Priest (7.10) An Introduction to Non-Classical Logic, ‘Supervaluations, Modality and Many-valued Logic,’ summary

 

by Corry Shores

 

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

 

 

 

 

Summary of

 

Graham Priest

 

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

 

Part I:

Propositional Logic

 

7.

Many-Valued Logics

 

7.10

Supervaluations, Modality and Many-valued Logic

 

 

 

 

Brief summary:

(7.10.1) We turn now to two matters that are related to Aristotle’s argument for truth-value gaps on the basis of future contingents. (7.10.2) We will probably not want all statements about the future to be valueless, as many statements can be determinable now as true or false. Thus we need excluded middle to hold in many cases for statements about the future. And since it does not hold in “K₃ or Ł₃, these logics do not appear to be the appropriate ones for future statements” (133). (7.10.3) We can use a technique called supervaluation to produce a logic that is better suited to accommodate both valued and valueless statements about the future. “Let v be any K₃ interpretation. Define v ≤ v′ to mean that v′ is a classical interpretation that is the same as v, except that wherever v(p) is i, v′(p) is either 0 or 1. (So v′ ‘fills in all the gaps’ in v.) Call v′ a resolution of v. Define the supervaluation of v, v⁺, to be the map such that for every formula, A: 

v⁺(A) = 1 iff for all v′ such that v ≤ v′, v′(A) = 1

v⁺(A) = 0 iff for all v′ such that v ≤ v′, v′(A) = 0

v⁺(A) = i otherwise

The thought here is that A is true on the supervaluation of v; just in case however its gaps were to get resolved (and, in the case of future contingents, will get resolved), it would come out true. We can now define a notion of validity as something like ‘truth preservation come what may’, Σ ⊨^S A (supervalidity), as follows:

Σ ⊨^S A iff for every v, if v⁺(B) is designated for all B ∈ Σ ⊨^S, v⁺(A) is designated

(where the designated values here are as for K₃),” namely, just 1 (pp.133-134). (7.10.4) “A fundamental fact is that Σ ⊨^S A iff A is a classical consequence of Σ. (In particular, therefore, ⊨^S A ∨ ¬A even though A may be neither true nor false!)” (134). (7.10.5) Classical validity and supervaluational validity hold when conclusions are understood to be a singular formula, but it does not hold for multiple-conclusion validity. For instance,

A ∨ B ⊨ A, B

is classically valid but not supervaluationally valid. (7.10.5a) Priest next shows how we can avoid the misalignment of classical and supervaluational validity for multiple conclusions by redefining supervaluational validity in the following way: “Define an inference to be valid iff, for every K₃ interpretation, v, every resolution of v that makes every premise true makes some (or, in the single conclusion case, the) conclusion true. Since the class of resolutions of all K₃ interpretations is exactly the set of classical evaluations, this gives exactly classical logic (single or multiple conclusion, as appropriate)” (134-135). (7.10.5b) To give an LP logic corresponding to the K₃ logic from supervaluation, we use a technique called subvaluation: “we will use ⊨_S instead of ⊨^S (and call this subvalidity). This time, A ⊨_S Σ iff the multiple conclusion inference from A to Σ is classically valid (and a fortiori for single conclusion inferences)” (135). (7.10.5c) Priest next notes that the above subvaluational technique of LP does not work for multi-premise inferences. For example, A, B ⊨ A ∧ B is classically valid but not subvaluationally valid. (7.10.5d) The different super/sub-valuational techniques render different notions of validity, and so we need to ask, “In the case of future contingents, for example, are we interested in preserving actual truth value, truth value we can ‘predict now’, or ‘eventual’ truth value?” (136) Priest notes that our answer can depend on why we think that gaps or gluts arise in such situations and on the sort of application we have in mind. (7.10.6) For Łukasiewicz, a statement about a future contingent says something that possibly may happen, but it can be otherwise. Thus that statement of the future contingent with the possibility operator is true but with the necessity operator is false.

 

f◊
1	1
i	1
0	0

 

Defining □A in the standard way, as ¬◊¬A, gives it the truth table:

 

f□
1	1
i	0
0	0

(136)

(7.10.7) The above definitions for the modal operators give us a modal logic that captures some of Aristotle’s thinking on future contingency, like p ⊨_Ł3 □p, but it betrays others, like ◊A, ◊B ⊨_Ł3 ◊(A ∧ B). (7.10.8) “[N]one of the modal logics that we have looked at (nor conditional logics, nor intuitionist logic) is a finitely many-valued logic” (137). (7.10.9) By using uniform substitution, we can render every logic into an infinitely many-valued logic. “A uniform substitution of a set of formulas is the result of replacing each propositional parameter uniformly with some formula or other (maybe itself). Thus, for example, a uniform substitution of the set {p, p ⊃ (p ∨ q)} is {r ∧ s, (r ∧ s) ⊃ ((r ∧ s) ∨ q)}. A logic is closed under uniform substitution when any inference that is valid is also valid for every uniform substitution of the premises and conclusion. All standard logics are closed under uniform substitution” (137). (7.10.10) “[E]very logical consequence relation, ⊢, closed under uniform substitution, is weakly complete with respect to a many-valued semantics. That is, ⊢A iff A is logically valid in the semantics” (137).

 

 

 

 

 

Contents

 

7.10.1

[Matters Related to Future Contingents]

 

7.10.2

[The Insufficiency of K₃ or Ł₃ for Determinable Statements about the Future]

 

7.10.3

[Supervaluation and Supervalidity]

 

7.10.4

[Supervaluational Consequence as Classical Consequence]

 

7.10.5

[The Misalignment of Classical and Supervaluational Validity for Multiple-Conclusions]

 

7.10.5a

[A Remedy for the Multiple-Conclusion Validity Misalignment]

 

7.10.5b

[LP and Subvaluation]

 

7.10.5c

[The Misalignment of Classical and Subvaluational Validity for Multi-Premise Inferences]

 

7.10.5d

[Philosophical Issuer Regarding Super/Sub-Valuational Validity]

 

7.10.6

[Evaluating Possibility and Necessity for Future Contingents]

 

7.10.7

[Non-Aristotelian Elements of the Modal Logic]

 

7.10.8

[Our Modal Logics So Far as Not Finitely Many-Valued]

 

7.10.9

[Uniform Substitution and Infinitely Many-Valued Logic]

 

7.10.10

[Uniform Substitution and Completeness]

 

 

 

 

 

 

Summary

 

7.10.1

[Matters Related to Future Contingents]

 

[We turn now to two matters that are related to Aristotle’s argument for truth-value gaps on the basis of future contingents.]

 

[Recall from the previous section 7.9 that we were discussing Aristotle’s argument for truth-value gaps on the basis of future contingents. Priest says now that we will consider two related matters.]

Let us finish with two other matters that arise in connection with Aristotle’s argument of the previous section, though they have wider implications.

(133)

[contents]

 

 

 

 

 

 

7.10.2

[The Insufficiency of K₃ or Ł₃ for Determinable Statements about the Future]

 

[We will probably not want all statements about the future to be valueless, as many statements can be determinable now as true or false. Thus we need excluded middle to hold in many cases for statements about the future. And since it does not hold in “K₃ or Ł₃, these logics do not appear to be the appropriate ones for future statements” (133).]

 

[I might have the next idea wrong. It could be the following. Some might claim that future contingents are exceptional cases that call for truth-value gaps, but they might also claim that not all statements about the future are valueless. (For, certain future events might be determinable in advance.) So that means we need excluded middle to hold in many cases regarding statements about the future. Now, it does not hold for K₃ or Ł₃. Thus “these logics do not appear to be the appropriate ones for future statements” (133).]

First, those who have taken future contingents to be neither true nor false, like Aristotle, have not normally taken all statements about the future to be truth-valueless – only statements about states of affairs that are as yet undetermined have that status. In particular, instances of the law of excluded middle, S ∨ ¬S, are usually endorsed, even if S is a future contingent. Since this is not valid in K₃ or Ł₃, these logics do not appear to be the appropriate ones for future statements.

(133)

[contents]

 

 

 

 

 

 

7.10.3

[Supervaluation and Supervalidity]

 

[We can use a technique called supervaluation to produce a logic that is better suited to accommodate both valued and valueless statements about the future. “Let v be any K₃ interpretation. Define v ≤ v′ to mean that v′ is a classical interpretation that is the same as v, except that wherever v(p) is i, v′(p) is either 0 or 1. (So v′ ‘fills in all the gaps’ in v.) Call v′ a resolution of v. Define the supervaluation of v, v⁺, to be the map such that for every formula, A:  v⁺(A) = 1 iff for all v′ such that v ≤ v′, v′(A) = 1 ; v⁺(A) = 0 iff for all v′ such that v ≤ v′, v′(A) = 0 ; v⁺(A) = i otherwise . The thought here is that A is true on the supervaluation of v; just in case however its gaps were to get resolved (and, in the case of future contingents, will get resolved), it would come out true. We can now define a notion of validity as something like ‘truth preservation come what may’, Σ ⊨^S A (supervalidity), as follows: Σ ⊨^S A iff for every v, if v⁺(B) is designated for all B ∈ Σ ⊨^S, v⁺(A) is designated (where the designated values here are as for K₃),” namely, just 1 (pp.133-134).]

 

[We will now consider supervaluations, which we saw in Nolt’s Logics section 15.3.1. We begin with a K₃ interpretation. Whenever there is an i valuation, we change it either to 0 or 1. “Define v ≤ v′ to mean that v′ is a classical interpretation that is the same as v, except that wherever v(p) is i, v′(p) is either 0 or 1. (So v′ ‘fills in all the gaps’ in v)” (133). Whenever we fill in those gaps, we call it a resolution of the interpretation. (For the next concepts, I may not follow so well. So let me appeal first to Nolt’s manner of explanation. The following comes from the brief summary of section 15.3.1:

A three-valued semantics was one way to deal with a number of situations where bivalence was unsatisfactory. A third value, I or indeterminate, was used. But instead of that third value, we can keep just T or F, although certain formulas can be assigned neither of these two values. These cases of no value are called truth value gaps. To evaluate the truth value of formulas with component truth value gaps, we can use a technique invented by Bas van Fraassen called supervaluation. First we begin by making a truth table and putting in the values we know, but we place gaps where there is no value. This is called a partial valuation. Next, we fill those gaps with T or F such that every possible combination of values is given. These are called classical completions. Finally, we make a supervaluation of these classical completions in the following way. If all the classical completions compute the formula as true, then the supervaluation is true. If all the classical completions compute the formula as false, then the supervaluation is false. And if not all the classical completions are either entirely T or F, then the supervaluation does not assign a value to the formula. Here is the formal definition of supervaluation:

DEFINITION A supervaluation model of a formula or set of formulas consists of

1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.

2. A supervaluation V_S of S that assigns truth values V_S (Φ) to formulas Φ according to these rules:

V_S (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;

V_S (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;

V_S  assigns no truth value to Φ otherwise.

(Nolt 415-416, section 15.3.1, boldface in his quotation is his, boldface in my summary is mine)

It seems like Priest might be saying something similar. So let us look at it:

v⁺(A) = 1 iff for all v′ such that v ≤ v′, v′(A) = 1

v⁺(A) = 0 iff for all v′ such that v ≤ v′, v′(A) = 0

v⁺(A) = i otherwise

Here he seems to be saying that we take all possible resolutions, which in Nolt’s terminology would seem to be all “classical completions.” The first rule seems to be that a formula is supervaluationally true if all the resolutions make the i be 1, and so on for false and i. It might seem odd that if we try out both values for the atomic formulas that we will still have consistent outcomes. What we noted in the Nolt section was that there will be cases of more complex formulas that will have the same value regardless of how the i values are assigned for constituent atomic formulas. So the example was:

(P ∨ Q) & (R ∨ S)

Here P and R are both true, but Q and S are i. So we begin with the partial valuation, where their values are left blank.

Next, we fill out all the possible combinations for classical values for Q and S. This gives us four resolutions, using Priest’s terminology.

So while Q and S do not have the same value in all resolutions, the complex formula ‘(P ∨ Q) & (R ∨ S)’ does, namely, it is always true. Thus, my guess is that what Priest is saying is that we would say:

v⁺((P ∨ Q) & (R ∨ S)) = 1

v⁺(P) = 1

v⁺(R) = 1

v⁺(Q) = i

v⁺(S) = i

Next we define validity, but I may summarize this incorrectly. It seems we keep the designated values (those that are preserved in valid inferences) from K₃, which is just 1 (see section 7.3.2). So an inference is valid if when the premises have the designated, so too does the conclusion.]

A logic better in this regard can be obtained by a technique called supervaluation. Let v be any K₃ interpretation. Define v ≤ v′ to mean that v′ is a classical interpretation that is the same as v, except that wherever v(p) is i, v′(p) is either 0 or 1. (So v′ ‘fills in all the gaps’ in v.) Call v′ a resolution of v. Define the supervaluation of v, v⁺, to be the map such that for every formula, A:

v⁺(A) = 1 iff for all v′ such that v ≤ v′, v′(A) = 1

v⁺(A) = 0 iff for all v′ such that v ≤ v′, v′(A) = 0

v⁺(A) = i otherwise

The thought here is that A is true on the supervaluation of v; just in case however its gaps were to get resolved (and, in the case of future contingents, | will get resolved), it would come out true. We can now define a notion of validity as something like ‘truth preservation come what may’, Σ ⊨^S A (supervalidity), as follows:

Σ ⊨^S A iff for every v, if v⁺(B) is designated for all B ∈ Σ ⊨^S, v⁺(A) is designated

(where the designated values here are as for K₃).

(133-134)

[contents]

 

 

 

 

 

 

7.10.4

[Supervaluational Consequence as Classical Consequence]

 

[“A fundamental fact is that Σ ⊨^S A iff A is a classical consequence of Σ. (In particular, therefore, ⊨^S A ∨ ¬A even though A may be neither true nor false!)” (134)]

 

[Let us next see what happens when we supervaluate the excluded middle formulation. We have that A is indeterminate.

⊨S A ∨ ¬A
A	A	∨	¬	A
i
i

 

Now let us see what happens when we supervaluate it. We consider all possible classical valuations.

⊨S A ∨ ¬A
A	A	∨	¬	A
1
0

 

So

 

zz⊨S A ∨ ¬A
A	A	∨	¬	A
1	1			1
0	0			0

 

With the negation.

 

zz⊨S A ∨ ¬A
A	A	∨	¬	A
1	1		0	1
0	0		1	0

 

But this still supervaluates as true, even though the term A begins as indeterminate!

 

zz⊨S A ∨ ¬A
A	A	∨	¬	A
1	1	1	0	1
0	0	1	1	0

 

Priest then gives the argument for why all supervaluational consequences are classical consequences. (See the quotation below.)]

A fundamental fact is that Σ ⊨^S A iff A is a classical consequence of Σ. (In particular, therefore, ⊨^S A ∨ ¬A even though A may be neither true nor false!) The argument for this is as follows. First, suppose that the inference is not classically valid; then there is a classical interpretation that makes all the members of Σ true and A false. But the only resolution of v is v itself. So every resolution of v makes all the premises true and the conclusion false. That is, for all B ∈ Σ, v⁺(B) = 1, and v⁺(A) = 0. Hence, Σ ⊭S A.¹⁰ Conversely, suppose that Σ ⊭^S A. Then there is a v such that for all B ∈ Σ, v⁺(B) = 1 and v⁺(A) ≠ 1. Consequently, there is some resolution µ ≥ v such that µ(A) = 0, but for all B ∈ Σ, µ(B) = 1. Since µ is a classical interpretation, the inference is not classically valid.

(134)

10. In certain contexts, there may be reason to suppose that not all resolutions of an evaluation are ‘genuine possibilities’. In that case, one may wish to restrict the supervaluation of an evaluation to an appropriate subclass of its resolutions. If one does so, this half of the proof may break down, and the inferences that are supervaluation valid may actually extend the classically valid inferences.

(134)

[contents]

 

 

 

 

 

 

7.10.5

[The Misalignment of Classical and Supervaluational Validity for Multiple-Conclusions]

 

[Classical validity and supervaluational validity hold when conclusions are understood to be a singular formula, but it does not hold for multiple-conclusion validity. For instance, A ∨ B ⊨ A, B is classically valid but not supervaluationally valid.]

 

[I will probably missummarize the next idea. It might be the following. The seeming alignment between classical validity and supervaluational validity that we saw above in section 7.10.4 holds only when conclusions are understood to be a singular formula. Were they to involve multiple-conclusion validity, then the two systems will not always align. For instance, A ∨ B ⊨ A, B is classically valid but not supervaluationally valid.]

The alignment between classical validity and supervaluation validity is not, in fact, as clean as 7.10.4 makes it appear. For any logic, including classical logic, one can define a natural notion of multiple-conclusion validity. For this, the conclusions, like the premises, may be an arbitrary set of formulas (not just a single formula) and the inference is valid iff every interpretation (of the kind appropriate for the logic) that makes every premise true makes some conclusion true. Thus, in classical logic (and ignoring set braces for the conclusions as well as the premises), A ∨ B ⊨ A, B. This inference is not valid for ⊨^S. To see this, just consider an interpretation, v, such that v(p) = i. Then v⁺(p ∨ ¬p) = 1, but v⁺(p) = v⁺(¬p) = i.

(134)

[contents]

 

 

 

 

 

 

7.10.5a

[A Remedy for the Multiple-Conclusion Validity Misalignment]

 

[Priest next shows how we can avoid the misalignment of classical and supervaluational validity for multiple conclusions by redefining supervaluational validity in the following way: “Define an inference to be valid iff, for every K₃ interpretation, v, every resolution of v that makes every premise true makes some (or, in the single conclusion case, the) conclusion true. Since the class of resolutions of all K₃ interpretations is exactly the set of classical evaluations, this gives exactly classical logic (single or multiple conclusion, as appropriate)” (134-135).]

 

[(ditto)]

A slightly different way of proceeding avoids this consequence. Define an inference to be valid iff, for every K₃ interpretation, v, every resolution of v that makes every premise true makes some (or, in the single conclusion case, the) conclusion true. Since the class of resolutions of all K₃ | interpretations is exactly the set of classical evaluations, this gives exactly classical logic (single or multiple conclusion, as appropriate).¹¹

(134-135)

11. Note that supervaluation techniques can be applied to the logic Ł₃, but are less appropriate. Supervaluation is essentially a gap-filling exercise. It should not destabilise things that already have a determinate truth. A resolution of a K₃ interpretation preserves classical truth values in the appropriate way. That is, if v ≤ v′, and v(A) is 0 or 1, v′(A) has the same value. The same is not true of Ł₃. Similarly, subvaluations (about to be defined) do not destabilise classical values in LP, but they may do so in RM₃. See 7.14, problem 4.

(135)

[contents]

 

 

 

 

 

 

7.10.5b

[LP and Subvaluation]

 

[To give an LP logic corresponding to the K₃ logic from supervaluation, we use a technique called subvaluation: “we will use ⊨_S instead of ⊨^S (and call this subvalidity). This time, A ⊨_S Σ iff the multiple conclusion inference from A to Σ is classically valid (and a fortiori for single conclusion inferences)” (135).]

 

[Priest next shows how a subvaluational technique gives us an LP rather than a K₃ logic as with supervaluation. I am not grasping the difference very well, so I will need to return to this later to fill out an explanation.]

It is worth noting that there is a technique dual to supervaluation for the logic LP. Given any LP interpretation, define ≤ and validity exactly as in 7.10.3 (remembering that the designated values have now changed). In this context, it is usual to use the term subvaluation rather than supervalutation; correspondingly, we will use ⊨_S instead of ⊨^S (and call this subvalidity). This time, A ⊨_S Σ iff the multiple conclusion inference from A to Σ is classically valid (and a fortiori for single conclusion inferences). The argument for this is as follows. First, suppose that the inference is not classically valid; then there is a classical interpretation that makes A true and every member of Σ false. But the only resolution of v is v itself. So every resolution of v makes the premise true and all the conclusions false. That is, for all B ∈ Σ, v⁺(B) = 0, and v⁺(A) = 1. Hence, A ⊭_S Σ.₁₂ Conversely, suppose that A ⊭_S Σ. Then there is a v such that v⁺(A) = 0, and for all B ∈ Σ, v⁺(B) = 0. Consequently, there is some resolution µ ≥ v such that µ(A) = 1, but for all B ∈ Σ, µ(B) = 0. Since µ is a classical interpretation, the inference is not classically valid.

(135)

12. Again, if one restricts the subvaluation to an appropriate class of its resolutions, this half of the proof may break down, and subvaluation validity may extend the classically valid inferences. (135)

[contents]

 

 

 

 

 

7.10.5c

[The Misalignment of Classical and Subvaluational Validity for Multi-Premise Inferences]

 

[Priest next notes that the above subvaluational technique of LP does not work for multi-premise inferences. For example, A, B ⊨ A ∧ B is classically valid but not subvaluationally valid.]

 

[(ditto)]

The result does not extend to multiple-premise inferences. Thus, in classical logic, A, B ⊨ A ∧ B. This inference is not valid for ⊨_S. Just consider an interpretation, v, such that v(p) = i. Then v⁺(p) = v⁺(¬p) = i, but v⁺(p ∧ ¬p) = 0. However, if validity is defined as in 7.10.5a, replacing K₃ with LP, then it coincides with classical validity, for the same reason.

(135)

[contents]

 

 

 

 

 

7.10.5d

[Philosophical Issuer Regarding Super/Sub-Valuational Validity]

 

[The different super/sub-valuational techniques render different notions of validity, and so we need to ask, “In the case of future contingents, for example, are we interested in preserving actual truth value, truth value we can ‘predict now’, or ‘eventual’ truth value?” (136) Priest notes that our answer can depend on why we think that gaps or gluts arise in such situations and on the sort of application we have in mind.]

 

[The different super/sub-valuational techniques render different notions of validity, and so we need to ask, “In the case of future contingents, for example, are we interested in preserving actual truth value, truth value we can ‘predict now’, or ‘eventual’ truth value?” (136) Priest notes that our answer can depend on why we think that gaps or gluts arise in such situations and on the sort of application we have in mind.]

Clearly, applying the super/subvaluation technique provides a number of different notions of validity. In deciding whether or not to apply the technique, and if so how, one has to decide what one wishes one’s notion | of validity to preserve: designated value under an interpretation, designated value under a super/subvaluation, or designated value under a resolution. In the case of future contingents, for example, are we interested in preserving actual truth value, truth value we can ‘predict now’, or ‘eventual’ truth value? Quite possibly, the answer may depend on why, exactly, gaps/gluts are supposed to arise in the application at hand. Conceivably, the answer may be different for different applications (e.g., future contingents and vagueness¹³).

(135-136)

13. For vagueness, see 11.3.7.

(136)

[contents]

 

 

 

 

 

 

7.10.6

[Evaluating Possibility and Necessity for Future Contingents]

 

[For Łukasiewicz, a statement about a future contingent says something that possibly may happen, but it can be otherwise. Thus that statement of the future contingent with the possibility operator is true but with the necessity operator is false.]

 

[I will probably missummarize the next idea. It might be the following. Łukasiewicz made his Ł₃ in response to the problem of future contingents. But he thought that when statements about the future have a truth value, those values are unalterable, and thus they are necessarily true or false. Now we will consider how to evaluate the modal operators, necessity and possibility. I cannot tell, but my guess is that these evaluations are especially for sentences regarding the future, but maybe it holds in all cases too. At any rate, statements about future contingents take the value i for Łukasiewicz. That means for him that whatever they are saying about the future, it is merely possible. So we would evaluate that future contingent statement with the possibility operator as true. But since what it says can be otherwise, it is not necessary, and thus that statement with the necessity operator would be false.]

Let us now turn to the second matter. This concerns the connection between modality and many-valued logic. Notwithstanding the issue concerning the law of excluded middle that we have just discussed, Łukasiewicz was motivated to construct his logic Ł₃ by the problem about future contingents. According to him, statements about the past and present are now unalterable in truth value. If they are true, they are necessarily true; if they are false, they are necessarily false. But future contingents, those things taking the value i, are merely possible. Things that are true are also possible, of course. He therefore augmented the language with a modal possibility operator, ◊, and gave it the following truth table:

 

f◊
1	1
i	1
0	0

 

Defining □A in the standard way, as ¬◊¬A, gives it the truth table:

 

f□
1	1
i	0
0	0

 

(136)

[contents]

 

 

 

 

 

7.10.7

[Non-Aristotelian Elements of the Modal Logic]

 

[The above definitions for the modal operators give us a modal logic that captures some of Aristotle’s thinking on future contingency, like p ⊨_Ł3 □p, but it betrays others, like ◊A, ◊B ⊨_Ł3 ◊(A ∧ B).]

 

[Priest notes some properties of the modal logic given above in section 7.10.6. He says that p ⊨_Ł3 □p, but he adds “This is not the Rule of Necessitation” (see section 4.4.6). And given Aristotle’s argumentation, this could be admissible. But there is something it validates that Aristotle would not allow, namely ◊A, ◊B ⊨_Ł3 ◊(A ∧ B). For, under this construction, we could make the ◊B be ¬◊A and thus ◊(A ∧ ¬A), which Aristotle would reject, because he would not allow exceptions to non-contradiction.]

These definitions give a modal logic that, in the light of modern modal logic, has some rather strange properties. For example, it is easy to check that p ⊨_Ł3 □p. (This is not the Rule of Necessitation.) Given the Aristotelian motivation, this may be acceptable. But there are other consequences that are certainly not. For example, it is easy to check that | ◊A, ◊B ⊨_Ł3 ◊(A ∧ B). This is not acceptable – even to an Aristotelian. It is possible that the first pope in the twenty-second century will be Chinese and possible that she will not. But it is not possible that she both will and will not be.

(137)

[contents]

 

 

 

 

 

 

7.10.8

[Our Modal Logics So Far as Not Finitely Many-Valued]

 

[“[N]one of the modal logics that we have looked at (nor conditional logics, nor intuitionist logic) is a finitely many-valued logic” (137). ]

 

[Priest’s next point is that “none of the modal logics that we have looked at (nor conditional logics, nor intuitionist logic) is a finitely many-valued logic” (137). He notes where the proof for this is, but we have not summarized any of those cited sections yet.]

In fact, none of the modal logics that we have looked at (nor conditional logics, nor intuitionist logic) is a finitely many-valued logic. The proof of this is essentially a version of the argument of 7.5.4, 7.5.5. The proof is given in 7.11.1–7.11.3.

(137)

[contents]

 

 

 

 

 

 

 

7.10.9

[Uniform Substitution and Infinitely Many-Valued Logic]

 

[By using uniform substitution, we can render every logic into an infinitely many-valued logic. “A uniform substitution of a set of formulas is the result of replacing each propositional parameter uniformly with some formula or other (maybe itself). Thus, for example, a uniform substitution of the set {p, p ⊃ (p ∨ q)} is {r ∧ s, (r ∧ s) ⊃ ((r ∧ s) ∨ q)}. A logic is closed under uniform substitution when any inference that is valid is also valid for every uniform substitution of the premises and conclusion. All standard logics are closed under uniform substitution” (137).]

 

[I do not follow the much of the next ideas. Somehow, by using uniform substitution, we can render every logic into an infinitely many-valued logic. See the quotation for details on how that works.]

There is a certain sense in which every logic can be thought of as an infinitely many-valued logic, however. A uniform substitution of a set of formulas is the result of replacing each propositional parameter uniformly with some formula or other (maybe itself). Thus, for example, a uniform substitution of the set {p, p ⊃ (p ∨ q)} is {r ∧ s, (r ∧ s) ⊃ ((r ∧ s) ∨ q)}. A logic is closed under uniform substitution when any inference that is valid is also valid for every uniform substitution of the premises and conclusion. All standard logics are closed under uniform substitution.¹⁴

(137)

14. The general reason is as follows. Suppose that some substitution instance of an inference is invalid. Then there is some interpretation, (appropriate for the logic in question), which makes the premises true and the conclusion untrue (at some world). Now consider the interpretation that is exactly the same as , except that it assigns to every parameter (at a world) the value of whatever formula was substituted for it (at that world) in . It is not difficult to check that the truth value of every formula (at every world) is the same in this interpretation as its substitution instance was in . Hence, the inference is invalid also.

(137)

[contents]

 

 

 

 

 

 

7.10.10

[Uniform Substitution and Completeness]

 

[“[E]very logical consequence relation, ⊢, closed under uniform substitution, is weakly complete with respect to a many-valued semantics. That is, ⊢A iff A is logically valid in the semantics” (137).]

 

[Like with the prior section, I am not following the ideas here. Please see the quotation.]

Now, it can be shown that every logical consequence relation, ⊢, closed under uniform substitution, is weakly complete with respect to a many-valued semantics. That is, ⊢A iff A is logically valid in the semantics. This is proved in 7.11.5. The semantics is somewhat fraudulent, though, since it involves taking every formula as a truth value. Moreover, the result can be extended to strong completeness (that is, to inferences with arbitrary sets of premises – not just empty ones) only under certain conditions.¹⁵

(137)

15. See Priest (2005b).

(137)

Priest, Graham, (2005b), ‘Many-Valued Logics’, in D. Borchet, ed., Encyclopedia of Philosophy, 2nd ed. (New York: Macmillan).

(597)

[contents]

 

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

 

 

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

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Deleuze Entry Directory]

[James Williams, Deleuze’s Philosophy of Time, Entry Directory]

 

[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.

 

.

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

 

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 K_FDE. “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 K_LP. If we ignore the value b in the non-modal case, we get the logic K₃. In the modal case, we get K_K3” (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) K_FDE has the same relational semantics but with world designations and with the following rules for the necessity and possibility operators:

v_w(A) = 1 iff Aρ_w1 and it is not the case that Aρ_w0

v_w(A) = b iff Aρ_w1 and Aρ_w0

v_w(A) = n iff it is not the case that Aρ_w1 and it is not the case that Aρ_w0

v_w(A) = 0 iff it is not the case that Aρ_w1 and Aρ_w0

 

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ0 or Bρ_w0

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

 

□Aρ_w1 iff for all w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some w′ such that wRw′, Aρ_w′0

 

◊Aρ_w1 iff for some w′ such that wRw′, Aρ_w′1

◊Aρ_w0 iff for all w′ such that wRw′, Aρ_w′0

(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 K_FDE we can get K_LP. (11a.4.8) By adding a possible worlds exclusion constraint to K_FDE we can get K_K3. (11a.4.9) By adding both the exhaustion and exclusion constraints to K_FDE, 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: K_FDEρ, K_LPρτ, K_K3σ and so forth. (11a.4.11) There are no logical truths (tautologies) in K_FDE and K_K3, because to be a logical truth means that the formula is true under all interpretations. But in K_FDE and K_K3, 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 K_LPρ, 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

[K_FDE, K_L, and K_K3]

 

11a.4.4

[The Relational Semantics of FDE]

 

11a.4.5

[The Relational Semantics of K_FDE]

 

11a.4.6

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

 

11a.4.7

[Obtaining K_LP]

 

11a.4.8

[Obtaining K_K3]

 

11a.4.9

[Obtaining K]

 

11a.4.10

[Extensions of Our Many-Valued Modal Logics]

 

11a.4.11

[The Lack of Logical Truths in K_FDE and K_K3.]

 

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 ↑ >↓

(greatest lower bound, reading upwards)

	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 ↓ < ↑

(least upper bound, reading downwards)

	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

[K_FDE, K_L, and K_K3]

 

[Our modal FDE logic is called K_FDE. “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 K_LP. If we ignore the value b in the non-modal case, we get the logic K₃. In the modal case, we get K_K3” (245).]

 

[Recall from section 11a.2.8 that a many-valued modal logic is called K_L. 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 v_w(p) = 1 (or v_w(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 K_L. The L is a many-valued logic, which has the structure:

a propositional many-valued logic is characterised by a structure ⟨V, D, {f_c : c ∈ C}⟩, where V is the set of semantic values, D ⊆ V is the set of designated values, and for each connective, c, f_c 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 f_cs; 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, {f_c : 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)

f_c; 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 K_L is more specifically K_FDE. 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 K_LP. 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, K_K3. (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)).]

K_FDE 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 K_LP. If we ignore the value b in the non-modal case, we get the logic K₃. In the modal case, we get K_K3.

(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 K_FDE]

 

[K_FDE has the same relational semantics but with world designations and with the following rules for the necessity and possibility operators: □Aρ_w1 iff for all w′ such that wRw′, Aρ_w′1 ; □Aρ_w0 iff for some w′ such that wRw′, Aρ_w′0 ; ◊Aρ_w1 iff for some w′ such that wRw′, Aρ_w′1 ; ◊Aρ_w0 iff for all w′ such that wRw′, Aρ_w′0 .]

 

[Priest next says that we can formulate K_FDE 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:

v_w(□A) = Glb{v_w′(A) : wRw′}

v_w(◊A) = Lub{v_w′(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:

□Aρ_w1 iff for all w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some w′ such that wRw′, Aρ_w′0

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 K_FDE formulated in the relational semantics.

 

v_w(A) = 1 iff Aρ_w1 and it is not the case that Aρ_w0

v_w(A) = b iff Aρ_w1 and Aρ_w0

v_w(A) = n iff it is not the case that Aρ_w1 and it is not the case that Aρ_w0

v_w(A) = 0 iff it is not the case that Aρ_w1 and Aρ_w0

 

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ0 or Bρ_w0

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

 

□Aρ_w1 iff for all w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some w′ such that wRw′, Aρ_w′0

 

◊Aρ_w1 iff for some w′ such that wRw′, Aρ_w′1

◊Aρ_w0 iff for all w′ such that wRw′, Aρ_w′0

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

]

K_FDE 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:

□Aρ_w1 iff for all w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some w′ such that wRw′, Aρ_w′0

◊Aρ_w1 iff for some w′ such that wRw′, Aρ_w′1

◊Aρ_w0 iff for all w′ such that wRw′, Aρ_w′0

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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 v_w(□A), that is Glb{v_w′(A) : wRw′}. This has four possible values.

1: In this case, for all w′ such that wRw′ the value of v_w′(A) is 1. So for all w′ such that wRw′, Aρ_w′1 and it is not the case that Aρ_w′0. In this case, the truth/falsity conditions give that □Aρ_w1 and it is not the case that □Aρ_w0, as required.

b: In this case, for all w′ such that wRw′, the value of v_w′(A) is 1 or b, and at least one is b. That is, for all w′ such that wRw′, Aρ_w′1 and for at least one such w′, Aρ_w′0. In this case, the truth/falsity conditions give that □Aρ_w1 and □Aρ_w0, as required.

n: In this case, for all w′ such that wRw′, the value of v_w′(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 Aρ_w′0 and for at least one such w′, it is not the case that Aρ_w′1. 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 v_w′(A) = 0, or there are w′ and w′′, such that wRw′, wRw′′, v_w′(A) = b and v_w′′(A) = n. In the first case, for all w′ such that wRw′, Aρw′0 and it is not the case that Aρ_w′1. In the second case, Aρ_w′′1 and Aρ_w′′0, and neither Aρ_w′′1 nor Aρ_w′′0. In either case, the truth/falsity conditions give that □Aρ_w0 and it is not the case that □Aρ_w1, as required.

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

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11a.4.7

[Obtaining K_LP]

 

[By adding a possible worlds exhaustion constraint to K_FDE we can get K_LP.]

 

[Recall the exhaustion constraint from section 8.4.9:

Exhaustion: for all p, either pρ1 or pρ0

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 K_FDE we add the world designations to the exhaustion constraint, we can get K_LP. But I am not sure if that means it would be:

Exhaustion: for all p, either pρ_w1 or pρ_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 K_LP.

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11a.4.8

[Obtaining K_K3]

 

[By adding a possible worlds exclusion constraint to K_FDE we can get K_K3.]

 

[Now recall the exclusion constraint from section 8.4.6:

Exclusion: for no p, pρ1 and pρ0

| 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 K_K3 by adding the possible worlds exclusion constraint to K_FDE. So maybe it is formulated:

Exclusion: for no p, pρ_w1 and pρ_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 K_K3.

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11a.4.9

[Obtaining K]

 

[By adding both the exhaustion and exclusion constraints to K_FDE, 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 K_FDE, 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.

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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: K_FDEρ, K_LPρτ, K_K3σ 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 K_L to derive stronger logics, like K_Lρ, K_Lσ, K_Lρτ, and so forth. Thus we can apply them to our specific many-valued modal logics to get such extensions as: K_FDEρ, K_LPρτ, K_K3σ 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 K_FDEρ, K_LPρτ, K_K3σ, etc.

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11a.4.11

[The Lack of Logical Truths in K_FDE and K_K3.]

 

[There are no logical truths (tautologies) in K_FDE and K_K3, because to be a logical truth means that the formula is true under all interpretations. But in K_FDE and K_K3, 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 (φ ⊨ A), that is, every interpretation makes A true.

(p.5, section 1.3.4)

Now, in K_FDE and K_K3, 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 K_FDE, K_K3, 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 pρ_w1 nor pρ_w0. An easy induction shows the same to be true for all formulas. (Details are left as an exercise.)

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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 ₁ ⪯ ₂ iff the two interpretations have the same worlds and accessibility relation, and, in addition, for all propositional parameters, p, and all worlds, w:

if pρ_1w1 then pρ_2w1

if pρ_1w0 then pρ_2w0

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

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11a.4.13

[Corollary]

 

[A corollary of this is “that ⊨_K A iff ⊨_KLPA (and similarly for Kρ and K_LPρ, 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 K_LPρ, etc.). From right to left, the result is straightforward, since any interpretation of K is an interpretation of K_LP. 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 Aρ_w01). Let ₁ be any classical interpretation obtained from ₂ simply by resolving contradictory propositional parameters one way or the other. That is, when pρ_2w1 and pρ_2w0, only one of these holds for ρ_1w. Then ₁ ⪯ ₂. By monotonicity, A does not hold at w₀ in ₁; and ₁ is an interpretation for K.

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From:

 

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