The Problem: What Does Relevance Consist In? Following Relevant Logic, we can avoid Classical Logic’s paradoxes (or at least weirdnesses) of Material Implication, according to which the following statements are true…
1) If Cliff lives in Houston, Texas, then the earth has just one moon
2) If Cliff lives in Orange County, California, then Paris, Texas is the capital of France
…by insisting that the antecedent p be relevant to the consequent q.
Two questions immediately becomes pressing: first: what does ‘relevance’ mean? Second, what is it that makes p relevant to q?
First Question: What Does ‘Relevance’ Mean? As I intend to use the term, ‘relevance’ in general is a relation/connection that exists between one situation/state of affairs and another and is important to our concerns. In the case of relevant implication, the aforementioned relation is important to us because it underwrites a guarantee that we can infer q from p. That we can legitimately make inferences is one of our concerns.
Implication is a relation between propositions. One infers one proposition from another. Following Roderick Chisholm, I will be identifying propositions with states of affairs. For example, the proposition that this cat, Felix, is sitting on this Persian mat with MAT_ID 1123581321 is identical with the state of affairs consisting in Felix sitting on the Persian mat with MAT_ID 1123581321. So I will alternate between referring to p and q as propositions and as states of affairs.
By ‘situation’ I mean, roughly, ‘a site comprising one or more connected states of affairs which are available from a possible perspective. A perspective is always limited and therefore does not have available to it other states of affairs. The room in which I am typing this constitutes one situation. In this situation the doorbell’s ringing, when it occurs, is available to me. The button which, when pushed, causes the doorbell to ring is on the wall outside. This state of affairs is hidden from me in my current situation. The immediate vicinity of a person who is about to press the doorbell button outside is another situation. The states of affairs comprised by the room inside are not available to this person.
Second Question: What Is It That Makes p Relevant To q? One at least initially attractive answer to the second question is the following: p is relevant to q when p is information that q.
Here is one issue that I want to bring out into the open from the very start. The careful reader will notice, as they go along, that I am vulnerable to the charge of circularity. I will be analyzing implication in terms of information and information in terms of situations, which in turn I analyze in terms of perspectives. But it would seem that perspectives need to be analyzed in terms of information. You, my gentle reader, my fearsomely implacable judge, will decide later whether I am successful in defending myself against the circularity charge.
In what follows, I will first state what makes treating relevance that way attractive. After dealing with a counter-example that, at first sight, seems completely devastating, I will argue that the INFORMATION THAT relation remains the basis for understanding relevance as it pertains to implication — at least for the examples that I present or link to in this post.
To state the matter a bit abstractly at first, p is information that q when a channel exists through which information flows from a source site, an at least partially-obscured situation s0 (which includes the state of affairs that q), to a reception site, a situation s2 (which includes the state of affairs that p), making the information that q available in s2. Such a channel exists when some state of affairs that c in a situation s1 renders the conditional probability that q given p 11.
The channel may open up between s0 and s1 because s1 is a physical situation comprising states of affairs whose obtaining during a certain stretch of time makes it impossible without violating physical laws for that p to obtain without that q‘s obtaining.
To bring up an example into which I am about to go into much greater detail shortly, during the time that the wiring to a doorbell is in a certain physical condition, it would be impossible for the doorbell to ring without the button outside getting pushed by someone or something. Suppose (as is surely the case) that the doorbell could ring without the button’s getting pushed only if a defective physical condition of the wiring, given the physical laws of the universe, could allow for events x, y, or z occurring (for example, an unwanted electrical pulse caused by a short the wiring). Currently, the wiring is not in this defective condition and will not be so for a stretch of time. (Nothing, for example, could cause a short, given the physical laws of the universe.) Given this current condition of the wiring, the doorbell could ring without the button outside getting pushed only were the physical laws of the universe violated.
In the case of the doorbell, the channel is opened up by the physical condition of the wiring, a condition that functions as a constraint disallowing any doorbell ringing occurring without the proper cause — the button’s getting pushed. This physical constraint underwrites, so to speak, a guarantee that the doorbell will never ring without the button outside getting pushed.
This is a physical, causal constraint. There may be other constraints as well. [Including knowledge, perhaps?]
Another factor that will turn out to be pertinent to p’s being information that q is one’s state of knowledge cum ignorance regarding q. I will be asking later whether this factor poses a problem for regarding p‘s being information that q as the relation that makes p relevant to q by making the truth of an IF THEN statement relative to one’s knowledge.
Initially, the following doorbell example, taken from Fred Dretske’s KNOWLEDGE AND THE FLOW OF INFORMATION2 made this account of p‘s relevance to q highly attractive to me. Warning: what follows will be a veritable operatic doorbell aria. Those who are not fans of operatic arias are advised to go elsewhere.
The Doorbell Aria: You are in a room (s2 ) in which you are able to hear the doorbell. The wiring of the doorbell comprises situation s1. The state of affairs c regarding this wiring is such that in all possible worlds in which the laws of physics of this actual world hold, the doorbell will never ring without someone or something depressing the button outside. (Situation s0 is the ‘outside’, including the button.) This never happens, ever, no matter how much time goes by.
The doorbell’s ringing guarantees that someone or something is depressing the button. There are no poltergeists inside the wiring, no sudden bursts of electrical energy ultimately caused by a butterfly flapping its wings in the Amazon, or anything like that, that will cause the doorbell to ring without the button outside getting depressed. If one takes each occasion on which the doorbell rings, rolls back the clock, then lets the clock roll forward again, but this time with just one tiny change in the world they find themselves in (say, the butterfly flapping its wings in the Amazon has an orange dot on its wings rather than a maroon dot), and if they repeat this exercise for each possible world whose physics is the same as our actual world, someone or something will be depressing the button outside each time. Rinse and repeat for each time the doorbell rings. 100% each time.
100% of the time, when the doorbell rings, the button outside is getting depressed by someone or something. Given the doorbell’s ringing, the conditional probability that the button outside is getting depressed is 1.
The wiring is burdened by a defect, however, that results in the doorbell’s occasionally failing to ring even when the button outside is getting depressed. Let’s say that this failure to ring occurs in 0.001 percent of all the possible worlds in which the laws of physics are identical with those of this actual world. Suppose that each time the button outside gets depressed the clock gets rolled back, then rolled forward again, but into a another possible world whose physics is the same as our actual world but has just one tiny change (for example, in the color of the spot on the wings of the butterfly in the Amazon). In 0.001 percent of these possible worlds, the doorbell fails to ring. Rinse and repeat for each time the button gets pushed. 0.001 percent each time.
0.001% of the time, the doorbell fails to ring when someone or something depresses the button outside. The conditional probability that the doorbell will fail to ring even when the button outside is getting depressed is 0.001. The button’s getting depressed does not guarantee that the doorbell will ring.
If we follow Dretske’s definition of informational content, we will see that the doorbell’s ringing is information that the button outside is getting depressed. We will also see that the button’s getting depressed is not information that the doorbell is ringing inside. This (to anticipate) mirrors the situation in which 3) is true, and 4) is false.
3) IF the doorbell is ringing, THEN someone or something is depressing the button outside.
4) IF someone or something is depressing the button outside, THEN the doorbell is ringing.
Back to 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
Let me linger a bit on “but given k alone, less than 1”. k must be your knowledge cum ignorance of the source situation s0 outside. At the moment, the doorbell is not ringing. You have zero knowledge of how things stand out there with regard to the doorbell’s getting pushed. The value of k is therefore zero. With just this “knowledge” aka ignorance, and in the absence of a signal that the doorbell is getting pushed, the conditional probability that this is happening will be the probability that the doorbell is getting depressed at any given time of the day multiplied by 0.001. This figure, whatever it is, will be considerably less than 1.
Now the doorbell is ringing. All of a sudden, the conditional probability that the button outside is getting pushed has leapt to 1. The doorbell’s ringing is therefore information that the button outside is getting pushed by someone or something.
Correlatively, when I am pushing the button, my knowledge of what is happening inside is zero, provided I am not able to hear the doorbell ringing in any case. Given this knowledge alone, the probability that the doorbell is ringing is 0.999. Given my knowledge plus the button’s getting pushed, that knowledge stays 0.999. Therefore, according to Dretske’s definition of informational content, my pushing the button in this case is not information that the doorbell inside is ringing.
If the INFORMATION THAT relation is what makes for the relevance of p to q in true IF p THEN q statements, then 3) is true because this relation exists between p and q, and 4) is false because this relation does not exist. Likewise, 1) is false because ‘Cliff lives in Houston’ is not information that the earth has just one moon, and 2) is false because even if Cliff moved to Orange County, California, that item would still not be information that Paris, Texas is the capital of France. 1), 2), and 4) are all false because in each statement the antecedent is not relevant to the consequent.
— “Wait a second!” I hear someone objecting. “You mean that ‘someone or something is depressing the button outside’ is not relevant to ‘the doorbell is ringing?” I do think that the notion of degrees of relevance — a relevance spectrum — needs to be introduced here. The truth of ‘Cliff lives in Houston, Texas’ presumably adds exactly 0 to the probability that the earth has a single moon. The truth of ‘I am pushing the button outside’ adds 0.999 to the probability that the doorbell is ringing inside. The truth of the former statement lacks any relevance at all to its consequent. The truth of the latter statement … well, it is not exactly completely irrelevant to its consequent. But I do think this is a matter of ‘close, but no cigar’. The truth of ‘I am pushing this button outside’ is not relevant enough to ‘the doorbell is ringing inside’ to make 4) a true statement.
Assume that an INFORMATION THAT relation exists between p and q in the following truth table except, of course, when the truth value of q makes it impossible for such a relation to exist. (When this happens, of course, the IF THEN statement is also false.) In that case, we would get a truth table for implication that is exactly like the one set forth by proponents of Classical Logic. Except now the truth table makes intuitive sense — even the last row. This is the row in Classical Logic’s truth table for implication that seems absolutely counter-intuitive to anyone sane.
|p||q||IF p THEN q|
Let’s consider the rows one by one:
- ‘Doorbell is ringing’ is true, as is ‘the button outside is getting pushed’ IF p THEN q is obviously true in this case provided that p really is information that q.
- ‘Doorbell is ringing’ is true, while ‘the button outside is getting pushed’ is false. That q is false while p true guarantees that p is in fact not information that q, so IF p THEN q is guaranteed to be false.
- The doorbell is not ringing even though the button outside is getting pushed. p remains information that q when that p is the case, so IF p THEN q is true.
- The doorbell is not ringing, and the button outside is not being pushed. Nonetheless, p would be information that q should that p obtain. So IF p THEN q is true because the INFORMATION THAT relation still exists between p and q.
In short, provided this treatment of relevance is correct (which it is not quite — but I will get to that later), IF p THEN q is true if and only if p is information that q. When (on this treatment of relevance) p is not information that q, then IF p THEN q is false no matter what the truth values of p and q are. This means of course that IF p THEN q cannot be treated in relevant logic as equivalent to NOT p OR q, as it is in Classical Logic.
This, then, is what makes treating relevance as consisting in INFORMATION THAT initially so attractive. First, the INFORMATION THAT relation at work in the doorbell example mirrors in a satisfyingly intuitive way the truth of 3) and the falsity of 4). Second, this treatment provides an intuitive explanation for the fourth row of the truth table for implication given above. Proponents of Classical Logic are notorious for coming up with nothing more satisfying in this regard than ‘If you believe a false statement, you will believe anything’.
As side note, I would like to add that what I discussed in this post is the INFORMATION THAT relation as stemming from physical laws. Here (but this needs to be re-worked) I discuss the INFORMATION THAT relation as stemming from what at first looks like logical principles but which, I think, may be more aptly described as the laws of probability. I do want, after all, to base logic ultimately on something similar to INFORMATION THAT in a non-circular way.
[To sum up: the relevance of p to q is a relation — a connection — between the state of affairs p and the state of affairs q which is important to us because it underwrites inference by guaranteeing q given p.]
Incidentally, the shell-game example discussed in the post just linked to clearly shows that the relevance-making relation cannot be the causal relation, at least not in all cases. Turning over shell #3 to reveal a peanut is a signal carrying information that the peanut is under shell #4, but this action does not cause the peanut to be under shell #4.
However, there is a fly in the intuitive ointment. How is one to deal with statements like the following:
5) IF there is a ruby exactly 2 kilometers underneath my feet, THEN there is a ruby exactly 2 kilometers underneath my feet
or, more generally, with:
6) IF p THEN p
It would be a bit strange to suggest that a channel exists between the situation s0 (the way things stand exactly two kilometers underneath my feet) and the exact same situation s0. It would seem, then, the relevant-making relation cannot be identical with the INFORMATION THAT relation after all.3 Although an identity relation clearly exists “between” s0 and s0, it would seem there is never an INFORMATION THAT relation between “them”.
However, while there are clearly cases in which no INFORMATION THAT relation exists between s0 and s0, adopting Roderick Chisholm’s notions of direct evidence and self-presenting states of affairs suggests that, in some other cases, we can treat that p as information that p. I won’t be implying that Chisholm is correct in thinking that there is such things as direct evidence and self-presenting states of affairs. If there is such a thing, however, it would suggest that sometimes INFORMATION THAT is not always a three-place relation(source, channel, receiver), but sometimes a one-place relation.
Let’s look at Chisholm’s (simpler) statement of what direct evidence consists in:
What justifies me in thinking I know that a is F is simply the fact that a is F.
Roderick Chisholm, THEORY OF KNOWLEDGE, SECOND EDITION, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., p. 21. Henceforth TOK.
For example, when I suffer a sharp pain in my shoulder to which I point and say ‘here’, what justifies me in thinking I know I am suffering a sharp pain here is simply the fact that I am suffering a sharp pain here.
Likewise, if someone asked me the (somewhat strange) question ‘how can you tell there you are suffering a sharp pain there?” I could only answer:
7) I can tell I am suffering a sharp pain here because I am suffering a sharp pain here.
But information consists in what one can tell. It follows, then, that:
8) My suffering a sharp pain here is by itself information that I am suffering a sharp pain here.
A knock at the door (to use something other than the doorbell example for once) announces that someone or something outside is impacting the door. Something not identical with this person or thing does the announcing. The pain, by contrast, is self-announcing. The information in this case doesn’t travel or flow from a source site to a reception site because the source and reception sites are identical.
If one insists that information has to travel from a source site to a reception site, so that self-announcing information cannot really be information, we still have something that is very much like information. For to have information, or at least something that is like information, it suffices that one be able to tell something (that someone or something is depressing the button outside, that the peanut is under shell #4, that I feel pain here). One is able to tell something in all these cases, including the self-announcing case.
This gives another twist to:
9) IF I suffer a sharp pain here, THEN I suffer a sharp pain here.
Here p is relevant to q because q (alternatively p) is a self-announcing state of affairs that is either a case of INFORMATION THAT, or is something very much like INFORMATION THAT.
Let me turn now back those cases in which s0 clearly is not information that s0. ….
I argue, however, that 5) (and, to generalize, 6) are true because, were a ruby to exist exactly two kilometers underneath my feet, the conditional probability that there is a ruby exactly two kilometers underneath my feet would be 1.
Compare with: were the doorbell to ring (given c described above), the conditional probability that the button outside is getting pushed is 1. The doorbell example describes a case of a signal carrying information that because two distinct situations are in play, a source situation that is at least partially concealed from those inhabiting a reception situation. The (at least partial) concealment of a source situation from the perspective of a reception situation concomitant with the (at least partial) ignorance that is inherent in k is required for an INFORMATION THAT relation to exist. Without this, any signal arising from s would be “old information”, that is to say, not information at all.
So I would like to revise Dretske’s definition of informational content to the following:
Informational content: A signal r in reception situation s2 carries the information that t in source situation s0 is F = Because c is G in situation s1, the conditional probability of t‘s being F, given r (and k in s2), is 1 (but, given k alone, less than 1)
This guarantees the truth of IF p THEN q when p is information that q. When there is only a single situation, s0, knowledge (ignorance) k drops out of the picture because there is no longer any situation s2 from whose perspective one has (at least partial) ignorance of what is happening in s0. The signal r also drops out of the picture because we are no longer talking about INFORMATION THAT. What remains, however, is:
The conditional probability of t‘s being F in situation s0 is, given t‘s being F in situation s0, 1.
I think it requires only a moderately keen grasp of the obvious to grasp this point.
So what is common to both the doorbell and the ruby examples is a conditional probability of 1. You get the ‘conditional probability is 1’ feature of the ruby IF p THEN p example by removing features from the INFORMATION THAT relation existing in the doorbell IF p THEN q (where p and q are about states of affairs in distinct situations).
I submit, then, that the two-place relation4 that makes p relevant to p in the statement IF p THEN p is a derivative, degenerate case of the INFORMATION THAT relation. It is what you get by removing features from the IF THEN relation in order to accommodate the drastic simplification of a richer, complex situation s comprising s0, s1, s2 (and k in s2 ) into a more impoverished, simple situation s comprising just s0. This relation is degenerate enough to no longer count as INFORMATION THAT; all that remains of the INFORMATION THAT relation is the ‘conditional probability is 1’ feature; nonetheless, INFORMATION THAT remains the touchstone for understanding all the cases of implication presented or linked to so far — the doorbell case, the shell-game case, and the ruby case.
Or so I am thinking at this moment. We will see if this conclusion will survive consideration of further examples of implication.
1 I think this is identical with the theory of relevance developed by Jon Barwise and later by Greg Restall, as presented in Edwin D. Mares, RELEVANT LOGIC A Philosophical Interpretation, Cambridge and New York, Cambridge University Press 2004, pp. 54-55. Henceforth RELEVANT LOGIC.
I mention situations because I have in mind the Routley-Meyer truth condition for implication, to wit:
‘A → B‘ is true at a situation s if and only if for all situations x and y if Rsxy and ‘A‘ is true at x, then ‘B‘ is true at y. (RELEVANT LOGIC, p. 28.)
What I, at least, am calling a situation is what comprises one or more states of affairs available to one (or more, if the situation is shared) sentient creatures whose limitations prevent them from having direct access (in the absence of a signal) to other states of affairs. The room inside which a person is able to hear the doorbell ringing is situation s2 — the reception situation. The area immediately outside, where another person may be pressing the doorbell, is situation S0. — the source situation. The wiring to the doorbell, which perhaps a gremlin or poltergeist is inhabiting, is situation s1 — the channel situation.
Of course, the fact I am bringing both situations and possible worlds into the discussion is probably a signal, that is to say, a dead-giveaway that I do not yet sufficiently understand the distinction between situations and possible worlds. Keep in mind that this post is an exercise in writing to learn. So I want to issue a warning to non-experts in the field: I probably know less about this stuff than may at first seem to be the case. Needless to say, the actual experts, won’t be fooled.
2 Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 54-55.
3 cf RELEVANT LOGIC, p. 55.
4 — “Wait”, you say. “This is a two-place relation? Isn’t p identical with p? So why isn’t this a one-place relation?” Yes, p is identical with p. But the relation in question is a two-place relation because p is getting stated twice.
Today’s homage to Plato’s SYMPOSIUM is Channing Tatum, who is welcome to fix my pickup truck anytime. (In fact, I think I will buy a pickup truck just so that I can invite him to fix it.)
To distort Plato’s SYMPOSIUM just a little bit, pining after Channing Tatum is the first step on the ladder of Beauty that leads shortly thereafter to appreciation of the beauty of Classical Logic and Relevant Logic, and then, finally, to the form of Beauty — Beauty itself. Of course, my enemies say that I should avoid logic altogether and stick to pining after Channing Tatum.