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.”
[Philosophical Issues of the Ternary Relation]
[Interpreting the Ternary Relation as Combining Information]
[The Pooling Interpretation as Validating an Irrelevant Formula]
[R as Information Flow]
[Some Problems with the Information Flow Interpretation]
[The Need for More Work on This Account]
Summary
[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:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, vx(B) = 1
The exception is that if w is a non-normal world:
vw(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if vx(A) = 1, then vy(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)
[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)
[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)
[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)
[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)
[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)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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