Priest (10.6) Introduction to Non-Classical Logic, ‘The Ternary Relation,’ 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

 

10.

Relevant Logics

 

10.6

The Ternary Relation

 

 

 

 

Brief summary:

(10.6.1) We turn now to philosophical issues regarding the meaning of the ternary relation and its use for giving the truth-conditions for the conditional. (10.6.2) One possible interpretation of the ternary relation is that we “read Rxyz as meaning that z contains all the information obtainable by pooling the information x and y. This makes sense of the truth conditions of →” (207). (10.6.3) This understanding of the ternary relation in terms of information systems x and y being pooled into z leads to the validation of the irrelevant B → (A → A). (10.6.4) Another interpretation of the ternary relation is that worlds are conduits of information and A → B is an information flow, like how “a fossilised footprint allows information to flow from the situation in which it was made, to the situation in which it is found. Rxyz is now interpreted as saying that the information in y is carried to z by x” (207). (So we might understand the Rxyz ternary relation as meaning something like: for all pieces of information A and B, if there is a flow of information A → B  at conduit x (which is a conduit of information from y to z), and A is true at y, then B is true at z.) (10.6.5) But this metaphor of information flow is not entirely transparent, and it may even lead to irrelevant inferences. For example, “if a situation carries any information at all, it would appear to carry the information that there is some source from which information is coming. Call this statement S. If this is the case, then the inference from A → B to A → S would appear to be valid. But this would seem to give a violation of relevance, since A itself may have nothing to do with S” (207-208). (10.6.6) As of right now, ternary relation semantics and this notion of information flow are both too new to sufficiently do more than simply provide “a model-theoretic device for establishing various formal facts about various relevant logics,” where in addition it should also “justify the fact that some inferences concerning conditionals are valid and some are not.” For this, we need “some acceptable account of the connection between the meaning of the relation and the truth conditions of conditionals.”

 

 

 

 

 

 

Contents

 

10.6.1

[Philosophical Issues of the Ternary Relation]

 

10.6.2

[Interpreting the Ternary Relation as Combining Information]

 

10.6.3

[The Pooling Interpretation as Validating an Irrelevant Formula]

 

10.6.4

[R as Information Flow]

 

10.6.5

[Some Problems with the Information Flow Interpretation]

 

10.6.6

[The Need for More Work on This Account]

 

 

 

 

 

 

 

 

Summary

 

 

10.6.1

[Philosophical Issues of the Ternary Relation]

 

[We turn now to philosophical issues regarding the meaning of the ternary relation and its use for giving the truth-conditions for the conditional.]

 

[In section 10.2, we discussed the ternary accessibility relation of the logic B, which is a relevant logic. In section 10.2.2 we saw that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. And in section 10.2.5, we saw how it is used for giving the truth conditions for the conditional.

at normal worlds:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

The exception is that if w is a non-normal world:

v_w(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_x(A) = 1, then v_y(B) = 1

(p.189, section 10.2.5)

Now we will consider some philosophical issues regarding the meaning of the ternary relation and its use for the conditional.]

Let us now turn to some philosophical issues. In particular, what does the ternary relation mean, and why might it be reasonable to employ it in stating the truth conditions of a conditional?

(206)

[contents]

 

 

 

 

 

 

10.6.2

[Interpreting the Ternary Relation as Combining Information]

 

[One possible interpretation of the ternary relation is that we “read Rxyz as meaning that z contains all the information obtainable by pooling the information x and y. This makes sense of the truth conditions of →” (207).]

 

[Let us recall some notions about intuitionistic logic. In section 6.2.4, we noted that mathematical realists hold that there is an extra-linguistic reality corresponding to the truths of mathematical formulations like “2 + 3 = 5;” they think for example that there are “objectively existing mathematical objects, like 3 and 5.” Intuitionists however think rather that we should not apply the correspondence theory of truth to mathematical formulations. In section 6.2.5, we saw that intuitionism expresses a statement’s meaning on the basis of its proof conditions, which are the conditions under which the sentence is proved; while the proof condition of a simple sentence is whatever we would take to be a sufficient proof (6.2.5). In section 6.3.3, Priest formulated a possible worlds semantic for intuitionistic logic, and it had in particular the heredity condition, which means that when a proposition is true in one world, it it is true in all other worlds that are accessible from it. Then in section 6.3.6, Priest explained how this interpretation captures intuitionist ideas: we conceive of the way that information accumulates over time as being like one world (like our world at one moment) as being a set of proven things and another world accessible from the first having the same proven things and maybe more (like our world progressing later into a world perhaps with more information).

[...] let us see how an intuitionist interpretation arguably captures the intuitionist ideas of the previous section. Think of a world as a state of information at a certain time; intuitively, the things that hold at it are those things which are proved at this time. uRv is thought of as meaning that v is a possible extension of u, obtained by finding some number (possibly zero) of further proofs. Given this understanding, R is clearly reflexive and transitive. (For τ: any extension of an extension is an extension.) And the heredity condition is also intuitively correct. If something is proved, it stays proved, whatever else we prove.

(p.106, section 6.3.6)

Similarly, Priest will now consider a world as a state of information. Again recall from section 10.2.2 that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. Priest says that we can consider z as all the information pooled from x and y. So suppose in one pool of information A → B holds, and suppose in another A holds, then when you pool these sets of information, we should expect B to hold too. (I am not sure I follow entirely. Is it that B is a piece of information found in x or y? Or is it simply that we can use modus ponens to derive it?). Also, suppose A → B does not hold in information set x. We can expect then if we add A from set y that we would not be able to derive B.]

It is difficult to give a satisfactory answer to this question. The most promising sort of answer seems to be to tie up the relation with the notion of information. Suppose, for example, that we think of a world as | a state of information (as we did with intuitionist logic in 6.3.6). Then we may read Rxyz as meaning that z contains all the information obtainable by pooling the information x and y. This makes sense of the truth conditions of →. For if A → B holds in the information x, and A holds in the information y, we should certainly expect B to hold in the information obtained by pooling x and y. Conversely, if A → B does not hold in the information x, then it would certainly seem possible that we might add the information that A without thereby obtaining the information that B. Hence, there would seem to be a state of information, y, such that A holds in y, but B does not hold in the information obtained by pooling x and y.

(206-207)

[contents]

 

 

 

 

 

 

10.6.3

[The Pooling Interpretation as Validating an Irrelevant Formula]

 

[This understanding of the ternary relation in terms of information systems x and y being pooled into z leads to the validation of the irrelevant B → (A → A).]

 

[(The next idea gets a little more complicated, so it is best to skip to the quotation. Recall from section 9.7.8 that

A propositional logic is relevant iff whenever A → B is logically valid, A and B have a propositional parameter in common.

(p.172, section 9.7.8)

Priest says now that B → (A → A) cannot be valid in a relevant logic. So I am guessing the problem would be that there is no A in the antecedent of the main conditional, which here is just B. But now we are looking at  a relevant logic, which means this should therefore not be valid. However, on account of the interpretation of ternary relation we gave above in section 10.6.2, this would be made valid. We suppose here that A is true at y. Since z combines information from x and y, that means A will be true in z too. And we also suppose that Rxyz holds at y. But again recall from section 10.2.2 that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (Priest next says that A → A would be true at every world, but I am not sure why yet. Is it because it somehow fulfills the definition of the ternary relation? Or is it simply that so long as A is true in y and z, then it will be true in x, and from any true formula A we can derive A → A? I am not sure. At any rate,) A → A is thus true at every world. And since we can make it the consequent of any conditional and the conditional will always be true, then we can have the irrelevant B → (A → A). (Or maybe we obtain this also from the definition of Rxyz, but I am not sure. You will have to read the quotation, sorry.)]

The problem with this interpretation is that it seems to justify too much. For example, it justifies the claim that if Rxyz and A is true at y it is also true at z. But if this were the case, A → A would be true at every world, and hence, for any B, B → (A → A) would be logically valid, which it cannot be if the logic is to be relevant.

(207)

[contents]

 

 

 

 

 

 

10.6.4

[R as Information Flow]

 

[Another interpretation of the ternary relation is that worlds are conduits of information and A → B is an information flow, like how “a fossilised footprint allows information to flow from the situation in which it was made, to the situation in which it is found. Rxyz is now interpreted as saying that the information in y is carried to z by x” (207). (So we might understand the Rxyz ternary relation as meaning something like: for all pieces of information A and B, if there is a flow of information A → B  at conduit x (which is a conduit of information from y to z), and A is true at y, then B is true at z.)]

 

[(The next interpretation of the ternary relation is fascinating, but I do not entirely grasp it yet. So please read the quotation below. I am guessing it is the following. So once again, please recall from section 10.2.2 that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. Previously the worlds were states of information. Now we think of worlds as conduits of information. But I am not sure how this works. It seems that we think of the conditional A → B as an information flow from information conduit y to conduit z by means of conduit x. Priest’s example is a fossilized footprint that allows information to flow from the situation of its imprinting to the situation of its discovery many years later. But I am not sure how to think of that example in terms of a conditional.  Also, I would think that if information is flowing, it remains the same and is being transferred, so I do not yet grasp why an information flow is A → B instead of A → A. Yet, I am off track anyway. Maybe the idea has more to do with understanding the Rxyz ternary relation as meaning something like: for all pieces of information A and B, if there is a flow of information A → B  at conduit x (which is a conduit of information from y to z), and A is true at y, then B is true at z. Please read the quotation below.)]

Another possibility for interpreting R is to suppose that worlds are not themselves states of information, but that they may act as conduits for information in some way. Thus, a situation that contains a fossilised footprint allows information to flow from the situation in which it was made, to the situation in which it is found. Rxyz is now interpreted as saying that the information in y is carried to z by x. If we think of A → B as recording the information carried, this makes some sense of the ternary truth conditions. For if A is information at y, and x allows the flow of information A → B from y to z, then we would expect the information B to be available at z. Conversely, if x does not allow the information flow A → B, then it must be possible for there to be situations, y and z, where A is available at y, but B is not available at z.

(207)

[contents]

 

 

 

 

 

 

10.6.5

[Some Problems with the Information Flow Interpretation]

 

[But this metaphor of information flow is not entirely transparent, and it may even lead to irrelevant inferences. For example, “if a situation carries any information at all, it would appear to carry the information that there is some source from which information is coming. Call this statement S. If this is the case, then the inference from A → B to A → S would appear to be valid. But this would seem to give a violation of relevance, since A itself may have nothing to do with S” (207-208).]

 

[But it is not obvious how exactly to make sense of this metaphor of information flow. Also, it is not certain to provide a fully relevant interpretation. (I will summarize the reasoning here inadequately, so simply skip to the quotation. I wonder if the idea if the following. In the case of the fossil, not only do we have information that the dinosaur stepped in the ground at that location, we also have information that this fact was communicated by means of a fossilization process. So we know that the dinosaur stepped there, and we know that the fossilization tells us this. So we have the information flow A → B (which I am still not sure how to illustrate, but maybe it is the information being made by the dinosaur stepping to the information that we gather that the dinosaur had stepped). But we also have the information that this knowledge comes by means of a source, the fossilization process (or the fossil itself). We will call the statement that our information has a source, S. Now, since any conveyance of knowledge must have a source, then from A → B we should be able to infer A → S. (I am not sure yet how to think of that, but I suppose it means that were there a flow of information of any kind, we can infer that information that there is a source of that knowledge also flows in the same stoke.) But, the contents of S may have no relevance to the A. ((So maybe, fossilization is a process that has nothing about it that is about dinosaurs, or maybe more simply, information sourcing is a sort of knowledge with little relevance to dinosaur stepping.)) Thus, “this would seem to give a violation of relevance, since A itself may have nothing to do with S.”]

The problem now is to make sense of the metaphor of information flow – hardly a transparent one. Moreover, it is not at all clear that, when articulated, it will provide what is needed. For example, if a situation carries any information at all, it would appear to carry the information that there is some source from which information is coming. Call this statement S. If this is the case, then the inference from A → B to A → S would appear to | be valid. But this would seem to give a violation of relevance, since A itself may have nothing to do with S.

(207-208)

[contents]

 

 

 

 

 

 

10.6.6

[The Need for More Work on This Account]

 

[As of right now, ternary relation semantics and this notion of information flow are both too new to sufficiently do more than simply provide “a model-theoretic device for establishing various formal facts about various relevant logics,” where in addition it should also “justify the fact that some inferences concerning conditionals are valid and some are not.” For this, we need “some acceptable account of the connection between the meaning of the relation and the truth conditions of conditionals.”]

 

[(ditto)]

The ternary relation semantics, and the study of information flow are both very new; and it may be the case that a satisfactory analysis of the two together will eventually arise. But if the ternary relation semantics is ultimately to provide anything more than a model-theoretic device for establishing various formal facts about various relevant logics, this is a task that must be discharged successfully. In particular, if the ternary relation semantics is to justify the fact that some inferences concerning conditionals are valid and some are not, then there must be some acceptable account of the connection between the meaning of the relation and the truth conditions of conditionals.

(208)

[contents]

 

 

 

 

 

 

From:

 

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

 

 

.

Priest (10.2) Introduction to Non-Classical Logic, ‘The Logic B,’ 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

 

10.

Relevant Logics

 

10.2

The Logic B

 

 

 

 

Brief summary:

(10.2.1) We can strengthen relevant logics like N₄ and N_∗ to accommodate certain intuitively correct principles regarding the conditional by incorporating non-normal worlds and a ternary accessibility relation on worlds, Rxyz. (10.2.2) The intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (10.2.3) We will focus on the ternary relation ∗ semantics, as they have been the ones studied historically speaking. (10.2.4) The ternary ∗ interpretation is a structure, ⟨W, N, R, ∗, v⟩, where “W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds)”, “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world,” and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (167; 170; 189). (10.2.5) Priest next gives the truth conditions for connectives.

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, otherwise it is 0.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, otherwise it is 0.

v_w(¬A) = 1 if v_w*(A) = 0, otherwise it is 0.

(p.151, section 8.5.3; p.169. section 9.6.6, see 9.6.2)

at normal worlds, the truth conditions for → are:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

The exception is that if w is a non-normal world:

v_w(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_x(A) = 1, then v_y(B) = 1

(189)

(10.2.6) Validity is truth preservation over all normal worlds. (10.2.7) This logic is named B, and it is a sub-logic of K_∗, while N_∗ is a sub-logic of B. (10.2.8) The normality condition is Rwxy iff x = y. By implementing it in the conditional rule, we can simply it so that it works for all worlds: v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, then v_x(B) = 1. (10.2.9) Finally Priest notes that “the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx” (190).

 

 

 

 

 

Contents

 

10.2.1

[Strengthening Relevant Logics with Non-Normal Worlds and a Ternary Accessibility Relation on Worlds, Rxyz]

 

10.2.2

[The Intuitive Sense of Rxyz]

 

10.2.3

[Focusing on the Ternary Relation ∗ Semantics]

 

10.2.4

[The Structure of Ternary ∗ Interpretations]

 

10.2.5

[Truth Conditions for Connectives]

 

10.2.6

[Validity]

 

10.2.7

[B]

 

10.2.8

[Simplifying the Conditional’s Truth Conditions Using the Normality Condition]

 

10.2.9

[The Two Halves of the Normality Condition]

 

 

 

 

 

Summary

 

 

10.2.1

[Strengthening Relevant Logics with Non-Normal Worlds and a Ternary Accessibility Relation on Worlds, Rxyz]

 

[We can strengthen relevant logics like N₄ and N_∗ to accommodate certain intuitively correct principles regarding the conditional by incorporating non-normal worlds and a ternary accessibility relation on worlds, Rxyz.]

 

[Recall from section 9.7.8 that “A propositional logic is relevant iff whenever A → B is logically valid, A and B have a propositional parameter in common” (p.172, section 9.7.8). Next recall from section 9.1.1 and section 9.2.1 that K₄ combines {1} the four value-situation semantics of First Degree Entailment (FDE), meaning that it uses the ρ relation to relate a formula to either just 1 (true), just 0 (false), both 1 and 0, or neither 1 nor 0 (see section 8.2) with {2} the possible worlds logics of normal modal logic Kυ. And recall from section 9.4 we can add to it impossible worlds to get N₄. In section 9.7.9 and 9.7.10 we saw that N₄ and N_∗ are relevant logics. Priest now notes that N₄ and N_∗ “are too weak, on the ground that there are intuitively correct principles concerning the conditional that they do not validate” (188). Priest now explains that we can incorporate these principles using non-normal worlds and a ternary accessibility relation on worlds, Rxyz.]

N₄ and N_∗ are relevant logics, but, as relevant logics go, they are relatively weak. Many proponents of relevant logic have thought that the relevant logics of the last chapter are too weak, on the ground that there are intuitively correct principles concerning the conditional that they do not validate. A way to accommodate such principles within a possible-world semantics is to use a relation on worlds to give the truth conditions of conditionals at non-normal worlds. Unlike the binary relation of modal logic, xRy, though, this relation is a ternary, that is, three-place, relation, Rxyz.¹

(188)

1. Using a binary relation would produce irrelevance, since p → p would be true at all worlds, and hence, q → (p → p) would be logically valid.

(188)

[contents]

 

 

 

 

10.2.2

[The Intuitive Sense of Rxyz]

 

[The intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. ]

 

[Priest next gives the intuitive sense of the ternary relation Rxyz: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (In other words, Rxyz is a particular sort of relation between worlds. We suppose that A → B is true in world x, and A is true in world y. On those conditions, when Rxyz holds, then also B must be true in world z. Priest will come back to the issue of understanding what this all means.]

Intuitively, the ternary relation Rxyz means something like: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. What philosophical sense to make of this, we will come back to later.

(188)

[contents]

 

 

 

 

10.2.3

[Focusing on the Ternary Relation ∗ Semantics]

 

[We will focus on the ternary relation ∗ semantics, as they have been the ones studied historically speaking.]

 

[We can apply this technique (maybe, the incorporation of non-normal worlds and a ternary accessibility relation on worlds, Rxyz) to the relational semantics (maybe K₄ and N₄) and the ∗ semantics (maybe K_∗ and N_∗). Recall from section 9.6.9 that K₄ and N₄ are not equivalent to K_∗ and N_∗. For example,  K_∗ and N_∗ validate  contraposition (p → q ⊨ ¬q → ¬p), but K₄ and N₄ do not, and from section 9.6.10 that K₄ and N₄ verify p ∧ ¬q ⊨ ¬(p → q), but K_∗ and N_∗ do not. At any rate, it happens that the ternary relation ∗ semantics are what have been more studied, so we will focus on them.]

The technique can be applied to both the relational semantics and the ∗ semantics. As we noted in 9.6.9 and 9.6.10, these semantics diverge once we add → to the language. Though the ternary relation relational semantics are perfectly good, it is, as a matter of historical fact, the logics with the ternary relation ∗ semantics that occur in the literature. Hence, we look only at those.

(189)

[contents]

 

 

 

 

10.2.4

[The Structure of Ternary ∗ Interpretations]

 

[The ternary ∗ interpretation is a structure, ⟨W, N, R, ∗, v⟩, where “W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds)”, “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world,” and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (167; 170; 189).]

 

[Priest now defines the ternary ∗ interpretation. It has a structure, ⟨W, N, R, ∗, v⟩. Here “W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds)” (p.167, section 9.4.7), “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world” (p.170, section 9.6.6), and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (189).]

A ternary (∗) interpretation is a structure ⟨W, N, R, ∗, v⟩, where W, N, ∗ and v are as in the semantics for N_∗ (9.6.6), and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)

(189)

[contents]

 

 

 

 

10.2.5

[Truth Conditions for Connectives]

 

[Priest next gives the truth conditions for connectives.]

 

[Recall from section 9.6.6 the truth conditions for connectives in N_∗. To these we will exclude the conditional rule for N_∗ and replace it with one specific for our new system:

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, otherwise it is 0.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, otherwise it is 0.

v_w(¬A) = 1 if v_w*(A) = 0, otherwise it is 0.

(p.151, section 8.5.3; p.169. section 9.6.6, see 9.6.2)

at normal worlds, the truth conditions for → are:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

The exception is that if w is a non-normal world:

v_w(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_x(A) = 1, then v_y(B) = 1

(189)

]

With one exception, the truth conditions for all connectives are as for N_∗. In particular, at normal worlds, the truth conditions for → are:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

The exception is that if w is a non-normal world:

v_w(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_x(A) = 1, then v_y(B) = 1

(189)

[contents]

 

 

 

 

10.2.6

[Validity]

 

[Validity is truth preservation over all normal worlds.]

 

[Recall from section 9.6.6 that in N_∗, “Validity is defined in terms of truth preservation at normal worlds” (p.170, section 9.6.6). It is the same here too.]

Validity is defined as truth preservation over all normal worlds, as in N_∗.

(189)

[contents]

 

 

 

 

10.2.7

[B]

 

[This logic is named B, and it is a sub-logic of K_∗, while N_∗ is a sub-logic of B.]

 

[We call this logic B, which stands for ‘basic’. Priest notes that B is a sub-logic of K_∗, and N_∗ is a sub-logic of B (see the quotation for details why.)]

The logic generated in this way is usually called B (for basic).² Clearly, B is a sub-logic of K_∗ (since any K_∗ interpretation is a B interpretation, with W − N = φ). Moreover, any B interpretation, , is equivalent to an N_∗ interpretation. We just take that N_∗ interpretation which is the same as , except that it assigns to each conditional at each non-normal world, w, whatever value it has at w in . Hence, N_∗ is a sub-logic of B.

(189)

2. We continue to use B as a letter for formulas, too. Context will disambiguate.

(189)

[contents]

 

 

 

 

10.2.8

[Simplifying the Conditional’s Truth Conditions Using the Normality Condition]

 

[The normality condition is Rwxy iff x = y. By implementing it in the conditional rule, we can simply it so that it works for all worlds: v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, then v_x(B) = 1.]

 

[Recall from section 10.2.2 that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. And recall from section  10.2.5 above that

at normal worlds, the truth conditions for → are:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

The exception is that if w is a non-normal world:

v_w(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_x(A) = 1, then v_y(B) = 1

(189, 10.2.5)

Let us take a closer look at the conditional rules. Suppose we are dealing with a normal world. A conditional is true only if for all other worlds whenever the antecedent is true in that other world, so too is the consequent true in that other world. I am not sure however if non-normal worlds are included in those other worlds. Suppose instead we are dealing with a non-normal world. Then a conditional is true in the non-normal world only if for any other pair of worlds (normal or not), where the ternary relation holds (that is to say,  whenever the conditional is true in the non-normal world in question, and the antecedent is true also in the first other world, then the consequent is true in the second other world), then the consequent indeed is true in the second other world. I am not following this at all really, so you will need to simply read the quotation below. And I cannot really summarize the rest of this section, so you will need to read it for yourself. The next point is that we can simplify the bipartite truth conditions above by defining R at normal worlds using the normality condition: Rwxy iff x = y. I am guessing that this is added to the conditions already given for the intuitive sense of the ternary relation. The final idea is that by means of this, we can reduce our truth conditions for the conditional to hold for all worlds and to be formulated simply as:

v_w(A → B) = 1 iff for all x ∈ W such that v_x(A) = 1, then v_x(B) = 1

]

The bipartite truth conditions of → can be simplified if one thinks of R as defined at normal worlds. Specifically, if w is normal, we specify R by the following condition:

Rwxy iff x = y

Call this the normality condition. If we define R at normal worlds in this way, we may take the ternary truth conditions to govern conditionals at all worlds. For, given this condition, the ternary truth conditions:

for all x, y ∈ W such that Rwxy, if v_x (A) = 1, then v_y(B) = 1

|

become:

for all x, y ∈ W such that x = y, if v_x(A) = 1, then v_y(B) = 1

And given the standard properties of =, this is logically equivalent to:

for all x ∈ W such that v_x(A) = 1, v_x(B) = 1

which gives the standard truth conditions of → at normal worlds. We adopt this simplification in what follows.

(189-190)

[contents]

 

 

 

 

10.2.9

[The Two Halves of the Normality Condition]

 

[Finally Priest notes that “the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx” (190).]

 

[Recall from section 10.2.8 above that the normality condition is: Rwxy iff x = y. Priest then notes the following, but I do not know what he means:]

Notice that the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx.

(190)

[contents]

 

 

 

 

 

 

 

 

 

 

 

 

From:

 

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

 

 

.

Priest (9.7) Introduction to Non-Classical Logic, ‘Impossible Worlds and Relevant Logic,’ summary

 

by Corry Shores

 

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

 

[Central Entry Directory]

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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

 

9.

Logics with Gaps, Gluts and Worlds

 

9.7

Impossible Worlds and Relevant Logic

 

 

 

 

Brief summary:

(9.7.1) We will now discuss philosophical matters regarding K₄ , N₄ , K_∗ , and N_∗. (9.7.2) We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different. (9.7.3) Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds. (9.7.4) We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.” (9.7.5) Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling. (9.7.6) In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to. (9.7.7) Logically impossible worlds can also in fact be ones where laws of logic indeed are broken. (9.7.8) Relevant propositional logics are ones where whenever “A → B is logically valid, A and B have a propositional parameter in common” (172). (9.7.9) But N₄ is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds. (9.7.10) In a similar way, N_∗ is also a relevant logic. (9.7.11) Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters. (9.7.12) There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173). (9.7.13) Relevance in our systems here however is not conditions added on top of classical validity. (9.7.14) If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.

 

 

 

 

Contents

 

9.7.1

[Moving to a Discussion on K₄ , N₄ , K_∗ , and N_∗]

 

9.7.2

[Logically Impossible Worlds]

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

9.7.8

[Relevant Logics and the Conditional]

 

9.7.9

[N₄ as Relevant]

 

9.7.10

[N_∗ as Relevant]

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

9.7.12

[Filter Logics]

 

9.7.13

[More Than Classical Relevance]

 

9.7.14

[Preserving Conventional Truth in This System]

 

 

 

 

 

 

 

 

Summary

 

9.7.1

[Moving to a Discussion on K₄ , N₄ , K_∗ , and N_∗]

 

[We will now discuss philosophical matters regarding K₄ , N₄ , K_∗ , and N_∗.]

 

[Recall the semantics and tableau constructions from previous sections: K₄ (9.2 and 9.3); N₄ (9.4 and 9.5) and K_∗ and N_∗ (9.6). Now Priest will discuss philosophical matters regarding these constructions.]

We are now in a position to make some comments on the import of the previous constructions.

(171)

[contents]

 

 

 

 

9.7.2

[Logically Impossible Worlds]

 

[We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different.]

 

[Recall from section 4.2.3 that for modal logics, non-normal worlds are ones where nothing is necessary and all is possible; for, at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. In section 9.2, we discussed a possible worlds First Degree Entailment (and thus four value-situationed) system called K₄. In section 9.4, we noted that K₄ has the following problematic valid formula: ⊨ p → (q → q). This was problematic, because we want to be able to say things about what would follow if certain laws of logic were suspended. So the following would be valid, even though we would want it to be invalid: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (p.167, section 9.4.3 ). But “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (p.167). We thus incorporated non-normal worlds into K₄ in order to get N₄. But we note here that in non-normal worlds, the normal laws of logic are different, and we will call such non-normal worlds “logically impossible worlds.”]

As we saw (9.4.4 – 9.4.6), non-normal worlds of the kind we have employed in this chapter are worlds where the laws of logic are different. Let us call these ‘logically impossible worlds’.

(171)

[contents]

 

 

 

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

[Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds.]

 

[In section 3.6.5 (not yet summarized), Priest discussed physically impossible worlds: “Something is physically necessary if it is determined by the laws of nature, and physically possible if it is compatible with the laws of nature. Thus, it is physically impossible for me to jump thirty metres into the air (though this is not a logical impossibility)” (46). Priest says now that there is no reason why there cannot likewise be logically impossible worlds.]

There seems to be no reason why there should not be logically impossible worlds, in whatever sense there are possible worlds. Physically impossible worlds, where the laws of physics are different, are entirely routine (see 3.6.5). And just as there are worlds where the laws of physics are different, there must be worlds where the laws of logic are different.

(171)

[contents]

 

 

 

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

[We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.”]

 

[In fact, when we discuss non-classical logics, we seem to suppose such “logically impossible” worlds (even though in fact the real world might be one where the laws of classical logic do indeed fail). It is implied for example when we say that “if intuitionist logic were correct, the law of double negation would fail.”]

After all, we seem to envisage just such worlds when we evaluate conditionals such as ‘if intuitionist logic were correct, the law of double negation would fail’ (true), ‘if intuitionist logic were correct, the law of | identity would fail’ (false). Even if one is a modal realist (2.6), why should there not be such worlds?

(171-172)

[contents]

 

 

 

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

[Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling.]

 

[Priest next deals with some objections to the idea that there can be logically impossible worlds. {1} Objection: Someone might say that logical laws should always hold at possible worlds, by definition. Reply: we are not dealing with possible worlds but rather impossible ones. {2} Objection: One might say that some proposed logically law for a possible world that breaks one of our normal logical laws cannot be the case, simply because it is breaking a normal logical law. For example, suppose someone claims that there is a world where it is a logical law that A → (B ∧ ¬B) holds and so does A. Then, by modus ponens we can infer that B ∧ ¬B. The objector can say that this is a contradiction and it cannot be the case. Reply: {2a} Some might have philosophical reasons to say that the normal laws can be broken. For example, a dialetheist would say that the law of non-contradiction is breakable. {2b} The objection assumes that modus ponens holds in this world. But as an impossible world, it may not.]

One might suggest that there can be no worlds at which logical laws fail: by definition, logical laws hold at all possible worlds. Maybe so. But it is precisely impossible worlds that we are dealing with here. Or one might say: take a world in which it is a logical law that A → (B ∧ ¬B) and in which A is also true. It would follow that B ∧ ¬B is true at that world, which cannot be the case. This argument is hardly likely to persuade someone who accepts the possibility of truth-value gluts. But in any case, it is fallacious. For who says that modus ponens holds at that world? In the semantics we have looked at, it is entirely possible to have both A and A → C holding at a non-normal world, without C holding there.

(172)

[contents]

 

 

 

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

[In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to.]

 

[Priest next notes that logically impossible worlds do not necessarily have cases of broken laws of logic, just like how physically impossible worlds may allow for certain alternate physical situations without them ever obtaining. However, one could specifically define logically impossible worlds as ones where the laws of logic are in fact broken.]

Note that one might take ‘logically impossible world’ to mean something other than ‘world where the laws of logic are different’. One might equally take it to mean ‘world where the logically impossible happens’. This need not be the same thing. If this is not clear, just consider physically impossible worlds. The fact that the laws of physics are different does not necessarily mean that physically impossible things happen there (though the converse is true). For example, even if the laws of physics were to permit things to accelerate past the speed of light, it does not follow that anything actually would. Things at that world might be accelerating very slowly, and the world might not last long enough for any of them to reach super-luminal speeds.

(172)

[contents]

 

 

 

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

[Logically impossible worlds can also in fact be ones where laws of logic indeed are broken.]

 

[I might be mistaken about this next point. It might be that we know there are logically impossible worlds where the laws of logic are broken, because we have already seen that there is a world where A and A → C are true, but C is not. In the footnote Priest mentions some inferences that do not hold in any impossible world. However, these are instances without conditionals, and it is conditionals that express the laws of logic (but I do not myself know why that is.)]

But logically impossible worlds, in the sense that these occur in the semantics we have been looking at, may be logically impossible in the second sense as well. For example, there are, as has just been noted, worlds where A and A → C are true, but C is not.⁶

(172)

6. There are no worlds at which A ∧ B is true, but A is not, or at which ¬¬A is true, but A is not. But it is conditionals that express the laws of logic, not conjunctions or negations. That is why it is their behaviour (and only theirs) that changes at non-normal worlds.

(172)

[contents]

 

 

 

 

9.7.8

[Relevant Logics and the Conditional]

 

[Relevant propositional logics are ones where whenever “A → B is logically valid, A and B have a propositional parameter in common” (172). ]

 

[Priest now defines relevant logic: “A propositional logic is relevant iff whenever A → B is logically valid, A and B have a propositional parameter in common” (172). This may seem odd, because A and B would seem to be propositional parameters, and surely we are not saying that conditions need to be of the form A → A to be relevant. So recall from section 1.2.3: “I use capital Roman letters, A, B, C, ..., to represent arbitrary formulas of the object language. Lower-case Roman letters, p, q, r, ..., represent arbitrary, | but distinct, propositional parameters” (4-5). So maybe we would need to look at the lower-case sorts of formulations, when looking for relevance. We will see some examples. Or maybe A → B is shorthand for more complex formulations, like we will see below, as with ⊨ A ⥽ (B ∨ ¬B). He says that conditionals that suffer from the paradoxes of implication, including the strict conditional, are not relevant. Let us look at the paradoxes of strict implication. In section 4.6.3 we saw that the following  are valid for the strict conditional.

⊨ A ⥽ (B ∨ ¬B)

⊨ (A ∧ ¬A) ⥽ B

We might fill them out with propositional parameters I am going to guess in the following way.

⊨ p ⥽ (q ∨ ¬q)

⊨ (p ∧ ¬p) ⥽ q

But I am not sure about much here yet. Yet we can see that the strict conditional is not part of a relevant logic. And these same forms are not valid in K₄.

⊭_K4 p → q ∨ ¬q

⊭_K4 (p ∧ ¬p) → q

Nonetheless, Priest says that neither K₄ and K_∗ are relevant. To see why, we first recall from section 9.4.2 that in K₄, ⊨ p → (q → q) is valid, and in section 9.6.6 we saw that it if valid in K_∗ too.]

A propositional logic is relevant iff whenever A → B is logically valid, A and B have a propositional parameter in common. Obviously, any conditional that suffers from paradoxes of implication (material implication, | strict implication, the intuitionist conditional) is not relevant. Neither are K₄ and K_∗ relevant, as we have seen (9.4.2 and 9.6.6).

(172-173)

[contents]

 

 

 

 

9.7.9

[N₄ as Relevant]

 

[But N₄ is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds.]

 

[Priest will now show that N₄ is a relevant logic. It gets very technical, and I am the wrong person to summarize this, so please skip to the quotation below. I will try to say some things still, but they probably will not help you. Recall that in N₄, there are non-normal worlds. And recall from section 4.2.5 and 9.4.9 that inferences are valid only if they preserve truth in all interpretations at all normal worlds. I might have this wrong, but I think that means it cannot be that the premises are at least true and the conclusion not at least true. If I am following even a little here (and probably not), Priest is going to do the following. Let me first note that I am not certain if we are dealing with an inference, like in his cited problem, or a simple conditional, like mentioned in the last line of this paragraph. I am also not sure if it makes a difference. Let us for now say that we are dealing with the conditional A → B, and we want to know if it would be valid/true in N₄ whenever there is no relevance of the antecedent A to the consequent B. Priest will make a model where it is false, even though in K₄ presumably it would be true. It seems that the way this will work will have to do with the fact that in normal worlds we evaluate conditionals on the basis of all other worlds, whether normal or not. But as we saw in section 9.4.6, we do not evaluate the conditionals in non-normal worlds compositionally in terms of the component terms’ values but rather we assign their values arbitrarily however we please. Priest will exploit those two features of N₄ in order to make a non-relevant A → B be false/invalid in a normal world. So our model will have two worlds, 0 and 1, and world 1 is the non-normal one. It is still unclear to me if we are dealing with an inference from A to B or a conditional, but I am guessing wildly that Priest is covering both options. (Sorry, please read the text). For non-relevance, we suppose that we have antecedent A and consequent B (or premise(s) A and conclusion B), but A and B share no propositional parameters in common. We also consider a propositional parameter (or conditional) called D, which can be included either in antecedent A or consequent B (or in premise(s) A or conclusion B); but it cannot be in both, because as we said, A and B share no parameters in common, thus if D is in one, it cannot be in the other. We will assign our values for the conditional in non-normal world 1 arbitrarily, as that is how it works in non-normal worlds (see section 9.4.6). So we say, if D is among the antecedent (or premises), then we assign it both as true and also as false. Or, if instead it is in the consequent (or conclusion), then we assign it neither true nor false. Now, let us stick with conditionals for a second. Recall from section 9.2.4 that this is how we evaluate conditionals in N₄:

A → Bρ_w1 iff for all w′ ∈ W such that Aρ_w′1, Bρ_w′1

A → Bρ_w0 iff for some w′ ∈ W, Aρ_w′1 and Bρ_w′0

(p.164, section 9.2.4)

We ask, A → Bρ_w0??? In other words, we want to determine the value of A → B in world 0, the normal world. Priest’s way of proving this uses the induction method, which I have not learned yet (see section 0.2). Were we to perform it, we would find that somehow, regardless of whether D is in the antecedent (or premises) or in the consequent (or conclusion), A will be both true and false (and thus at least true) and B will be neither true nor false (and thus not at least true). And hence the formula will be false/invalid in world 0, the normal world. Please read the quotation, as I am not grasping this one very well at all.]

But N₄ is a relevant logic. This can be seen by modifying the argument of 8.10, problem 5. Suppose that A and B share no propositional parameters, and consider an interpretation ⟨W, N, ρ⟩, where W = {w₀, w₁}; N ={w₀}; if D is a propositional parameter or a conditional in A, Dρ_w11 and Dρ_w10; if D is a propositional parameter or a conditional in B, neither Dρ_w11 nor Dρ_w10. (D cannot occur in both, since A and B have no parameters in common.) It is easy to check that Aρ_w11 and Aρ_w10, but neither Bρ_w11 nor = Bρ_w10.⁷ In particular, A is true at w₁ and B is not. Hence A → B is not true at w₀.

(173)

7. Proof: For the first, what we show is that every formula made up from the propositional parameters occurring in A – and so, in particular, A – the result holds. Similarly for B. This is proved by induction on the construction of sentences, but an induction slightly different from the normal kind. Note that every formula can be built up from conditionals and parameters using the extensional connectives. Hence, the result may be proved by induction, with parameters and conditionals as the basis case, and induction cases for the extensional connectives. The basis case is true by definition. The induction cases are as in the notes to 8.4.6 and 8.4.9.

(173)

[contents]

 

 

 

 

9.7.10

[N_∗ as Relevant]

 

[In a similar way, N_∗ is also a relevant logic.]

 

[Priest next shows how in a similar way N_∗ is a relevant logic. Please consult the text for the details.]

A similar argument shows that N_∗ is a relevant logic. Take a ∗ interpretation ⟨W, N, ∗, v⟩, where W = {w₀, w₁, w₂}; N = {w₀}, w^*_o  = w₀, w^*₁ = w₂, w^*₂ = w₁; for every propositional parameter or conditional, D, in A, v_w1(D) = 1 and v_w2(D) = 0; for every propositional parameter or conditional, D, in B, v_w1(D) = 0 and v_w2(D) = 1. One can check that v_w1(A) = 1, and v_w1(B) = 0. Hence v_w0(A → B) = 0. Details are left as an exercise.

(172)

[contents]

 

 

 

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

[Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters.]

 

[We have the intuition already that “for a conditional to be true there must be some connection between its antecedent and consequent” (172). But it is not always obvious how to do that in a formalized way. Yet we saw in section 9.7.8 that one way is to require shared parameters.]

It is a natural thought that for a conditional to be true there must be some connection between its antecedent and consequent. It was precisely this idea that led to the development of relevant logic. A sensible notion of connection is not so easy to spell out, however (as we saw, in effect, in 4.9.2). The parameter-sharing condition of 9.7.8 gives some content to the idea.

(172)

[contents]

 

 

 

 

9.7.12

[Filter Logics]

 

[There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173).]

 

[Priest then notes another sort of relevant logic called filter logics. Here “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter.” But it is a different sort of logic than the ones of this book. Priest notes that often times filter logics break the principle of transitivity.]

There are some approaches to relevant logic where a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter. (These are | sometimes called filter logics, since the extra constraint filters out ‘undesirables’.) Characteristically, such approaches give rise to relevant logics of a kind different from those considered in this book. For example, if the parameter-sharing filter is used, (p ∧ (¬p ∨ q)) → q is valid, which it is not in the relevant logics of this, and subsequent, chapters. Typically (though not invariably), a feature of filter logics is the failure of the principle of transitivity: if A ⊨ B and B ⊨ C then A ⊨ C (thus breaking the argument of 4.9.2).

(173-174)

[contents]

 

 

 

 

9.7.13

[More Than Classical Relevance]

 

[Relevance in our systems here however is not conditions added on top of classical validity.]

 

[The way that we are dealing with relevance here “is not some extra condition imposed on top of classical validity” (174). Rather, it is something else (but I am not sure I understand what it is and what the distinction is. So see the quote below.)]

In the present approach, relevance is not some extra condition imposed on top of classical validity. Rather, relevance, in the form of parameter sharing, falls out of something more fundamental, namely the taking into account of a suitably wide range of situations.

(174)

[contents]

 

 

 

 

9.7.14

[Preserving Conventional Truth in This System]

 

[If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.]

 

[Priest then notes some additional concerns. We might want within this four value-situationed non-normal worlds logic to reserve certain conventional properties of truth for the “real” or actual world. Priest then explains how this would work. We symbolize @ as the actual world, and we say truth in this restricted sense is truth at @, and validity is truth preservation at @ for all interpretations. Then, we can add constraints in @ to model the properties we think truth should have in the actual real world. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts (see section 8.4.6 and section 8.4.9).]

One final comment: one might hold that truth – real truth, not just truth in some world – has some special properties; that unlike truth in an arbitrary world, truth itself can have no gaps or gluts. To accommodate this view, one could take an interpretation to include a distinguished normal world, @ (for actuality), such that truth (simpliciter) is truth at @. Validity would then be defined as truth preservation at @ in all interpretations.⁸ The special properties of truth would be reflected in semantic constraints on @. Thus, if it be held that there are no truth value gluts in @, one would impose the constraint that ρ@ satisfy the condition Exclusion of 8.4.6. If it be held that there are no truth-value gaps in @, then one would impose the constraint that ρ@ satisfy the condition Exhaustion of 8.4.9.⁹ Or in a ∗ interpretation, one might require that @ = @∗, which rules out gaps and gluts. But from the present | perspectives, these conditions would require justification by some novel considerations.

(174-175)

8 One could, in fact, set up all the possible-world semantics that we have had till now in this way. But since these semantics contain nothing to distinguish @ from any other normal world, this would have had no effect on validity.

9 Strictly speaking, these conditions are not sufficient. To rule out truth-value gluts and gaps with formulas containing → s, we need to make another change as well. Specifically, to rule out truth-value gaps, the falsity conditions for A → B at @ have to read:

A → Bρ_@0 iff (for some w′, Aρ_w′1 and Bρ_w′ 0) or (it is not the case that A → Bρ_w@1) and to rule out truth-value gluts, they have to read:

A → Bρ_@0 iff (for some w′, Aρ_w′1 and Bρ_w′ 0) and (it is not the case that A → Bρ_w@1).

(174)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

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