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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).
[Matters Related to Future Contingents]
[The Insufficiency of K3 or Ł3 for Determinable Statements about the Future]
[Supervaluation and Supervalidity]
[Supervaluational Consequence as Classical Consequence]
[The Misalignment of Classical and Supervaluational Validity for Multiple-Conclusions]
[A Remedy for the Multiple-Conclusion Validity Misalignment]
[LP and Subvaluation]
[The Misalignment of Classical and Subvaluational Validity for Multi-Premise Inferences]
[Philosophical Issuer Regarding Super/Sub-Valuational Validity]
[Evaluating Possibility and Necessity for Future Contingents]
[Non-Aristotelian Elements of the Modal Logic]
[Our Modal Logics So Far as Not Finitely Many-Valued]
[Uniform Substitution and Infinitely Many-Valued Logic]
[Uniform Substitution and Completeness]
Summary
[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)
[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)
[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 VS of S that assigns truth values VS (Φ) to formulas Φ according to these rules:
VS (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;
VS (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;
VS 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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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.
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[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
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15. See Priest (2005b).
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Priest, Graham, (2005b), ‘Many-Valued Logics’, in D. Borchet, ed., Encyclopedia of Philosophy, 2nd ed. (New York: Macmillan).
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From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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