But What Does ‘Relevance’ Mean? If (at the time of this writing) one googles for a definition of the word ‘relevance’, the gist of what they will get will be something like: a state of affairs1 p is relevant to a state of affairs q when p is connected to q in some way and that connection is important to us in some way. The connection matters.
Any given state of affairs will of course bear a very large (perhaps indefinitely large) number of connections to any other state of affairs. I am trivially connected for example to all people in the world whose last name begins with ‘W’ (I bear a W connection to each of them); and I am trivially connected to everyone else in the world whose last name does not begin with ‘W’ (I bear a non-W connection to each of them).
But some connections matter to us, perhaps in relation to some particular goal, or in relation to some highly pervasive desire. The importance of the connection selects out those cases in which p is relevant to q.
The Ice Example: Warning — I Intend To Use This As A Metaphor For Implication: For example, the thickness/thinness (or even complete absence) of the ice covering a river (state of affairs p) is connected to my reaching the other bank of the river (state of affairs q) by way of enabling/hindering/rendering-impossible my reaching that other bank. This connection matters to me when I have the goal of reaching the other side alive, or at least in some reasonable approximation thereto. (And I have this goal because of something else that matters to me. I need, say, to evade the secret police on this side, or the only food there is exists only on the other side.) The importance of this connection, the place it has in the web of my goals, renders p relevant to q.
So when the Relevant Logician insists that p be relevant to q in propositions of the form IF p THEN q, they can plausibly be construed as asserting that there is some connection between p and q, and this connection is important to us. What this connection is and why it is important to us may be suggested by the following examples. The first example to follow (Madame Olensky) does not quite get us to this connection, but it is suggestive enough to put us on the right track leading to it (The Doorbell).
The Matter Regarding Madame Olensky And Professor Plum: When Madame Olensky is caught standing over the body of Professor Plum with a smoking gun in her hand, this state of affairs (p) bears a definite connection to another (quite) possible state of affairs, namely, that Madame Olensky murdered Professor Plum (q). This connection consists in the fact that p‘s obtaining/being true increases the probability (in this case drastically) that q obtains/is true. That probability is now somewhere greater than 0 but equal to or less than 1. The connection matters to us whenever we are concerned enough to ask (say, out of a desire for justice, I should hope, or at least out of a general desire to get things right): Did Madame Olensky murder Professor Plum? Because this increases-the-probability connection matters to us, it renders Madame Olensky’s standing over the body of Professor Plum (whose last twitches ceased just one second ago) with a smoking gun relevant to the possible state of affairs comprising Madame Olensky’s just having murdered Professor Plum.
But the Relevant Logician will want something a bit stronger for the connection between p and q that will make p relevant to q in propositions of the form IF p THEN q. For in propositions of that form, the obtaining/being true of q is guaranteed should p obtain/be-true. In other words, the probability of q, given p, needs to be 1. Not 0.86, not 0.9999, but 1. Implication needs to be completely reliable.
In other words, the ice needs to be so solid that the chances of falling through, of losing one’s footing and plunging into deep cold water while trying to cross to the consequent q are zero.
Although Madame Olensky’s standing over the body of Professor Plum with a smoking gun definitely increases the probability that she is the murderer of Professor Plum beyond 0, that probability is doubtlessly not 1. For a sufficiently competent writer of mystery novels can invent a scenario just barely within the realm of possibility in which, despite the bald fact that Madame Olensky is standing over the body of Professor Plum with a smoking gun in her hand, she is in fact not the actual murderer of Professor Plum. The probability is, say, a mere 0.99999999999.
In the matter regarding Madame Olensky and Professor Plum, there is a minuscule, but real chance that one might fall through the ice, lose their footing, plunge into the deep cold swift water while crossing to the other bank of the river.
So the statement
1) IF Madame Olensky is standing over the body of Professor Plum with a smoking gun, THEN Madame Olensky is the murderer of Professor Plum
is false. It is false because, although the state of affairs comprising Madame Olensky’s standing over the body of Professor Plum with a smoking gun is definitely relevant to the possible state of affairs comprising Madame Olensky’s being the murderer of Professor Plum, the connection which generates this relevance is not the right relevance-making connection.
The Doorbell (In Perfect Working Order): The right relevance connection does exist, I think, taking a cue from Fred Dretske, in the case of a doorbell whose wiring is in perfect condition. Given the condition of the wiring, the probability, when the doorbell is ringing (p), that someone outside is pushing the doorbell button, or that, at least, something is depressing that button (q), is 1. The constraint created by the perfect condition of the wiring makes p a completely reliable indicator of q. So this IF THEN statement:
2) IF the doorbell is ringing THEN someone or something is depressing the button outside
is true. That someone or something outside is depressing the doorbell button is guaranteed by the doorbell’s ringing inside.
This particular increases-the-probability (to 1) connection between the doorbell’s ringing and someone-or-something’s depressing the button outside matters to (most of) us because there is, I should think, a pervasive desire to get things right, to know how things actually stand outside the room, to know what is actually the case among the things that are not immediately present to us, to be able to tell what is happening. This mattering selects out this particular connection as a relevance-making connection between p and q. Because of this relevance of p to q, 2) above is true.
The doorbell’s ringing (when the condition of the wiring is perfect) is, of course, the classic example of Information That, of informational content. The ringing (r, for reception) is information that the button outside is getting depressed (s, for source), if we follow Dretske’s definition of informational content:
Informational content: A signal r carries the information that s is F = The conditional probability of s‘s being F, given r (and k), is 1 (but, given k alone, less than 1)
Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, p. 65
I will dwell on the knowledge k part of this definition in some detail later.
That the conditional probability of the button outside’s getting depressed increases to 1 when the doorbell rings is both what makes the ringing a signal, information that the button outside is getting depressed and what makes p relevant to q in 2) above. Therefore, it is tempting to identity the relevance-making relation between p an q with the information-that relation. Implication, it is tempting to say, is always information that. The following:
3) IF Cliff lives in Houston THEN the earth has just one moon
fails to be a true implication because Cliff’s living in Houston is not information that the earth has just one moon. I will be returning to this point later.2
To revert back to the river ice metaphor, the antecedent in 3) is ice that never formed in the first place. There is no chance one can cross to the consequent q on the basis of p. One cannot even lose their footing here, because there was only ever swift cold water to plunge into.
However, there are of course a number of rather severe challenges to the notion that implication is always information. I will consider some of these in the snippets that follow.
At the time of this writing, I am suffering under the delusion that once all the challenges that I have considered so far have been dealt with, one ends up with the concept of relevant implication as always to be made sense of in terms of the concept of information — sometimes as full-blooded information, sometimes as degenerate or denatured information, and sometimes as the radical absence of information. Whichever is the case, there is always the reference to the concept of information. We will see if I end up having to eat crow on this point.
Some Housekeeping: First, however, I want to do some housekeeping. The careful reader will notice that I keep shifting back and forth between talking about p and q as states of affairs and as propositions. I will continue to shift back and forth because I will be following Roderick Chisholm in treating propositions as a subspecies of states of affairs.3 The state of affairs comprising this cat, Munti sitting on this Persian mat can obtain or not obtain at different times. The state of affairs comprising ‘Munti is sitting on on this Persian mat on October 31 at 12:00 am’ either always obtains or never obtains according as it was true or not true October 31 at 12:00 am that Munti was sitting on the Persian mat. The latter is a state of affairs (obtaining or not obtaining) that is also a proposition (true or false); the former is a state of affairs (obtaining or not obtaining) that is not also a proposition.
Propositions are true or false; a proposition can follow from another or fail to follow from it. Implication, therefore, is a relation between sets of states of affairs obtaining/failing to obtain being true/failing to be true at particular times (the doorbell is ringing at times t0, t1, t2, t3 … tn) and the button outside is getting pushed at times t0, t1, t2, t3 … tn).
One Final Point: I have defined relevance in terms of mattering. Since in Relevant Logic p has to be relevant to q in implication propositions in order for those implications to be true, does this mean that no implication statement was true before any sentient creature existed to whom anything could matter? (I don’t think so, but this still needs to be shown, of course.) If so, is this a weirdness that is off-putting enough to make one prefer Classical Logic to Relevant Logic?
1 I will leave ‘state of affairs’ as an undefined primitive.
2 One reason p is not information that q here is, of course, that the earth has just one moon is “old information” and therefore not information at all. But the more important reason is that even if this were not “old information”, Cliff’s living in Houston would still not be information that the earth has just one moon because the former, by itself, leaves the probability of the latter at 0. This ‘even if’ is pertinent to my claim that implication is to be understood in terms of information even if a particular example of an implication proposition is not an instance of information that.
3 Roderick Chisholm, THEORY OF KNOWLEDGE SECOND EDITION, Englewood Cliffs, Prentice-Hall, Inc., 1977, pp. 87-88.
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Edit Log: June 04, 2017: Made some fairly minor edits in an always-ongoing and never-fully-accomplished effort to avoid complete and total embarrassment.