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

 

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.

 

 

 

 

 

 

Nolt (15.3.1) Logics, '[Basic technique of supervaluation],’ summary

 

by Corry Shores

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

 

[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[John Nolt, entry directory]

[Nolt, Logics, entry directory]

 

[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 5: Nonclassical Logics

 

Chapter 15: Mildly Nonclassical Logics

 

15.3 Supervaluations

 

15.3.1 [Basic technique of supervaluation]

 

 

 

 

Brief summary:

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 supervaluational 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, boldface his)

 

 

 

Summary

 

Nolt recalls that we can have certain motivations to reject simply two logical values, true and false, and to add some third one [see section 15.2]. But we have further motivations that would make us want to hold on to just the truth and falsity without adding a third or more. One solution would be to say that propositions can be either true, false, or neither true nor false. Sentences with this third option “are said to exhibit truth-value gaps” (Nolt 414 boldface his). [The idea here might be the following. We have reason to think that certain propositions do not fall neatly under the valuations of true or false. But we also have the intuition that truth is something that admits of no quantitative variation. It is absolute in some sense. So the notion of a third value might be mathematically modelable, but philosophically speaking it is too difficult to conceive and believe strongly in the reality of. So this solution keeps the idea that there are no other logical values than true and false. But it deals with the problematic cases by saying that they simply have neither of these values, rather than saying they have a third value.] Nolt will now examine how we can calculate truth values when a component proposition lacks a value (414).

 

[Recall Bochvar’s three-valued semantics from section 15.2 where any instance of a constituent indeterminate value would make the whole formulation indeterminate.] Nolt notes that one option “would be to declare all such formulas truth-valueless. The result would be just like Bochvar's three-valued semantics, except that in the truth tables where Bochvar’s logic has an ‘I’, this semantics would have a blank” (415). [There were other ways to arrange the truth values for three-valued semantics,] or instead we “could create truth tables like those for other forms of three-valued logic” (415). But, Nolt says, “this approach is hardly novel, differing in no essential respect from three-valued logics themselves” (415). [The problem with that might be the following. Suppose we simply mimicked the three-value truth tables, only replacing the ‘I’ value with a blank. We might then ask what really the difference would be between them other than how we philosophically interpret the meaning of the third option. In other worlds, this “gap” option would be little more than a different way to interpret the indeterminately three-valued semantics.]

 

Nolt says that instead we should consider Bas van Fraassen’s technique of supervaluation, which “unlike most multivalued logics, preserves the validity of all classical inference patterns” (Nolt 415). The first thing we note about supervaluations is partial value assignment, where we assign to sentences letters either T, F, or no value.

In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.)

(Nolt 415, boldface his)

[In other cases, we compute the truth value of complex formulas on the basis of the truth values of their component formulas.] In supervaluations, we will calculate the values of complex formulas by means of classical completions of partial valuations, rather than by calculating by means of “truth tables directly from the truth values of their components” (415).

 

[The basic idea with this technique seems to be the following. Although we might have truth gaps for certain constituent parts of a formula, that may not in the end have any effect on its overall value, were we to give it every possible classical valuation. And thus for some cases, we can determinately say that it maintains classical truth or falsity, despite the gaps in its internal values. So we begin with a partial valuation of a complex formula, meaning that some of the sentence letters will not originally be assigned a T or F value. The second step is a classical completion, where we exhaust every possible arrangement for these blank values to receive either T or F. The next thing we do is we see if regardless of every possible substitution, does the whole formula compute as either T or F, in all cases? If so, we say that the formula is T or F, depending on what all the evaluations compute it as.  If instead they are not all of one or the other value, then we say that it has no value. Giving it this final valuation based on the classical completions of its partial valuation is called its supervaluation.]

A more interesting method – and one that, unlike most multivalued logics, preserves the validity of all classical inference patterns-is the technique of supervaluations invented by Bas van Fraassen. In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.) The truth values of complex formulas, however, are not calculated by truth tables directly from the truth values of their components. Rather, the calculation takes into account all of what are called the classical completions of a given partial valuation.

 

Let S be any partial valuation. Then a classical completion of S is a classical valuation (one that assigns each sentence letter in the relevant formula or set of formulas one and only one of the values T or F) that fills in all the truth-value gaps left by S. In other words, a classical completion of S does not change any truth-value assignment to a sentence letter that has already been made by S, but merely supplements assignments made by S, giving each sentence letter of the given formula or set of formulas a truth value. Since each truth-value gap can be filled in by the assignment either of T or of F, each partial valuation, unless it is classical to begin with, has more than one classical completion. Consider, for example, the formula ‘(P∨Q)&(R∨S)’, and let S be the partial valuation of this formula that assigns the value T to both ‘P’ and ‘R’ but no value to ‘Q’ or ‘S’.

(Nolt 415, emphases his)

[Let us first try to give the partial valuation for this situation in a table format.

As we can see, we have only entered the data that we are sure of. The rest is left out, and thus this is a partial valuation. In the next step, we will fill in every possible combination of values for Q and S, and then we evaluate them on the basis of normal classical valuation rules, in this case for disjunction and finally for conjunction. The result will be that all possible classical completions of this partial valuation will make it true, which means that the gaps would not interfere with classical evaluation of this formula, as it will not matter what values the gaps might have.]

Then S has with respect to this formula four classical completions, corresponding to the four ways of assigning truth value to the sentence letters ‘Q’ and ‘R’. Each classical completion of S is represented by a horizontal line in the truth table below:

The columns under ‘P’ and ‘R’ list only T’s, because these are the values assigned by the partial valuation S, and they are retained in each classical completion.

 

To determine the truth value of the compound formula ‘(P∨Q)&(R∨S)’, we expand S into a new nonclassical valuation V_S called the supervaluation of S. | This is done by calculating the truth value of ‘(P∨Q)&(R∨S)’ on each of the classical completions of S, using the valuation rules of classical logic, as in the table above. If ‘(P∨Q)&(R∨S)’ is true on each of these classical completions (as the table shows that it is), then ‘(P∨Q)&(R∨S)’ is true on the supervaluation V_S. If it had been true on none of them, then it would have been false on V_S. And if it had been true on some but not others, then it would have been assigned no truth value on V_S.

 

A supervaluation, then, is constructed in two stages. First, we define a partial valuation S, which assigns to each sentence letter the value T, or the value F, or no value. Next, at the second stage, we construct all the classical completions of S and use the classical valuation rules to calculate the truth values of complex formulas on each of these classical completions. A formula Φ (whether atomic or complex) is then assigned a truth value by the supervaluation V_S if and only if all the classical completions of S agree on that truth value; if not, V_S assigns no value to Φ. More formally:

DEFINITION A supervaluational 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, boldface his)

 

As we can see, supervaluation may leave truth-value gaps in complex formulas. For example, if we have a disjunction, and one term is assigned false but the other a gap, then the classical completion will not all have the same value for the disjunction (416).

 

[For the next point, recall Kleene’s three-valued semantics from section 15.2.

(Nolt 412)

] We might be tempted to say that supervaluations follow the pattern of Kleene’s semantics, where we can simply take the value ‘I’ to be equivalent to a gap. So consider disjunction in the Kleene system, where both disjuncts are indeterminate. That makes the whole disjunction indeterminate. Similarly, the classical completion for a partial evaluation where both disjuncts are valueless would one giving false (where both disjuncts are false) and the rest giving true, and thus the whole would be valueless, just as in Kleene the whole is indeterminate. However, this does not hold in all cases, as for example with the formula ‘P∨~P’, where ‘P’ is either indeterminate or has no truth-value. [As we see in the Kleene system, when a formula is I, then its negation is I. Thus both P and ~P would be I in our example. And when we have a disjunction where both disjuncts are I, as in our case here, then the whole disjunction is I. However, for supervaluations, the negation of a formula with no value also has no value. For, supposing it is true, then the negation is false. And supposing it is false, then the negation is true. There is not a consistent outcome, so P has no value and ~P has no value. Then, when we make a classical completion of ‘P∨~P’, we see that it is always true, no matter what we assign it (given that both P and ~P are gaps).

Thus supervaluations are not equivalent to Kleene’s three-valued semantics.]

 

Nolt then defines validity and consistency for supervaluations. “a formula is valid iff it is true on all supervaluations; and a sequent is valid iff there is no supervaluation on which its premises are true and its conclusion is untrue. (And, of course, there are two ways of being untrue: being false and being truth-valueless.)” (Nolt 417).

 

Nolt then shows how “the logic that results from these stipulations is just classical propositional logic – even though, as we have seen, its semantics differs significantly from that of classical logic. Thus, for example, an inference is valid on supervaluational semantics if and only if it is valid on classical bivalent semantics, despite the fact that the former, but not the latter, permits truth-value gaps” (Nolt 417).

 

[Nolt next proves this with a complicated metatheorem and proof (417-418). After that he extends supervaluation semantics to predicate logic (418-419). These parts are left for another summary.]

 

 

 

 

 

 

From:

 

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

 

 

.