Category Archives: Informational Content And Relevance In Relevant Logic

A Question For Edwin D. Mares

This is the question I have at the moment for Edwin Mares: Doesn’t the relevant conditional ‘if p then p‘ (I use ‘conditional’ to include implication and entailment — more on this below) pose as much of a challenge to an interpretation of the relevance-making relation Rstu in which s is an ‘information link’ enabling ‘situated inference’ as it does to the interpretation in which s is a channel of information? What precisely is the difference between an ‘information link’ and a ‘channel of information’ anyhow? [Doesn’t implication require some sort of necessity (This is the insight behind strict implication) even if not logical necessity? We can model different kinds of necessity based on different accessibility relations. One of these surely apply. Is it possible to do this with situations rather than possible worlds? Can we make an implication both necessary and defeasible? (Yes, with epistemically accessible possible worlds)

As I explored this issue in an attempt to make my understanding of Relevant Logic a bit less shallow, I came to the conclusion that ‘information link as part of situated inference’ does better account for the implication than does ‘information channel’. But of course more questions arose in my mind.

Let me start motivating these questions by explaining the interpretation of Rstu in which s is a “situation” serving as an information channel connecting two other situations, t and u. Much of what can be said about s as ‘information channel’ will also hold true for s as information link — or at least to what degree this is so will be one of my questions. And in the course of doing that, I hope to make sufficiently clear what in the heck Rstu is in the first place — sufficiently clear, I hope, to make it reasonable to think — reasonable at least by my own lights even even if by no one else’s — that I have some vague inkling of what I am talking about.

After exploring the ‘channel of information’ and ‘situated inference supported by information link’ interpretations of the relevance-making relation Rstu, I will briefly look at one more interpretation, which I will call the ‘persistent lack of counterexamples’ interpretation.

A Problem That Arises When The Relevance-Making Relation R Gets Interpreted In Channel Theoretic Terms: On pages 54-55 of his RELEVANT LOGIC A Philosophical Interpretation Mares says:

Barwise … and Restall … have suggested treating the Routley-Meyer semantics in channel theoretic terms. We should think of situations as being sites of information. One site, for example, contains the news reporter at Ground Zero. Another site contains me and my lounge and television set.

Some situations are also channels of information that connect sites. On their interpretation, we read Rstu as saying that s is a channel between t and u.

This interpretation motivates a slightly weaker logic than R. Here is the problem: on the channel theoretic reading, Rsss says that s is a channel from itself to itself. It does not seem intuitive to require of all situations that they be [at least?] channels from themselves to themselves, at least not if the notion of a channel is a generalization of cases like that of the link between television cameras and a television.

Edwin D. Mares, RELEVANT LOGIC A Philosophical Interpretation (Cambridge, Cambridge University Press, 2004), pp. 54-55. Henceforth RL.

But if we look at the matter more closely, we end up talking anyway about situated inference enabled by information links. And in the process we come up with an alternative to having to posit a situation that serves as a channel of information to itself. Finally, while I don’t know about you, dear reader, I at least end up with a somewhat less shallow understanding of Relevant Logic.

[Situated inference via information link, channel of information which I take to be a special case of this; lack of counterexamples through time.]

We end up talking anyway about situated inference enabled by information links: What is an information channel anyhow, and why should we identify the situation s with it?

Consider the doorbell to my apartment. My favorite doorbell repairman, who has 25 years experience repairing doorbells, has just given it a health check and declared that it is, without the slightest shadow of doubt, functioning properly. (I feel obligated to ensure that the examples I adduce in my philosophical thought experiments are in good working order.) 100% of the time, when my doorbell rings, someone (so far it had not been — alas! — Channing Tatum) or something (for example, a twig blown onto it with gale force by Hurricane Ike or Hurricane Harvey) is depressing the button outside. The connection between the doorbell’s ringing and the button’s getting pressed seems absolutely reliable. The reliability of the connection seems perfect. (But note the weasel word ‘seems’.)

Because of this reliability, of this 100%, I regard the doorbell’s ringing as telling. It counts as information that someone or something is pressing the button outside. If, somehow, someone or something was pressing the doorbell just 99% of the time when the doorbell rang (say, I had good reason to suppose, on the basis of a camera directed a the button, that no one or no thing was depressing the button during 1% of the times), that the doorbell is ringing would no longer be telling. I would both be in a twilight-zonish mood, and I would have to consider the possibility, when the ringing sound happens, that no one or nothing was pressing the button outside. The doorbell’s ringing would no longer have the status of information that.

If I may be allowed to wade for a moment into some murky waters, the ringing sound has to increase to 1 the probability that the button outside is getting depressed for that sound to count as information that the button is getting depressed. (Yes, I know that it is sometimes supposed that only logical truths can achieve a probability of 1, but bear with me for a while. Maybe for a long while.)

Because the doorbell apparatus is in a state such that, at least when the apparatus is in this state, the ringing sound raises to 1 the probability that the button outside is getting depressed, that apparatus counts as a channel of information linking the former to the latter. Let me call this channel, the situation of the doorbell apparatus, situation s. This situation, I submit, renders true the following implication:

1a) If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

Note that s has to be in place for 1a) to be true. If a different situation were in place the implication would not be true. If, for example, the apparatus were completely disfunctional so that the causal chain from button to sound never occurred, my hearing the doorbell sound, through whatever bizarre, non-normal freak circumstances (say, aliens were producing it somehow), would obviously not raise to 1 the probability the button outside was getting pressed, and would not be information that this is so. Similarly, if the apparatus were completely functional, but someone created a wireless connection to the part that actually creates the sound so that the sound can be produced bypassing the causal chain from the button to that component, a different situation s would be in place, one in which the previous channel of information no longer existed. Likewise, the previous situation s would no longer be in existence were some physical peculiarity of the apparatus to arise (not really imaginable by me) that made it possible for a wave of electricity to form independenttly of the button’s getting depressed and triggering the ringing sound.

In other words, the apparatus has to be in such a condition — s has to be such — that, as long as s is in place, the button’s getting depressed is the only “possible” cause of the ringing sound’s getting produced. Showing how this could be can get a little tricky; one needs, I think, to mark off s from other situations and isolate it, I believe, conceptually from them. (Consider this to be promissary note 1.) But before discussing this, let me turn to the twin brother of 1a):

1b) If someone or something is depressing the button outside, then the doorbell inside is ringing

As long as the doorbell apparatus is in good condition (situation s), pressing the button outside will result in the ringing sound getting generated inside. That is to say, provided that the laws of physics don’t change. Let me venture, then, the claim that the production of the sound is nomically necessary, given s, when the button is pushed. In all possible worlds which obey the laws of physicals of this actual world (now), pressing the button will generate the ringing sound provided situation s holds. If, say, three months from now the laws of physics abruptly change so that, in addition to grand pianos falling upward, pushing the button no longer generates the sound even given situation s, that world would not be nomically accessible from the actual world now. It would be excluded from the set of all possible worlds nomically accessible from this actual world.

Because the production of the sound is nomically necessary in s, that the doorbell is being pushed is, I venture, information that the ringing sound is getting generated inside. This is so even if one does not hear this sound as they push the button. That pressing the button serves as information (given s) that the ringing sound is happening inside renders 1b) a true implication and gives one a license to infer that this is so.

Should s change so that, say, pressing the button generates the ringing sound just 99% of the time, the button’s getting pressed would no longer be information that the sound was happening inside. Let’s say that a wire has come loose in such a way that the connection is made just most of the time, not always. In those 99% of the cases, the causal relation would be there, but, because of the 1% of the cases, the informational relation is not there. Even when the pressing the button does result in the sound’s getting generated, pressing the button is not information that the sound getting generated. To be a channel of information, s has to result in perfect reliability. In the following passage, Mares is talking about informational links, which is, I take it, different from a channel of information (I take the latter to be a special case of the former); nonetheless, what he says here about informational links also applies to channels of information:

To be an informational link [or a channel of information] a relation needs to be perfectly reliable. As Barwise and Seligman argue, causal relations are often not reliable enough to be considered informational links. They use the example of a flashlight with an unreliable connection between the button and the light ((Barwise and Seligman 1997), p. 17). Sometimes pushing the button turns on the light, but sometimes it does not when other factors come into play, such as a wire’s coming loose.

RL, p. 44

Nomic necessity renders 1b) true. I think, however, that more than nomic necessity is needed to render 1a) true. I now turn to 1a) again (making good promissary note 1).

When the doorbell apparatus is not perfectly reliable, 1a) may still be true. For even though pressing the button does not produce the ringing sound 1% of the time, it remains true that 100% of the time someone or something is depressing the button outside when it does ring. This is why affirming the consequent is a fallacy. One cannot go from:

If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

Someone or something is pressing the button outside

Therefore the doorbell is ringing

For if s doesn’t hold because the apparatus is faulty, while there may be a flow of electricity from button to the part producing the sound, there is no flow of information. Certainly in this particular case the 1% failure rate precludes the consequent from being information that the antecedent is true. So no conclusion that the sound is getting produced for you.

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he circumstance in which the condition of the doorbell apparatus is in this state of the doorbell apparatus situ 100% of the time when the ringing sound is produced, it serves as a channel of information linking the latter to the former. is functioning properly, it is able to serve as a Let me start with an example that, I believe, will make the channel of information conception intuitive.

The following example makes the case for treating s in information-theoretic terms powerfully intuitive, at least for me. When the doorbell apparatus of my apartment is in good condition, pressing the button outside reliably generates a ‘doorbell’ sound inside my apartment; and when I hear the doorbell sound inside my apartment, 100% of the time someone or something is depressing the button outside. Given the good condition of the apparatus, and given the laws of physics, that apparatus serves as a channel of information s through which information flows from the button’s getting depressed (situation u) to the doorbell sound inside my apartment (situation t), such that when I hear the doorbell sound, I have received information that the button outside is getting depressed. This flow of information supports the following two implications (assuming these are implications and not a defeasible natural-language indicative conditional, but I will ignore the difference for now).

1a) If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

1b) If someone or something is depressing the button outside, then the doorbell inside is ringing

It is the information, not the causal relation that supports these two implications. (I will assume they are implications and not natural-language conditionals.) Provided that 100% of the time — provided there are no counterexamples, ever — someone or something is depressing the button outside when the ringing sound is produced inside, the latter is information that the former is occurring. This of course is not a causal relation going from sound to button, though it is supported by a causal relation from button to sound. If a counterexample ever occurs — a ringing sound without the button outside getting depressed — the sound can no longer count as information that the button is getting depressed. This has to be 100%. It has to be 100%. This presupposes that the causal chain starting with the button’s getting pushed is the “only possible” cause of the the sound, in a sense of “only possible” I am about to get to.

1b) because they would not be true if something were wrong with the doorbell apparatus. If, say, some of the wiring in that apparatus came loose so that (full disclosure: I know precious little about doorbell apparati) pressing the button outside resulted in the ringing sound inside just 99% of the time, 1b) would be false because the button’s getting pushed would no longer be information that the doorbell was sounding inside. This is so even though there is a causal relation between the antecedent and the consequent.

But given the good condition of the doorbell apparatus, there is no way — in a sense of ‘no way’ that I am about to explore – that the doorbell can ring and the button outside not get depressed. Likewise, there is “no way” that the button outside can be pressed and the doorbell inside not ring. That is to say, the doorbell must be a reliable conduit of information between u (the outside of my apartment) and t (the inside of my apartment, where I hear the ring sound). This conduit must be perfectly reliable, Mares says:

To be an informational link a relations needs to be perfectly reliable. As Barwise and Seligman argue, causal relations are often no reliable enough to be considered informational links. They use the example of a flashlight with an unreliable connection between the button and the light ((Barwise and Seligman 1997), p. 17). Sometimes pushing the button turns on the light, but sometimes it does not when other factors come into play, such as a wire’s coming loose.

RL, p. 44

[The reader should compare this concept of information with Dretske’s dicussion of that concept in his KNOWLEDGE AND THE FLOW OF INFORMATION. ]

1b above is contingently true because it is true only when and only as long as the apparatus is perfectly reliable in transforming the energy of the button’s getting pushed into the ringing sound. There are possible worlds — say, the actual world three months from now — in which 1b is false. Three months, from now, the apparatus might not be in good condition because some of the wiring has come loose (full disclosure: I am completely ignorant of how doorbell apparati work), so that pushing the button outside resulted in the ringing sound inside just 99% of the time, not 100%. The causal connection would not be perfectly reliable, and therefore could not ground an information-that relation between button and ringing sound.

Reliability is a property attaching (or failing to attach) to equipment and to signals. The doorbell’s ringing is a reliable signal that the button outside is getting pushed because the doorbell apparatus is functioning reliably. Contingency, on the other hand, is a property attaching to propositions and to relations between propositions. That the doorbell is ringing is a contingent proposition because it is possible for it not to ring. The proposition expressed by the English sentence ‘the button is getting pushed’ is related contingently to the proposition ‘a ringing sound is occurring inside’ because the latter’s following from the former hinges upon the reliability of the doorbell apparatus. Because this relation is contigent, 1b is a contingent proposition. Likewise, 1a) is contingent because, even when the doorbell apparatus is functioning reliably, someone could conceivably set up some wireless Rube Goldberg setup producing the ringing sound in such a way that bypasses the button.

But the apparatus also has to be reliable — indeed, it has to be perfectly reliable. If there even one case occurred in which s held (the doorbell apparatus is in good condition), the button outside was pushed, but no ringing sound ensued — and how could that happen if all of a sudden the laws of physics no longer held? — then 1b would not be a true implication. Likewise, if there were even one case in which the doorbell rang (t), and the apparatus was in good condition (s), but no person or thing was depressing the button outside (u) — suppose, quite implausibly, that a poltergeist is at work inside the apparatus, in no way degrading its condition — then 1a) would not be a true implication.

The connection between antecedent and consequent in 1a and 1b) is contingent (RL, p. The antonym for ‘contingent’ is ‘necessary’.

In these cases, 1a) and 1b) would be defeasible. A proposition is defeasible if extra information can come to light showing that it is false. But implication, Mares says, is indefeasible:

A defeasible inference is one that may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, pl 45

But isn’t it the case that all contingent propositions (among which empirical propositions occupy pride of place) are defeasible? The proposition expressed by the English sentence ‘every body with mass warps the space around it’ could — in some sense of ‘could’ — turn out to be false if we found a body with mass that does not warp the space around it. That the doorbell’s ringing always means the button outside is getting pushed could turn out to be false if we find out that poltergeists exist after all (maybe it is is some weird quantum entanglement thing), or, a bit closer to home, there is some physical law that would allow for a burst of energy suddenly appearing leading to the production of the ringing sound in a way that bypasses the button. In that case, it would seem, all “implications” would be defeasible, that is to say, there would be no implications at all. At best, whenever it seems extremely implausible that further information would make the falsity of an implication inescapable, we can posit the indefeasibilty of certain if-then propositions as an ideal. Implications in the narrowest sense of the term would exist in an idealized realm in which, in contrast to all the murky uncertainty of the world we live in, there is not even the faintest shadow of a doubt that there are no poltergeists.

Let’s say that non-idealized, defeasible implications are a species of natural language conditionals. Natural-language implications, as opposed to ‘in the strictest sense of the term implications’, have defeasibility as one of their properties. They also have as one of their properties non-transitiveness which make them resistant to regimentation. But these are outside of the scope of this post, and I will not be considering them any further. Henceforth, unless otherwise noted, I will be using ‘implication’ to mean ‘implication in the strictest sense of the term.’ I now turn back to these.

[To be true in the strictest sense of ‘true’, and not just an idealization, aOfn implication has to indefeasible. And of course, ‘indefeasible’ is a synonym for ‘necessary’. As is indeed, to come full circle, ‘perfectly reliable’ — as in there is no way, given situation s, that when the doorbell is ringing there is someone or something outside not depressing the button. Deductive inference also implies necessity]

It would seem that we have fallen into a contradiction. Implication, as opposed to entailment, is contigent, not necessary. But contingent implications — if any exist — are indefeasible. Indefeasibility is necessity. Extra information cannot show up later that would render the indefeasible implication false. How can this apparent contradiction be resolved?

But before I try to resolve this contradiction I want to point out that I have just arrived at situated inference. Implication is inextricably tied to inference because we must be able to infer the consequent from the antecedent.

Mares ties this reliability to deductive inference. If I may add yet one more plate to the stack, I will be exploring the connections between reliability, information, and deductive inference shortly. In a nutshell, implication needs to mirror deductive inference. For now, however, I would like to note the connection Mares wants to draw between deduction, which is non-defeasible, and the ‘perfect reliability’ of the informational connection between sites required by implication.

The problem [posed by unreliable causal connections] is that unreliable connections do not warrant deductive inference. At best, they can be used to justify defeasible inference. A defeasible inference is one that may or may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, pp. 44-45

Add to the mix the idea that implication is a contingent, not a necessary conditional (it is not entailment), and we have a number of threads that need to be untangled. Let me start with untangling the threads the ‘conditional simpliciter’ (Mares does not include the word ‘simpliciter’), to implication, to entailment, and to ‘natural-language conditionals. Unless otherwise noted, I will be restricting my discussion to relevant versions of all of these. (Complications, complications.)

Conditionals simpliciter comprise any If Then proposition that is (at least) transitive and obeys whatever other rules which I will leave unlisted for now. Conditionals simpliciter come in two flavors: implications and entailments. The relation of implication is contigent; the relation of entailment is necessary. To say that the relation of implication is contingent is to say (roughly) that things could have been otherwise, for which we can advance as evidence that we can imagine them to have been otherwise. The doorbell apparatus for example is currently in good condition, but we can easily imagine it to be defective such that pressing the button results in the ringing sound just 99% of the time. So the implication relation expressed by 1b) above is contigent — things could easily have been otherwise such that the relation would not have held. Similarly, to use Mares’ example, the following implication held (at least at the time of his writing):

In New Zealand, if one does not pay income tax on honoraria given for presenting seminars at other universities, then one is in violation of the tax code.

Here the relationship between not paying income tax on an honorarium and a violation of the New Zealand tax code is not a necessary connection. We can easily imagine a world in which the tax code were different such that it made honoraria tax exempt income. When we make claims like the one above, we do so assuming other facts that connect the failure to pay tax and a violation of the law, that is, particular facts about the New Zealand tax code. Thus the way in which the violation of the law and the failure to pay tax are connected is contingent.

RL, pp. 10-11

The relations between the axioms of arithmetic and particular true arithmetical propositions are, on the other hand, entailments. They are “necessary” — where for the moment at least “necessary” means metaphysically aka logically necessary, i.e. true in all worlds metaphysically/logically accessible from the actual world. For now, I will be touching metaphysical and logical possibility very lightly, except to say that, like Mares “I follow the mainstream in the metaphysics of modality in equating metaphysical and logical possibility” (RL, p. 14).

We say that the relationship between the axioms of arithmetic and 2 + 2 = 4 is that of entailment and the relationship between the failure to pay tax and the violation of the law is that of implication.

RL, p. 11

So far …. natural language conditionals those implications that refuse to be regimented into the rules governing implications and which are defeasible and no move is made to idealize …. and In some sense of ‘absolute’, this “no way” must be absolute. In some sense of “must” the button outside must be getting depressed if the doorbell is ringing and pressing the button must result in the doorbell’s ringing. In some sense of ‘non-defeasible’, the relation must be non-defeasible. Because of this reliability, this ‘absolute no way’, this ‘must’, this non-defeasible character, the ringing sound serves as information that the button is getting depressed, and the button’s getting depressed serves as information (whether this information gets picked up or not) that there is a ringing sound inside.

But the relation between antecedent and consequent in 1a) and 1b) is a contingent relation. In a possible world — say, the actual world three months from now — in which the wiring in the doorbell apparatus came loose in such a way that pressing the button outside resulted in just an intermittent production of the ringing (say, it produces the sound just 99%, no longer 100% of the time), pressing the doorbell would no longer be information that the doorbell was ringing inside. 1b) would then no longer be a true implication. Likewise, if someone constructed some sort of Rube Goldberg contraption hooked up to the doorbell apparatus such that the doorbell could be gotten to ring without the aid of the button, then 1a) would no longer be true, and the doorbell’s ringing inside would no longer be information that the button outside was getting depressed.

The truth of 1a) and 1b) is hinges upon situation s, the situation comprising the doorbell in its current good condition. It is contingent upon that situation; it holds only given that situation; 1a) and 1b) are true only for that situation. Put a bit more opaquely, 1a) and 1b) are true only “ins.

‘Contingent’ is usually opposed to ‘necessary’. But I asserted above that the relation between situations connected by a channel of information is in some sense a necessary one. What are we to make of this necessity?

The following attempt to say what ‘necessary’ means in the sentence ‘It is necessary for Jones to pay his bills’ may prove illuminating. Consider all the possible worlds that are accessible from the actual world by virtue of having Jones both present in all of them and enmeshed in the same social and economic institutions as he is in the actual world. In all of these possible worlds, unpleasant things happen to Jones if he does not pay his bills. Now restrict these to all those possible worlds in which Jones has the means to pay his bills. Now, again, narrow these worlds down further to those in which nothing impedes Jones’ paying his bills (the internet goes down, the postal workers go on strike, Jones has a masochistic desire to be a martyr to capitalism). Again, remove Jones from those worlds in which, suffering from a crippling psychotic anxiety revealing ala Heidegger the world as a totality, finds that nothing has meaning and finds himself unable to engage in the world as average everyday bill-paying Dasein. Again, remove from those worlds the worlds in which Jones, desiring to be a Sartrean hero gloriously exemplifying the same absolute freedom and spontaneity of consciousness exhibited by someone, who, standing before the yawning expanse of the Grand Canyon, decides to jump for no other reason than that they have the freedom to do so; simply decides not to pay his bills.

In other words, remove from all of these possible worlds anything that might remove Jones from the normal, mundane, boring course of paying his bills in order to avoid unpleasant things from happening. In all the possible worlds accessible this way from the actual world, Jones pays his bills. It is necessary that he pay his bills, meaning, he does so in all possible worlds that are accessible from the actual world in the way just described.

Resolving the contradiction: In a similar, in attempting to say in what sense it is ‘necessary’ that the button outside be depressed when the doorbell is ringing inside, or that the doorbell be ringing when the button outside is depressed, we can try to define an accessibility relation by starting with all possible worlds nomically accessible from the actual world (shares with the actual world the same laws of physics), then paring those down. We remove all those possible worlds in which the doorbell apparatus is not in good working order. We then require that, in addition to being nomically accessible, all the remaining worlds be epistemically accessible. A possible world is epistemically accessible from the actual world for a knower S when S knows nothing that would rule out the identity of the actual world with that possible world. For example, I know nothing that would rule out the identity with the actual world of the following possible world: at such and such GPS coordinates specifying a spot in the Amazon (I leave the production of these coordinates as an exercise for the reader), a violet butterfly is flapping its wings. This possible world is epistemically accessible from the actual world relative to me.

Now at the time of this writing I do not know anything that would rule out the identity with the actual world of a possible world in which poltergeists do not exist. This would rule out any scenario in which a poltergeist disrupted the reliability of 1a) by suddenly causing the doorbell to sound without the button’s getting pushed, or disrupted the reliability of 1b) by cutting off the normal chain of events from button to sound. Likewise, I do not know anything at the time of this writing that would rule out the identity with the actual world of the possible world in which freak occurrences such as a sudden burst of electrical energy coming out of the blue (maybe ultimately caused by that violet butterfly flapping its wings in the Amazon) having no causal link to the button but still in accordance with the laws of physicals causes the doorbell sound. Nor do I know anything that, still in accordance with the laws of physics, would cause the normal stream of causal events to suddenly stop, even though the doorbell apparatus is still in perfect condition, right after the button outside was pushed.

Obviously there is some risk that researchers might someday come to find with certainty that poltergeists do exist and do influence physical events (maybe through some kind of quantum entanglement, if I may be permitted to violate, at least this once, my rule that one should not bring up quantum mechanics in a philosophical discussion unless they have have completed at least eight graduate courses in quantum mechanics with no grade lower than a B+ in any of them, and that B+ is allowed in just one of them.) Were these researchers to impart their knowledge to me, the possible world in which poltergeists do not exist would no longer be epistemically accessible to me from the actual world.

Likewise, I am always at risk of finding out that researchers might uncover with certainty the occurrence of the freak physical events just described (the sudden burst of electricity in the wiring coming out of the blue; the strange sudden halting of the causal chain from button to sound). And were we to discover this could still eliminate from our possible worlds those physical conditions which lead to these freak events. In this way, I hope to define a realm of the normal mundane but still recognize that, since we don’t know everything, there are possibilities at the margin.

In this way, we can define a sense in which the relation between antecedent and consequent in 1a) and 1b) is, in a limited sense, necessary, but also contingent and defeasible. This is the sense that I hope to give to the “must” that is involved.

Jones paying his bills in order to prevent unpleasant things from happening is part of the ‘normal unimpeded stream.’ The stream of causal events going from button getting pushed to doorbell sound is the ‘normal unimpeded stream.’ The opposite of ‘normal’ is ‘freak’.

Entailment is implication plus necessity. Situated inference explains why we need Rsss and not just Rss. Because implication requires some kind of necessity, there is no sharp dividing line between implication and entailment. Conditionals in general are divided into two groups: implications and entailments. Natural-language conditions are broader than implications.

Given the situation s comprising the apparatus in its current state, situation u in which the doorbell is ringing inside my apartment, and situation t in which the button outside is getting depressed, 1) is supported by the relation Rstu. The implication is true given s, or for s. Put a bit more opaquely, 1) is true in s.

Unless it is clear from the context that something else is going on, I will put the ‘relevance-making’ situation (in this case, the situation comprising the channel) in the first position after the ‘R’. Apart from my wanting to maintain a certain convention, the position of the variable naming the situation does not matter.

Rstu is an accessibility relation, where the concept ‘accessibility’ is borrowed from ‘possible world’ talk. Here, one situation (the doorbell’s ringing inside my apartment) is giving me access to another situation (the outside of my apartment) that would otherwise be closed off from me. This particular example, I submit, makes interpreting Rstu in terms of channels of information connecting sites powerfully intuitive.

All of this is very well and good for the implication that the button outside is getting depressed if the doorbell is ringing inside. But what about the implication:

2) If the doorbell is ringing, then the doorbell is ringing

? This of course is the sort of proposition uttering which tends to lead to the exclamation “You do have such a wonderfully intuitive sense for the blazingly obvious” or “You are so smart! What does your boyfriend feed you?”. But this does not make the proposition any less true. It is of course necessarily true in a way that makes it an example of entailment, not of contingent implication. But (at least I am assuming for the present) that it is a case of relevant entailment, which, if I understand correctly, is for Mares relevant implication plus necessity. So we still need to understand what makes p relevant to p here.

The ternary relation seems to be necessary for relevance. Why? Well, be patient — I will get to that eventually. But here there is just a single situation — let’s call it s — namely, the inside of my apartment (in, say, the actual world), where I hear the doorbell sound. If the ternary supports relevance in this case, then, it would have to be Rsss. In other words, the same situation is getting named three times. So the ‘relevance-making’ situation, the information channel, has to be, somehow, a channel from itself to itself. I take it this is the sort of thing Mares means when he says: “…on the channel theoretic reading, Rsss says that s is a channel from itself to itself.”

Well, that certainly sounds weird. There does not seem to be any ‘initial s‘ that can serve as the mediating s in Rsss — nothing that serves the physical function of the doorbell apparatus as a channel of information. But surely the doorbell’s ringing in my apartment is relevant (at the very least in a ‘you have a wonderfully intuitive sense for the glaringly obvious’ sort of way!) to the question whether the doorbell is ringing in my apartment, just as its ringing in my apartment is relevant to the question whether someone or something is depressing the button outside. But if a channel of information does not account for the relevance here, what relation does?

One possible answer that comes to mind is ‘the binary relation that holds between possible worlds in strict implication’. That is what makes the antecedent p relevant to the consequent p in ‘if p then p‘:

The obvious way to model implication in this [Kripke-style possible world semantics] is to use the following truth condition: ‘A –> B‘ is true at a world w if and only if for every world w’ such that w’ is accessible from w, either ‘A‘ is false in w’ or ‘B‘ is true in w’

RL (pp. 26-27

So if we add modal necessity to material implication do we make p relevant to p? Does given ‘A‘ then necessarily we have ‘A‘ makes ‘A’ relevant to ‘A’. Doesn’t necessity automatically give relevance? There is no glue between p and q in Material Implication, which is why p’s relevance to q is of no import in Material Implication. Might necessity provide a strong enough glue? Maybe we don’t need all this information channel stuff to account for relevance after all.

Well, no, certainly not if necessity is interpreted as what holds in strict implication. For the following is true as a strict implication even though the antecedent is clearly not relevant to the consequent:

3) If (if the doorbell is ringing then the doorbell is ringing) then (if the earth has just one moon then the earth has just one moon)

Label the antecedent (if the doorbell is ringing then the doorbell is ringing) as p. Label ‘if the earth has just one moon then the earth has just one moon’ as q. In every possible world accessible from the actual world, either q is true or it is false. In classical logic, which is getting used here to talk about this stuff, this is equivalent to ‘if q then q‘. So in every possible world accessible from the actual world either p is false in that world or ‘if q then q‘ is true in that world. So given the truth condition for strict implication given above, ‘If p then (if q then q)’ is true in the actual world as a strict implication. But as we can see from the example given in 3), this is clearly unsatisfactory,

To switch metaphors from ‘glue’ to ‘sieves’, the necessity in strict implication is not fine-meshed enough to filter out all irrelevant implications. Relevance is one thing; necessity seems to be a different thing.

On the other hand, some sort of necessity seems to be a … well … necessary … condition for relevant implication. For a true implication cannot be defeasible:

A defeasible inference is one that may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, p. 45

And non-defeasibility seems to imply some sort of necessity. I hope to explore what this sort of necessity in further writing-to-learn exercises. For now, however, I would like to return to the difficulties that ‘if p then p‘ poses for an interpretation of relevant implication.

Necessity by itself can’t constitute the relevance-making relation. So we are back to the ternary relation — in the case of 2) Rsss, which, interpreted in information-theoretic terms, seems to pose the bizarre requirement that a situation be a channel of information from itself to itself. But why are we stuck with the ternary relation? Why can’t the relation be a binary one, such as the identity relation? Why can’t the identity with itself of ‘the doorbell is ringing inside’ suffice to make ‘the doorbell is ringing inside’ relevant to ‘the doorbell is ringing inside’?

If p then p: The reason the relevance-making relation has to be ternary, not binary is that it supports not just implication, but deductive inference as well. Consider the following deductive chain, which proves the truth of ‘if p then p‘:

  1. p {1} hyp
  2. p {1} 1, reit,
  3. p –> p {1} 1 – 2, –> I

The three subscripts {1} each refer to one situation s (say, the doorbell’s ringing inside my apartment) that gets named three times. Employing the subscripts forces us to make sure that the premises actually are used to reach the conclusion — in other words, that the premises be actually relevant to the conclusion. If we take this deductive argument to be implicit in ‘p –> p‘ — if we take the implication to be a condensed form of the argument — then the relation that makes p relevant to p in that implication has to be ternary. But we get a bizarre result if we try to force an information channel interpretation on this ternary relation in the case of ‘p –> p‘, because we end up with situations having to be information channels from themselves to themselves.

So ‘if p then p‘ poses a challenge for a ‘channel of information’ interpretation of relevant implication. Might another interpretation fare better? Is their an alternative to making a situation a channel to itself? Mares proposes using the concept of an information link. Let’s see how this concept holds up.

An information link is itself contained as information in situations:

On my view, informational links are themselves contained as information in situations, and vary from situation to situation. For example, the information that a particular convention is in place may be contained in one situation, but not in another.

RL, p. 44

For example, if the city council of Salt Lake City, Utah has passed an ordinance requiring the motorists stop at the white line on the road in front of a stop sign facing a certain direction, that law is information that is part of the situation described by ‘in Salt Lake City’ and is available in that situation. It is not information contained in and available in, say, a desert island owned by some eccentric billionaire with artistic pretensions who has set up stop signs there purely as a form of conceptual art.

I take it that Mares’ phrase ‘contained as information in a situation’ means ‘in that situation the information is available’ whether that information has actually been received or not. That the doorbell apparatus in my apartment is in good working order is information contained in the situation described by ‘the doorbell apparatus’ and is available in that situation say, to a competent doorbell repair person, whether any such person has actually received that information or not.

I take it, then, that an information channel would be a special case of an an information link. ‘Information link’ lets us talk about implications going beyond the narrow cases of physical as opposed to, for example, legal or conventional) situations linked together by channels of information that meet whatever requirements are imposed by information theory. Perhaps exposing thereby my vast ignorance of information theory, I will assume that information theory is best suited for purely physical situations. If so, the looser concept of an ‘information link’ would be better suited to elucidate the phenomena than the stricter concept of a ‘channel of information’, which in many cases, I think, would be trying to fit them in a Procrustean bed.

An information link gives us a license to infer. The laws of nature, for example, once uncovered, give us a license to infer that a body will, because it has mass, warp the space around it. The laws of Salt Lake City will, once one has arrived at the ordinary condition of being aware of those laws pertaining to stop signs, give one a license to infer that one is to stop on the white line in front of the stop sign. Packed, so to speak, into an implication is this inference license.

But does the concept ‘information link’ suffice to deal with ‘if the doorbell is ringing then the door bell is ringing’? What would a ternary relation between situations look like in that case? As shown by the proof offered above, ‘if the doorbell is ringing then the doorbell is ringing’ has packed into it, is a condensed form of that 3-step proof. So implicit in the implication is a three-fold structure in which a particular situation gets named three times. A supposition (the hypothesis ‘were the doorbell ringing in my apartment’), a statement (not requiring the ability to track long arguments or hold a complex thesis in one’s mind) of the obvious, a repetition (in that case a doorbell would be ringing in my apartment), and finally a conclusion (so if a doorbell were ringing in my apartment, then a doorbell would be ringing in my apartment’ — I am a very stable genius!). To unpack the implication would be to go through a process in which the same situation gets named three times. Thus Rsss. Of course, as I state all of this I am wondering how dubious or contrived it sounds. Nonetheless, this seems to me to be a more supple way of accounting for the ternary relation in ‘if p then p’ than trying to force on that relation an information channel that is somehow a channel to itself, something that seems (at least to my untutored eyes) appropriate only for physical implications anyhow.

So yes, for now I will go out on a limb and assert that the concept ‘information link’ succeeds in accounting for ‘if p then p‘ whereas the concept ‘information channel’ does not.


The Problem

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.

But what is it that makes p relevant to q?  What is relevance anyhow?

 

 

 

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Next Snippet:  What Is Relevance Anyhow?

 

 
 
 
Edit Log: June 04, 2017: Made some minor changes.


What Is Relevance Anyhow?

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, t… tn) and the button outside is getting pushed at times t0, t1, t2, t… 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.