{vw(A) : wRw′}

vw(◊A) = Lub{vw(A) : wRw′}

(242)

1. Semantically, □ and ◊ are forms of (respectively) universal and particular quantifiers over worlds. The following truth conditions are the obvious analogues of the truth conditions for these quantifiers in many-valued logic. (See Part II, 21.3.)

(242)

[contents]

11a.2.7

[Validity]

[Validity is defined in the following way: “Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ ” (242).]

[Validity seems to still be defined as preservation of the designated values in all worlds and interpretations.]

Validity is naturally defined as follows:

Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ .

(242)

[contents]

11a.2.8

[KL and Its Constraints]

[Our many-valued modal logic is called KL, and we can apply the accessibility relation constraints on it to derive stronger logics, like KLρ, KLσ, KLρτ, and so forth.]

[Recall the two-valued modal semantics we examined in section 2.3. We said in section 2.1 that this modal logic is called K (for Kripke). Priest says that our many-valued version is an analog to it, and we call our many-valued modal logic KL. Next recall from section 3.2.3 the constraints on the accessibility relation that generate variations of a modal logic:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.

τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.

η (eta), extendability: for all w1, there is a w2 such that w1Rw2.

(p.36, section 3.2.3)

Priest says now that we can apply these constraints to KL as well.]

This gives the analogue of the two-valued modal logic K. Call it KL. Stronger logics can be obtained by the addition of constraints on the accessibility relation, such as reflexivity (ρ), symmetry (σ), transitivity (τ), giving the logics KLρ, KLσ, KLρτ, etc. (See ch.3.)

(242)

[contents]

From:

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