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.

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[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.

 

 

 

 

 

 

Priest (11a.1) An Introduction to Non-Classical Logic, ‘Introduction [to ch.11a: Many-valued Modal Logics],’ 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.1

Introduction

 

 

 

 

Brief summary:

(11a.1.1) In this chapter we will examine many-valued modal logics. (11a.1.2) First we examine the general structure of a many-valued modal logic, illustrated with Łukasiewicz continuum-valued modal logic. (11a.1.3) We will also examine many-valued modal First Degree Entailment logics, including modal K3 and modal LP. (11a.1.4) We will end the chapter with a discussion of future contingents.

 

 

 

 

 

 

 

Contents

 

11a.1.1

[Many-Valued Modal Logics]

 

11a.1.2

[The General Structure]

 

11a.1.3

[FDE Modal Logics]

 

11a.1.4

[Future Contingents]

 

 

 

 

 

 

Summary

 

11a.1.1

[Many-Valued Modal Logics]

 

[In this chapter we will examine many-valued modal logics.]

 

[Normally in modal logics, worlds are two-valued, meaning that a formula can take one of two values, true or false. However, we can also formulate modal logics where worlds are many-valued (also see ch.7 and ch.9).]

In standard modal logics, the worlds are two-valued, in the following sense: there are two values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case: the worlds could be many-valued. This chapter looks at many-valued modal logics.

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[contents]

 

 

 

 

 

 

11a.1.2

[The General Structure]

 

[First we examine the general structure of a many-valued modal logic, illustrated with Łukasiewicz continuum-valued modal logic.]

 

[(ditto)]

We will start with the general structure of a many-valued modal logic. To illustrate the general structure, we will look briefly at modal logic based on Łukasiewicz continuum-valued logic.

(241)

[contents]

 

 

 

 

 

 

11a.1.3

[FDE Modal Logics]

 

[We will also examine many-valued modal First Degree Entailment logics, including modal K3 and modal LP.]

 

[(ditto)]

We will then look at one particular many-valued modal logic in more detail, modal First Degree Entailment (FDE), and its special cases, modal K3 and modal LP. In particular, tableau systems for these logics will be given.

(241)

[contents]

 

 

 

 

 

 

11a.1.4

[Future Contingents]

 

[We will end the chapter with a discussion of future contingents.]

 

[(ditto)]

Modal many-valued logics engage with a number of philosophical issues. The final part of the chapter will illustrate by returning to the issue of future contingents.

(241)

[contents]

 

 

 

 

 

 

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

 

 

Priest (11.4) An Introduction to Non-Classical Logic, ‘The Continuum-valued Logic Ł,’ 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

 

11.

Fuzzy Logics

 

11.4

The Continuum-valued Logic Ł

 

 

 

 

Brief summary:

(11.4.1) One way to construct a fuzzy logic is as a many-valued logic with a continuum of values from 0 (completely false) to 1 (completely true), including all real number values between, such that 0.5 is half true, and so on. (11.4.1a) To formulate the semantics for the connectives, we for now will use the oldest and most philosophically interesting means to do so. (11.4.2) Priest next gives the semantics for the connectives in our continuum many-valued logic.

f_¬ (x) = 1 − x

f_∧(x, y) = Min(x, y)

f_∨(x, y) = Max(x, y)

f_→(x, y) = x ⊖ y

where Min means ‘the minimum (lesser) of’; Max means ‘the maximum (greater) of’; and x ⊖ y is a function defined as follows:

if x ≤ y, then x ⊖ y = 1

if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)

| Note that we could say ‘x ≥ y’ instead of ‘x > y’ in the second clause, since if x = y, 1 − (x − y) = 1. Note, also, that we could define x ⊖ y equivalently as Min(1, 1 − x + y).

(225-226)

(11.4.3) The formulations of the semantic evaluations for the connectives in fuzzy logic hold to the basic intuitions we have about how they should operate. (11.4.4) Priest next notes that: if x ≤ y, then y ⊖ z ≤ x ⊖ z  ; and if x ≤ y, then z ⊖ x ≤ z ⊖ y . (11.4.5) Our continuum-valued fuzzy logic “is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic;” for, if we use only 1 and 0, we get the outcomes for classical semantics, and if we use just 0, 0.5, and 1 (with 0.5 understood as i), we get the outcomes for Ł3. (11.4.6) The designated value is context dependent, and so “any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226). (11.4.7) Validity is defined as: “Σ ⊨_ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε” (226). (In other words, an inference is valid under the following condition: whenever the premises are at least as high as the ((context-determined designated fuzzy)) value ε, then so too is the conclusion at least as high as ε. (11.4.8) Our fuzzy logic is called Ł, and its context-independent definition of validity is: Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨_ε A . (11.4.9) A set of truth-values X can be listed in descending numerical order. Suppose it is in an infinite set following a pattern like {0.41, 0.401, 0.4001, 0.40001, . . .}. Even though there would be no least member, there is still however a number that would be the greatest possible figure that is still less than or equal to all the members, in this case being 0.4. And it is called the greatest lower bound of set X, abbreviated as Glb(X). (11.4.10) A simpler characterization of validity would be: Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A). (11.4.11) Given the semantic evaluation for conjunction and the conditional, we can formulate validity in the following way: {B₁, . . . , B_n} ⊨ A iff for all v, v((B₁ ∧ . . . ∧ B_n) → A) = 1. “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨₁. The logic with just 1 as a designated value is usually written as Ł_ℵ, and it is called Łukasiewicz’ continuum-valued logic” (227).

 

 

 

 

Contents

 

11.4.1

[Fuzzy Logic as a Continuum Many-Valued Logic]

 

11.4.1a

[Ways to Formulate the Semantics for Connectives]

 

11.4.2

[The Semantics for Connectives]

 

11.4.3

[The Thinking Behind the Semantic Formulations for Connectives]

 

11.4.4

[Some Conditional Formulations with a z Value]

 

11.4.5

[Continuum Semantics as a Generalization of Classical Semantics and Ł3]

 

11.4.6

[The Designated Value]

 

11.4.7

[Validity]

 

11.4.8

[The Context-Independent Formulation of Validity in Ł.]

 

11.4.9

[The Greatest Lower Bound]

 

11.4.10

[Simplified Characterization of Validity]

 

11.4.11

[Validity in Terms of the Conditional in Ł_ℵ]

 

 

 

 

 

 

 

Summary

 

11.4.1

[Fuzzy Logic as a Continuum Many-Valued Logic]

 

[One way to construct a fuzzy logic is as a many-valued logic with a continuum of values from 0 (completely false) to 1 (completely true), including all real number values between, such that 0.5 is half true, and so on.]

 

[In section 11.3.9, we noted that to deal with sorites situations where there are continuous changes but no apparent discrete truth-value breaks, that we could use fuzzy logics, which employ continuous gradations of truth value. Now Priest notes that we can construct a fuzzy logic as a many-valued logic with a continuum of truth values. We take 1 as completely true, and 0 as completely false. And truth values can take take all real numbers between 1 and 0 such that 0.5 would be half true, and so on.]

A natural way to construct a fuzzy logic is as a many-valued logic with a continuum of truth values. Let the truth values, V, be the set of real numbers (decimals) between 0 and 1, {x: 0 ≤ x ≤ 1}. This is often written as [0,1]. 1 is completely true; 0 is completely false; 0.5 is half true; etc.

(224)

[contents]

 

 

 

 

 

 

11.4.1a

[Ways to Formulate the Semantics for Connectives]

 

[To formulate the semantics for the connectives, we for now will use the oldest and most philosophically interesting means to do so.]

 

[We will now look at ways to formulate the semantics for connectives. Later in the chapter Priest will return to this matter with a discussion of something called a t-norm. But for now, we will use the oldest and most philosophically interesting way to formulate the connectives.]

What are the semantic functions that correspond to the connectives ∧, ∨, ¬ and →? There are various ways to answer this question, based on the general notion of something called a t-norm. Details can be found in the technical appendix to this chapter, 11.7a. For the rest of this chapter, we will concentrate on the oldest and, perhaps, most interesting answer for philosophical purposes.

(224)

[contents]

 

 

 

 

 

11.4.2

[The Semantics for Connectives]

 

[Priest next gives the semantics for the connectives in our continuum many-valued logic.]

 

[Priest next gives the semantics for the connectives. Let us consider them each on the basis of our intuition. Negation we often think of as giving the flip side or “other” of the value under negation. Here it is formulated: f_¬ (x) = 1 − x. So suppose that we want the negation of something that is true. We would want 0. So 1 – 1 would give us 0. Suppose instead we want the negation of something false. We would want 1. And 1 – 0 is in fact 1. Suppose something is only a little bit true, like 0.1 true. It negation should be very true, in a proportional way. And in fact, 1 – 0.1 gives us 0.9, which fits our intuitions about how negation should function. All other cases of negation should match our intuitions for the same reasons. For the next two connectives, we need a notion of “minimum (lesser) of” and “maximum (greater) of”. These notions remind me a little bit of the diamond lattices from section 8.4.3. For conjunction, normally when at least one of the conjuncts is false, that “drags down” the whole conjunction’s value to false, regardless of the truth-value of the other conjunct. So with our continuum-valued logic, we will “drag” the whole conjunction’s value down to whatever the lowest of the two conjunct values are, ignoring the other value. (The formulation is: f_∧(x, y) = Min(x, y) .) We may have a similar intuition for disjunction. Normally if one of the disjuncts is true, that “lifts up” the whole disjunction’s value to true. So for the continuum-valued logic, we take the higher of the two disjunct values, ignoring the other, thereby “lifting” the disjunction’s value up to it. (The formulation is: f_∨(x, y) = Max(x, y) .) It gets more complicated of course for the conditional. Normally the cases where a conditional is true is when {1} the antecedent is 1 and the consequent 1 (so here they are equal), {2} when the antecedent is 0 and the consequent 0 (so here again they are equal) or {3} when the antecedent is 0 and the consequent 1 (so here the antecedent is less than the consequent). In all cases of the conditional being true, the antecedent is either less than or equal to the consequent. And, it is false if the the antecedent is 1 and the consequent is 0 (thus when the antecedent is more than the consequent). But we still need to puzzle through the formulation and consider our intuitions. The formulation will make it such that when the antecedent is less than or equal to the consequent, the whole conditional is 1 rather than some partial value, whereas when the antecedent is greater than the consequent, then it can take a partial value. More specifically, in that case we calculate it in the following way. We subtract the consequent’s value from antecedent, and that value we subtract from 1. Suppose we have the simple values that the antecedent is 1 and the consequent is 0. We would want the outcome to be 0. Here we have 1 – (1 – 0) which equals 0, as we expected. But suppose that the antecedent is simply true but consequent is only a little true, like 0.25 true. Then we have 1 – (1 – 0.25), or 0.25. Or suppose the antecedent is simply true and the consequent is mostly true, at 0.75. Then the conditional is 0.75 true. Now, suppose that the antecedent is mostly true at 0.75 and the consequent is only a little true, at 0.25. Then the conditional is 0.5. So we should consider two issues with regard to our intuitions about conditionals. The first question is, why would the first cases not admit of fuzzy values? I guess the idea would be that part of our intuition of the conditional is that there is a certain allowance that is absolute. But why is it ok for example for something 0.5 to follow conditionally from something 0.49? Why intuitively is that conditional a 1, being the same value as a conditional where 1 follows conditionally from 0? This is not so obvious to me. My best guess is that whenever the antecedent is less than the consequent, then we think that the conditions are not being met sufficiently to make it false, and so it is true with no further qualification to say it is less than fully true. Furthermore, it is already tricky to understand why in classical logic when the antecedent is false the whole conditional is thereby a true conditional. For the same intuition or reasoning that would go so far as to hold this might also be at work in saying that simply being a tiny fraction less true than the consequent is enough to make the whole conditional true (the example we consider later will illustrate this possible intuition). Now what is the intuition that says when the antecedent is more true than the consequent, the conditional should have fuzzy values that are proportional to their numerical difference? Here maybe (and probably not) the idea is that a conditional is all about something following conditionally from something else, and when the consequent is less than the antecedent, then the “following conditionally” has been tampered with. When the antecedent is less than or equal to the consequent, then the “following conditionally” is not being tampered with. Consider, “if they are an adult, then they are responsible.” And suppose they are 0.49 an adult but 0.5 responsible. We would think that if they are 0.49 an adult, then they would need to be at least 0.49 responsible for this to be true. But in fact, we went over that value by a little, and so for even more reason it would be true when they are 0.5 responsible but only 0.49 an adult. However, suppose instead they are 0.9 an adult but they are 0.1 responsible. The conditions would have called for them to be at least 0.9 responsible. Yet they are not. They are much less responsible than that. But they are still a little responsible, so the conditional is still a little true (being 0.2 true). Still, why do we not say that it is 0 true? My guess would be that it would go against our intuitions for the borderline cases. Suppose someone is 0.5 an adult but 0.49 responsible. It would seem to go too far to say that the conditional here is 100 percent false. It seems instead to have nearly met the conditions to be completely true, and so we should not evaluate the conditional as completely false. That is the best I have at the moment for making these notions intuitive. (In fact, Priest will give the actual reasoning later in section 11.4.3, but I wanted to work through it now when they are presented.)]

According to this:

f_¬ (x) = 1 − x

f_∧(x, y) = Min(x, y)

f_∨(x, y) = Max(x, y)

f_→(x, y) = x ⊖ y

where Min means ‘the minimum (lesser) of’; Max means ‘the maximum (greater) of’; and x ⊖ y is a function defined as follows:

if x ≤ y, then x ⊖ y = 1

if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)

| Note that we could say ‘x ≥ y’ instead of ‘x > y’ in the second clause, since if x = y, 1 − (x − y) = 1. Note, also, that we could define x ⊖ y equivalently as Min(1, 1 − x + y).

(225-226)

[contents]

 

 

 

 

 

 

11.4.3

[The Thinking Behind the Semantic Formulations for Connectives]

 

[The formulations of the semantic evaluations for the connectives in fuzzy logic hold to the basic intuitions we have about how they should operate.]

 

[Priest next discusses the thinking behind the connective formulations. As we went through similar thinking in 11.4.2, I will defer to the quotation below.]

The truth functions for negation, conjunction and disjunction are fairly natural. As the truth value of ‘Mary is a child’ goes down, the truth value of ‘Mary is not a child’ would seem to go up coordinately. A conjunction would seem to be just as good as its least true conjunct; and a disjunction would seem to be just as good as its most true. The truth function for → is anything but obvious. Here is its rationale. Consider A → B. If A is less true (or, better, no more true) than B, then the truth value of A → B is 1. That’s how it works, after all, with the standard 2-valued material conditional. If A is more true than B, then there is something faulty about the conditional: its truth value must be less than 1. How much less? The amount that the truth value falls in going from A to B. In particular, if it falls all the way from 1 to 0, then the value of A → B is 0. All this is exactly what ⊖ means.²

(225)

2. Fuzzy logic should not be confused with probability theory. Though fuzzy truth values and probability values are both real numbers in [0, 1], fuzzy truth values are truth functional – that is, the value of a compound is determined by the values of its components – whilst probabilities are not. Given a die, let A be ‘you roll 1, 2, or 3’, and B be ‘you roll 4, 5, or 6’. Then if P(A) is the probability of A, P(A ∧ A) = P(A) = 0.5, but P(A ∧ B) = 0, even though P(A) = P(B).

(225)

[contents]

 

 

 

 

 

 

11.4.4

[Some Conditional Formulations with a z Value]

 

[Priest next notes that: if x ≤ y, then y ⊖ z ≤ x ⊖ z  ; and if x ≤ y, then z ⊖ x ≤ z ⊖ y .]

 

[Priest next notes some other numerical relations for the conditional that are formulated with a z value, but I am not sure yet what they are for. See them below.]

Note that:

if x ≤ y, then y ⊖ z ≤ x ⊖ z

if x ≤ y, then z ⊖ x ≤ z ⊖ y

For the first of these, suppose that x ≤ y (and so, that −y ≤ −x): if x ≤ z, then x ⊖ z = 1, so the result follows. If z < x ≤ y, then y ⊖ z = 1 − y + z ≤ 1 − x + z = x ⊖ z. The second conditional is left as an exercise.

(225)

[contents]

 

 

 

 

 

 

11.4.5

[Continuum Semantics as a Generalization of Classical Semantics and Ł3]

 

[Our continuum-valued fuzzy logic “is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic;” for, if we use only 1 and 0, we get the outcomes for classical semantics, and if we use just 0, 0.5, and 1 (with 0.5 understood as i), we get the outcomes for Ł3.]

 

[Priest then brings to our awareness that if we stick to the classical 1 and 0 values, then the formulations will give us the classical outcome values. (We saw this in our bracketed comments in section 11.4.2). Also, if we think of 0.5 as i, then our continuum-valued semantics are the same as Łukasiewicz’ 3-valued logic Ł3 (see section 7.3.2 and section 7.3.8).]

Notice that if we restrict ourselves to just the values 1 and 0, then the truth functions of 11.4.2 are exactly the same as those of classical truth tables. It is less obvious, but is easy to check, that if we restrict ourselves to just the values 1, 0.5 and 0, then the truth functions are exactly the same as those of Ł3 (7.3.2 and 7.3.8), thinking of → as ⊃, and 0.5 as i. In this sense, the logic is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic.

(225)

[contents]

 

 

 

 

 

 

11.4.6

[The Designated Value]

 

[The designated value is context dependent, and so “any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226).]

 

[Recall the notion of the designated value from section 7.2. Designated values are “the values that are preserved in valid inferences” (p.120, section 7.2.2). In classical logic, it is just the 1 value that is preserved in valid inferences. Now we wonder, what will be the designated value in our fuzzy logic? Priest says that context will decide how true something needs to be: “If you buy a new car, you expect it not to have been driven at all. (So ‘this is a new car’ needs to have truth value 1.) But you would still describe it as a new car to a friend, even if you had bought it and driven it around for a few weeks. (So in this context, ‘this is a new car’ need have truth value only 0.95, say.)” (226). Thus: “if A is acceptable as true, and B is truer than A, then B is acceptable as true as well. What all this means is that any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226).]

What of the designated values of the logic? In general, things do not have to be completely true to be acceptable. If I ask for a red apple, and you give me one with a very small patch of green (so that ‘this is red’ is, say, 0.95 true), that’s probably good enough. How true something has to be to be acceptable will depend on the context. If you buy a new car, you expect it not to have been driven at all. (So ‘this is a new car’ needs to have truth value 1.) But you would still describe it as a new car to a friend, even if you had bought it and driven it around for a few weeks. (So in this context, ‘this is a new car’ need have truth value only 0.95, say.) But at any rate, if A is acceptable as true, and B is truer than A, then B is acceptable as true as well. What all this means is that any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε.

(226)

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11.4.7

[Validity]

 

[Validity is defined as: “Σ ⊨_ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε” (226). (In other words, an inference is valid under the following condition: whenever the premises are at least as high as the ((context-determined designated fuzzy)) value ε, then so too is the conclusion at least as high as ε.]

 

[In section 11.4.6 above, Priest introduced the ε value for giving the designated values in our fuzzy logic. It is a value between 0 and 1 whose exact quantity is determined by the context. But once established, it means that an inference is valid whenever the premises are at least as high as ε, then so too is the conclusion at least as high as ε.]

Correspondingly, for every such ε, taking the set of designated values, D_ε, to be {x: x ≥ ε}, will define a notion of validity. Thus Σ ⊨_ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε.

(226)

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11.4.8

[The Context-Independent Formulation of Validity in Ł.]

 

[Our fuzzy logic is called Ł, and its context-independent definition of validity is: Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨_ε A .]

 

[(ditto)]

Each logic defined in this way is a perfectly good many-valued logic. But in logic, it makes sense to abstract from context and consider a notion of validity that is context-independent. Hence, it is natural to define the central notion of logical consequence as follows:

Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨_ε A

We will call this logic Ł.

(226)

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11.4.9

[The Greatest Lower Bound]

 

[A set of truth-values X can be listed in descending numerical order. Suppose it is in an infinite set following a pattern like {0.41, 0.401, 0.4001, 0.40001, . . .}. Even though there would be no least member, there is still however a number that would be the greatest possible figure that is still less than or equal to all the members, in this case being 0.4. And it is called the greatest lower bound of set X, abbreviated as Glb(X).]

 

[The next ideas get more mathematical, and I will likely missummarize them, so please consult the quotation below. We have a set of truth values, called X. And this set can be either infinite or finite. The members of the set are listed in descending numerical order. Suppose it is simply the finite set: {0.41, 0.401, 0.4001}. We now think of a value, either in or not in the set, that is less than or equal to all the members in the set. But the idea is that this number should be as high as possible. In our case here, that number would be 0.4001. Now suppose the set is infinite, and it follows the same pattern, but without terminating in some particular figure: {0.41, 0.401, 0.4001, 0.40001, . . .}. As such, it has no least member. However, we can still say that there is the highest possible number that is less than or equal to all the members in this infinite set, being namely 0.4. (I do not know much math, but intuitively the idea seems to be the following. The numbers seem to tend to a limit at 0.4. Even if they never attain it exactly in a single terminal member, 0.4 would still be the highest we can say that is less than or equal to all the members of the set.) It is called, the greatest lower bound of set X, abbreviated as Glb.]

A set of truth values, X, may have no least member. (Consider, for example, {0.41, 0.401, 0.4001, 0.40001, . . .}.) But there will always be a greatest number that is less than or equal to every number in the set. (In this case, the number is 0.4.) This is called the greatest lower bound of X (Glb(X)). If the set is finite, then the Glb of the set is, of course, its least member. Notice that, by definition, if x ∈ X, x ≥ Glb(X); and if for all x ∈ X, x ≥ y, then Glb(X) ≥ y.

(226)

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11.4.10

[Simplified Characterization of Validity]

 

[A simpler characterization of validity would be: Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A).]

 

[Priest next gives a simplified formulation of valid inference. (See the quotation for details).]

⊨ has, in fact, a very simple characterisation. If Σ is a set of formulas, let v[Σ] be {v(B): B ∈ Σ}. Then:

Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A)

| Proof: Suppose that Σ ⊭ A. Then there is some ε, such that Σ ⊨_ε A. That is, for some v, and for all B ∈ Σ, v(B) ≥ ε, and v(A) < ε. But if every member of v[Σ] is ≥ ε, Glb(v[Σ]) ≥ ε. Hence, for this v, it is not the case that Glb(v[Σ]) ≤ v(A). Conversely, suppose that for some v, Glb(v[Σ]) > v(A). Let ε = Glb(v[Σ]). Then for all B ∈ Σ, v(B) ≥ ε, but v(A) < ε. That is, Σ ⊭_ε A. Hence, Σ ⊭ A.

(226-227)

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11.4.11

[Validity in Terms of the Conditional in Ł_ℵ]

 

[Given the semantic evaluation for conjunction and the conditional, we can formulate validity in the following way: {B₁, . . . , B_n} ⊨ A iff for all v, v((B₁ ∧ . . . ∧ B_n) → A) = 1. “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨₁. The logic with just 1 as a designated value is usually written as Ł_ℵ, and it is called Łukasiewicz’ continuum-valued logic” (227).]

 

[This final notion is quite complex, and I will missummarize it. So please consult the quotation below. The first notion is one we encountered above in section 11.4.9, namely, that for a finite set of truth-values, the greatest lower bound will be its lowest value. Next recall the semantic evaluation for conjunction from section 11.4.2 above.

f_∧(x, y) = Min(x, y)

(p.225, section 11.4.2)

So given what we have said in the prior sections about validity, the inference is valid when the least of the premises is less than or equal to the conclusion. With what we say about conjunction, this is the same as: v(B₁ ∧ . . . ∧ B_n) ≤ v(A), (when premises are B and conclusion is A). Next we need to recall the semantic evaluation for the conditional again from section 11.4.2 above.

f_→(x, y) = x ⊖ y

if x ≤ y, then x ⊖ y = 1

if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)

(p.225, section 11.4.2)

Priest will now formulate validity under the form of a conditional, but again it is best to skip to his rendition. We notice in the semantic evaluation that the only way the conditional can be exactly 1 is if the antecedent is less than or equal to the consequent. And keep in mind that an inference is valid when all the premises are less than or equal to the conclusion. (I get confused with this, but maybe we need to keep in mind also the designated value here.) So combine what was said about the semantic evaluations of conjunction and the conditional, then whenever we have:

v((B₁ ∧ . . . ∧ B_n) → A) = 1

that means the antecedent terms are all less than the consequent value. Now consider the B formulas as premises and the A as the conclusion. Given that under the above evaluation they would fulfill the requirement for validity, we also have:

{B₁, . . . , B_n} ⊨ A

Priest then concludes: “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨₁.” We call such a continuum logic with just 1 as the designated value Ł_ℵ, and we call it: Łukasiewicz’ continuum-valued logic.  (But I am not following so well the notion of infinity here. I also was not able to put all of these ideas together. So please read the quotation.)]

For a finite set, the Glb is its minimum. So if Σ = {B₁, . . . , B_n}, then Σ ⊨ A iff for all v, Min(v(B1), . . . , v(B_n)) ≤ v(A) iff v(B₁ ∧ . . . ∧ B_n) ≤ v(A).³ A little thought concerning ⊖ suffices to show that v(C) ≤ v(A) iff v(C → A) = 1. Hence:

{B₁, . . . , B_n} ⊨ A iff for all v, v((B₁ ∧ . . . ∧ B_n) → A) = 1

Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨₁. The logic with just 1 as a designated value is usually written as Ł_ℵ, and called Łukasiewicz’ continuum-valued logic. Hence, to investigate Ł further, we may investigate Ł_ℵ.⁴

(227)

3. Strictly speaking, the conjuncts should be bracketed in some way, since conjunction is a binary connective. But, however one inserts brackets, the value of the iterated conjunction is the same: the minimum of the values of the conjuncts. It therefore does no harm to omit the brackets.

4. ℵ is the Hebrew letter aleph, and, following Cantor, is used by logicians to denote a size of infinity.

(227)

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From:

 

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