Apple Math, Comprising Some Basic (Doubtlessly Ninth-Grade Level) Probability Theory

Nota Bene:  This little bit of math is the keystone in my attempt here (still in draft status)  to provide a sharp, clear articulation of the concept of relevance as that concept pertains to Relevant Logic.  Here I invited members of the online Physics Forum to point out any mistakes in the math should I have made any.  Since no one there pointed out any such mistakes, I will assume that the math is correct.  Naturally, should it turn out that I did make mistakes in the math, I will be royally pissed.  ūüôā

This post belongs to the ‘I invite anyone and everyone to tear this to pieces, should they uncover any missteps’ category.

The subject here isn’t roses (this is an obscure allusion to a movie I saw in my childhood), but wormy and non-wormy red and yellow apples.

In discussing the subject of apples, I will be using the following terms: ‘set’ (which I will leave as an undefined primitive); ‘sample space’ (which term is I think self-explanatory); ‘event’ (which I will be using in an extremely narrow and a bit counter-intuitive technical sense, following the standard nomenclature of probability theory); ‘experiment’ (ditto); ‘state of affairs’ (which I will be leaving as a primitive); and ‘proposition’ (which I will define in terms of states of affairs).

Wormy Red Apple Image courtesy of

First Situation:  All Of The Red Apples Are Wormy; Only Some Of The Yellow Apples Are:  Let’s start with the following situation (henceforth ‘situation 1’):  There is an orchard in Southwest Iowa, just across the border from Nebraska. In the orchard there is a pile of apples comprising 16 apples.  Eight of the apples are red.  All of the red apples are wormy.  Eight of the apples are yellow.  Of these yellow apples, four are wormy. 

Let’s suppose that the DBA in the sky has assigned an identifying number (doubtlessly using the Apple Sequence Database Object in the sky) to each apple. This lets us write the set of apples in the pile — the Sample Space ő© — as follows:

The Sample Space ő© =

ő© = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13yw, a14yw, a15yw, a16yw }

where a1…an indicate the numbered apples, and the superscripts r, y, w, and w indicate a red apple, a yellow apple, a wormy apple, and a non-wormy apple respectively.

An ‘event’ is a (not necessarily proper) subset of this set. It represents the set of possible outcomes should one draw an apple from the pile. This particular red apple is drawn; this other particular red apple is drawn; this particular yellow apple is drawn, and so on. Contrary to the ordinary sense of ‘event’, an ‘event’ here is not something concrete, happening in space and time, but abstract — a set.

Eyes shut, someone has randomly drawn an apple from the pile. They have not yet observed its color. Why their having not yet/having observed the apple matters will become apparent later [promissory note]. Following the standard nomenclature, I will call actually drawing an apple — a concrete outcome that has come forth in space and time — an ‘experiment’.

Now I show that….

E is the event ‘a red apple gets drawn from the pile’, which =

E = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw }

F is the event ‘a wormy apple gets drawn from the pile’, which =

F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw,a9yw, a10yw, a11yw, a12yw}

And of course the intersection of E and F, E ‚ą© F, the set of apples that are both red and wormy =

{ a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw}

I will be assuming that each apple in ő© has an equal probability of being drawn.

The conditional probability that the apple drawn from the pile is wormy given that it is red is 1, as you can see from the following steps:

P( F | E ) = P( E ‚ą© F ) / P(E)

P( E ‚ą© F ) = |E ‚ą© F| / |ő©| = 8/16 = 1/2

P(E) = |E| / |ő©| = 8/16 = 1/2


P( E ‚ą© F ) / P(E) = 1/2 / 1/2 = 1


P( F | E ) = 1

The conditional probability that an apple drawn from this pile is wormy given that it is red is 1.

Now P(F) = 12/16 = 3/4.  Since P(E) = 1/2, P(E) * P(F) = 1/2 * 3/4 = 3/8.  So in this case P(E ‚ą© F) != P(E) * P(F),  since 1/2 != 3/8.  But two distinct events are independent of one another if and only if

P(E ‚ą© F) = P(E) * P(F)

So in this case E and F are not independent events.   The probability that the apple is wormy given that it is red increases to 1 from the 3/4 probability given just the draw from the pile, before observing whether the apple drawn is red or yellow.  (Conversely, the probability that the apple is red given that it is wormy increases to 2/3 from 1/2 given just the draw from the pile.)

When the probability of an event is 1, that event is certain, as opposed to ‘just likely’. The concept of certainty is, of course, intimately bound up with the concept of knowledge, an entanglement I hope to examine shortly. But whatever the relation is, the event of this apple’s turning out to be red moves the event of its being wormy from a mere likelihood to a certainty. And whatever the relation of certainty to knowledge is, this certainty surely provides a foundation for knowing that this apple is wormy. In this limited situation (“situation 1”), the apple’s turning out to be red is potentially telling — namely, that it is wormy. It increases our (potential) knowledge.

When this apple drawn at time t0 (the experiment that takes place at that time) turns out to be red , the state of affairs ‘this apple is red’ obtains at t0. I will label this state of affairs ‘p’. Similarly, I will call q the state of affairs that obtains at t0 when this apple is wormy. In situation 1, the fact that the probability of F given E is 1 means there is no way that p can obtain at t0 and q fail to obtain at t0. For the moment, at least, I will refrain from unpacking ‘cannot fail to obtain’, except to link this notion to the probability of an event being 1.

I like to identify propositions with states of affairs that obtain at a particular time. So p is the proposition that the apple is red, and q is the proposition that the apple is wormy. States of affairs obtain or fail to obtain; propositions are true or false. So I am now moving from talking about states of affairs obtaining (failing to obtain) to propositions being true or false. If, gentle reader, you would rather not identify propositions with states of affairs obtaining at some time, just add whatever verbiage is necessary to identify a proposition that corresponds to the state of affairs just mentioned.

In situation 1, whenever p is true q cannot fail to be true. This means that the proposition If p Then q is true, for it satisfies the truth table in Classical Logic for If Then propositions. In situation 1, If p Then q remains true even when p is false (the apple is yellow) and q is false (the apple is not wormy); when p is false and q is true (the apple is wormy); and of course the proposition is true when p is true and q is true. The only time the proposition is false is when p is true and q is false.

What is more, in situation 1, p is relevant to q. For p maps to the event E given which the probability of F, to which q maps, [talk some more about this mapping business] increases from 3/4 to 1, i.e., from mere likelihood to certainty. p inherits this ‘increasing q to certainty’ property. That one proposition/state of affairs (that the apple is red) p increases the probability of another proposition/state of affairs (that the apple is wormy) q surely renders p relevant to q. It is a sufficient condition for p’s relevance to q. It therefore renders If p Then q true in both Relevant Logic (which demands that the antecedent be relevant to the consequent) and in Classical Logic.

I submit, then, ‘increasing the probability of q to 1’ as a candidate for the relevance-making relation that p bears to q when p is relevant to q. This relation is a candidate, that is, for those If Then propositions that can be treated in a probabilistic manner. It is not a candidate for the relevance of the antecedent to the consequent in the proposition ‘If the length of side A of this right triangle is 2 and the length of side B is 3 (neither A nor B being identical with the triangle’s hypotenuse), then 13 is the length of the hypotenuse.’ For even though the antecedent here excludes any other possibility other than the hypotenuse having a length of 13 (just as the apple’s turning out to be red excludes in situation 1 the possibility of it’s not being wormy), there is nothing in the mathematical proposition that invites treatment in terms of chance and draws.

That the probability increases to 1 renders the proposition ‘If E then F’ true — at least in this circumscribed situation (this particular pile in this particular orchard for this particular stretch of time, which stretch of time will come to an end should a non-wormy red apple happen to roll into the pile). Within this situation, the apple will always be wormy should it turn out to be red. The ‘all’ in ‘all the red apples are wormy’ guarantees the truth of the conclusion as long as this ‘all’ lasts. Taking the increase in probability combined with the guarantee (the increase is to 1) together suffice to make ‘If this apple is red, it is wormy’ a true proposition in relevant logic, since the conclusion meets the truth-table standard of classical logic and meets the additional condition demanded by relevant logic, namely, that the antecedent be relevant to the conclusion. F will never fail to be true should E turn out to be true, a state of affairs that is a sufficient condition for the proposition ‘If E then F’ to be true.

I submit, then, that at least in those states of affairs that allow for a probabilistic treatment, the relevance of p to q consists in p’s increasing the probability of q to 1. [tie p and q to E and F.] Naturally, not all p’s and q’s will allow for a probabilistic treatment. Mathematical propositions don’t allow for such a treatment, for example. We should perhaps not assume that what makes p relevant to q is the same in all cases of IF THEN propositions is just one type of relation. But at least in the case of those propositions that do allow for a probabilistic treatment, we can see that increasing the probability of q to 1 given p is a strong candidate for the relevance-making relation, given that this increase suffices to render p relevant to q.

At least in those cases that do admit of a probabilistic treatment, increasing the probability of q to 1 is also a necessary condition for p’s being relevant to q.

Second Situation:  All Of The Red Apples Are Wormy, As Are All Of The Yellow Apples

When all the apples are wormy, the color, either red or yellow, of the apple becomes independent of its worminess. Thus the aforementioned sufficient condition for relevance is absent. Maybe some other relation could render p relevant to q here, but I am at a loss for what it could be. So until someone can point out such a relation, I will therefore go out on a limb and say that dependence is a necessary, as well as a sufficient, condition for the relevance of p to q in cases similar to the wormy apple case. This provides support — though clearly not support achieving the level of certainty — for the original intuition. vvggggg

A paradox or at least weirdness comes to the fore. I deal with this by examining the nature of probability. Assuming a deterministic universe (at least on the post-quantum level) probability is perspectival — on either a global or a local level. The example can seem paradoxical because one is assuming the position of someone who knows everything about the apples. A local orchard god, so to speak. But that is just one perspective. Thus the original intuition is vindicated.

If just a credence, there are no relevant IF THEN propositions from a God’s-eye’ point of view. (Actually, no perspective at all). Possible worlds (complete) vs. situations (partial).

Today’s homage to Plato’s SYMPOSIUM is this image of a young boxer appearing on the cover of a computer book.


I have to admit that this is the only computer book I have ever bought just for its cover.

How can anyone get anything done, much less study computer science and ninth-grade math, with beauty like this walking the earth?

Update 11/12/2018:  Made one revision for the sake of clarity.


The Difference In A Nutshell Between Medieval and Modern Philosophy

From a commenter on Ta-Nehisi Coates’ blog in the ATLANTIC:

Aristotle, like Hobbes, did think that knowledge came from the senses, but he had a very different view of how senses worked. Aristotle believed that every physical object has a form or essence, and a substance. So a clay model of a tree and real tree share commonalities of form, although their substances are totally different. Aristotle also thought that the psyche is an instrument whereby we can receive the form of objects without the substance. He compares sensation to a signet ring making an impression of wax.
Hobbes, however, does not really believe that the concept of “essence” is useful in explaining the world. He is basically a materialist. He believes that the only things worth talking about are matter and its interactions. Therefore, his account of how we obtain knowledge through the senses has to rely on interaction between matter.
This might sound like an obscure difference, but it has a lot of consequences for how one studies the world. If you agree with Aristotle, the implication is that by observing the world, you can get an idea of the real essence of things. Acquiring theoretical knowledge is then a matter of thinking rationally about the implications of this knowledge. Thus physical science is a matter of everyday observation followed by rigorous thinking.
However, if the information you get from the senses is just a bunch of particles bouncing off of your sensory organs, as Hobbes believes, then there’s good reason to be worried that the senses are unreliable, and you need to spend time carefully tweaking the information you get from the senses to make sure you have it right. This gives rise to an experimental model (which Hobbes’ contemporary, Francis Bacon, focused on far more than Hobbes did).
As for how commonplace it was – Aristotelianism was basically the dominant philosophy from the time of Thomas Aquinas (1200s) up until the 1600s. Hobbes is writing around the time of transition away from Aristotle’s position as the preeminent thinker on matters such as this. I actually am not sure how dominant the view still was among academics by the time of the Leviathan.
As an aside, the reason Hobbes talks about mediate and immediate interaction is that, at the time, people who subscribed to this materialst view did not believe that matter could interact with other matter at a distance. The only interactions allowed into the theory were direct ones. The view of no interaction at a distance was thrown out after Newton’s theory of gravity became the consensus view – since gravity is interaction at a distance.

The Quality Quest

[The following is a letter I wrote a while ago to the editor of Chicago’s NEW ART EXAMINER responding to an article by Betty Ann Brown.  Betty Ann Brown’s article is badly vitiated, if I may say so, by the sort of sloppy reasoning peculiar to postmodern political flimflam.  As might be expected from the low quality of Brown’s article, Brown’s only response was to engage in some perfunctory hand waving.]

Betty Ann Brown (“A community self-portrait,” NAE, December, 1990) would have us retire the word “quality” because she believes that the concept the word expresses has built into it standards which improperly and objectionably tend to exclude women and artists of color from museums, galleries and exhibitions.¬† (I will put “quality” in double quotes when I am talking about the word, and in single quotes when I am talking about the concept.)¬† That is to say, the concept is constructed along class/race/gender lines.¬† She seems to identify ‘quality’ with the concept of formalistic quality, i.e., a work’s excellence or lack of excellence considered as hinging on such factors as line quality, touch, handling, composition, spatial balance, relations between forms, relations between colors, and so on.¬† ‘Quality’ interpreted as ‘formalistic quality’ is the concept, she asserts, whose use excludes women and artists of color.¬† Instead of the word “quality,” she would have us use “worthy.”¬† According to Brown, a work is worthy when its content “…authentically [accurately?] reflects the artist’s social/historical/political moment.”¬† She prefers work that grates on her, reflects experiences beyond her own, and concerns issues of race, gender, and class.

I very much doubt whether Brown is really rejecting the concept of quality at all.  If she uses “worthy” in such a way that “This work is good or excellent” follows from “This work is worthy” (surely the word means nothing if this does not follow), then the concept of quality has not been done away with.  For if a work is high in quality, it is good or excellent, and if it is good or excellent, it is high in quality.  Thus I suspect Brown is really just advancing a different theory of what artistic quality (worth, merit, excellence, being good) consists in.  She thinks that a work’s quality hinges not on its formalistic values, but on its authentically reflecting an artists’s social/historical/political moment.

However, Brown’s theory of quality (or worth, merit, excellence, or whatever) is obviously false.  Consider all the dull, heavy-handed, poorly observed works stemming from the nineteenth century that use vicious stereotypes to depict African Americans, male and female.  Surely these works reflect their artists’ social/historical/political moment in the most authentic way possible.  They even grate on me, reflect on experiences beyond my own, and concern the issues of race, gender, and class that Brown holds so dear.  Brown is not about to value them as worthy.  If her theory of quality is true, however, there is no way one could escape the conclusion that they are worthy, their shoddiness and viciousness notwithstanding.  Brown could try to avoid this unappetizing conclusion by claiming that the content of  work must reflect the correct politics if it is to count as excellent, but such a move would be clearly ad hoc, if not laughable.  The only reason to make such a move would be to save Brown’s theory.

In the absence of any plausible alternative, one is left with the formalistic theories of quality.  Do these theories in fact have built into them standards that improperly and objectionably tend to exclude women and artists of color?  Consider the following theory, and see if it has any such standards built in.  I submit that the concept I describe below is the one operative in most critical discourse.

A work of art is a symbol that both expresses and sometimes denotes (to use Nelson Goodman’s terms) a content or subject matter.  The work’s excellence or lack of excellence is a function of both its formalistic values and what it expresses.  If what the work expresses is of low value, the work itself is of lesser value, even if (and in fact partly because) its formalistic values express its content perfectly.  Suppose, for example, that Jones, a critic, becomes convinced that Jackson Pollock’s drip paintings express the same types of feelings expressed by New Age music.  Since Jones holds those feelings in low esteem, she values the paintings less than they are usually valued.  Similarly, Smith, a curator at the Metropolitan Museum of Art, holds in low esteem what Anne Ryan’s collages express, namely, a sense of intimacy and pleasure (usually regarded as feminine) in materials and fabrics.  The fact that the formalistic values of the collages expresses those things perfectly hardly commends them to him.  He therefore places the works in storage.

Clearly, Smith’s application of the concept ‘quality’ has been guided by his gender attitudes.  He regards feminine stuff as minor and of lesser value.  I take it this is the sort of case Brown has in mind when she claims that ‘quality’ has built into it standards that improperly and objectionably tend to exclude women.  In what follows, I argue that the claim is nonetheless false.  The argument focuses on the expressive content of an artwork.

There are two possibilities concerning the value of what an artwork expresses.  1) Conventional, relativistic, folk wisdom is correct.  Conventional folk wisdom would like to relativize value the way Einstein relativizes motion.  In Einstein’s theory, of course, the speed of an object is relative to a frame of reference.  In one frame of reference, the speed is 60 mph, and in another it is 1 mph.  Folk wisdom treats Smith and Jones as one-person frames of reference.  In the Smith frame of reference, what Ryan’s work expresses has a low value, while in the Jones frame of reference, say, it has a high value.  Just as there is no absolute measure of speed, but only the speed in this frame of reference and the speed in that one, there is no absolute measure of value for what Ryan’s work expresses.  There is only its value for Smith, and its value for Jones.  2)  What an artwork expresses has a value that is not relative to particular individuals, and Smith and Jones can measure that value accurately or inaccurately, correctly or incorrectly.

Assume that 1) is right.  Suppose also that Jones is a feminist who wants to believe that Smith’s exclusion of Ryan (and the exclusion of other women artists on similar grounds) is improper and objectionable.  Jones, however, cannot cogently criticize or object to Smith’s exclusion of Ryan’s work.  For surely the following thesis is true:

A) If an artwork is of low value (is not good, excellent, worthy, etc.), excluding it (putting it into storage in a museum, not exhibiting it in a show, not buying it, and so on) is not objectionable or improper.

This is, I suspect, an intuition everyone shares.  Even Brown’s view commits her to it, since if a work is worthy, it is surely not low in value.  Now in the Smith frame of reference, Ryan’s collages are low in value.  It follows from A), then, that Smith’s putting her work into storage is not improper or objectionable.  The mere fact that in the Jones frame of reference the collages have a high value does not make the exclusion objectionable.  For disputing the exclusion on those grounds would be like disputing a measure of speed made in another frame of reference on the grounds that it does not match the measure one has made in his own frame of reference.

So if the relativism outlined in 1) is correct, Smith’s exclusion of Ryan’s work is not objectionable.  I assume, by the way, that Brown objects to ‘quality’ because it allegedly leads to cases of objectionable exclusion.

Assume now that 2) is right.  Smith has either correctly or incorrectly valued the expressive content of Ryan’s work.  If he has valued that content correctly, then Ryan’s work is of lesser quality and therefore of lesser value.  It follows from A), then, that Smith’s exclusion of Ryan’s work is not objectionable or improper.  Smith’s exclusion has not resulted from biases and prejudices that have prevented him from valuing the work correctly.  So the concept ‘quality’ is not open to criticism in this case because it has not led to an improper or objectionable exclusion.

Suppose now that Smith has valued the expressive content of Ryan’s work incorrectly (presumably because of gender biases).  He was wrong to put it in storage.  (This is, incidentally, the view I hold, and I suspect Brown would prefer to hold it as well.)  In this case, however, the fault does not lie with the concept ‘quality,’ but with a bad and misguided application of that concept to a particular case.  The application of ‘quality’ went afoul because prejudice prevented Smith from valuing correctly the expressive content of Ryan’s work.  In cases like these, then, the concept ‘quality does not have built into it standards that improperly and objectionably exclude women; rather, it is particular application of the concept that can objectionably exclude women (not all women, by the way) when the expressive content of a work gets wrongly valued.

In each case, then, either the concept ‘quality’ is not the culprit, or the exclusion in question is not objectionable.  Contrary to Brown, it turns out that ‘quality’ does not have built into it (through some kind of white male conspiracy) standards which improperly and objectionably exclude women.  If women are underrepresented in museums relative to their population, the fault lies not with ‘quality,’ but with other factors, including bad applications of the concept (assuming that relativism is false and that female concerns are incorrectly assigned a low value — if relativism is true and female concerns are correctly given a low value, cases of the sort discussed above, which I take to be bad applications of the concept, are in fact not objectionable), prejudice, and social discouragement.  The same analysis applies mutatis mutandis to artists of color.

Cliff Engle Wirt                                                                                                                                        Chicago, IL

Today’s homage to Plato’s SYMPOSIUM takes the form of James Dean and Sal Mineo.


‘Look at me the way I look at Natalie Wood,’ James Dean reportedly told Sal Mineo during the filming of REBEL WITHOUT A CAUSE.  Mineo, having a crush on Dean, needed very little prompting to heed this instruction.  Homoerotic expression is, I dare say, something that in the past has been given an incorrect valuation.

“How Can We Know What The Probability Is?” And Other Objections And Remarks

The following, of course, is not yet developed.

“Where did you get that 99% probability from?” someone may object. ¬†“Did you pull it from your ass?” ¬†Well, I did stipulate it. ¬†But the general objection remains valid nonetheless: ¬†it would seem that there is no way to come up with an objective evaluation of what the probability actually is unless it is 0 or 100%, these figures being based on the physical laws of the universe or the laws of probability. ¬†Deal with this. ¬†See if Dretske’s discussion of this works.

Inductive:  probability of less than 100% but greater than 0.  Deductive (or what supports deduction):  conditional probability is 100%.  Absolute reliability, absolute safety.  What makes the transmission a case of information is also what makes it something supporting deduction.

IF p THEN p — either a complete lack of transmission of information or the exact opposite — a complete surfeit of “transmission” (quote unquote) at the “zero point”.

Update: 09/08/2018: Graying this out because it is too revealing of my vast ignorance of subjective vs. objective probability.


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?




Back To The Main Page

Next Snippet:  What Is Relevance Anyhow?


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

What Is Relevance Anyhow?

But What Does ‚ÄėRelevance‚Äô 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.



Back To Main
Back To The Problem









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.

The Role Of Informational Content In Establishing Relevance In Relevant Logic

“This current version of the notes is not yet complete, but meets I think the
usual high standards for material posted on the internet.”¬† (Link. ¬†No, I have not read the paper apart from this snippet.) ¬†Please feel free to comment if you have any corrections or objections to the disquisition below, or email me at¬†


Logic first became interesting to me when I entered the DBA field and started reading the works of C.J. Date, Hugh Darwen, and Nikos Lorentzos on the foundations of relational databases. ¬†While reading in logic, I became intensely interested in Edwin D. Mares’¬†book RELEVANT LOGIC A PHILOSOPHICAL INTERPRETATION, which seemed to tie in — I am apparently not the first to notice this! — in a very natural way with Fred I. Dretske’s classic work, KNOWLEDGE AND THE FLOW OF INFORMATION. ¬†As an exercise in writing to learn¬†Mares’ book I have been for a while entering posts on this blog on the topic of Dretske’s theory of informational content as it relates to Relevant Logic.

Up until now, these posts have been nothing except an effort to decide what my position is on the topic.  They pretend to be nothing more than efforts to get my own thoughts in order.  As a result, I have not been terribly afraid to be (just occasionally, I hope!) simply mistaken and (worse) unclear.

Basically, I was thinking out loud in order to decide what I do think about the topic. ¬†Even though these exercises in thinking out loud were both tentative and preliminary, I have found it to be a useful discipline in performing them in public, where there is always the possibility that someone actually engaging with the posts (in other words, someone who is not merely a troll) may legitimately, pointing to specifics, exclaim ‘THIS IS SIMPLY WRONG!!!!’ or ‘THIS IS CONFUSED!’

Lions And Trolls Oh My!¬†But now that I am suffering under the delusion that I do have my thoughts on the topic in something vaguely resembling order, I am now actively throwing them out to the lions in order to see what survives intelligent, informed criticism. ((I am assuming there are lions out there who are not only hungry, but also intelligent and informed. As regards lions I keep thinking about Ned Rorem’s LIONS (A DREAM)¬†which I once heard on WFMT in Chicago… but I digress.)) Please consider this post and the the posts linked to here as a request for comment.

As each section of this disquisition takes (almost) final shape, the link to it will become active. Each section will be kept as short as possible partly as an troll-control device: the brevity of each piece makes it easier to force the troll to state a specific objection to a specific assertion ((has the troll misstated the assertion (most of the time intentionally but sometimes not)? If so, challenge them to state it in their own words — honestly this time. Once the troll has correctly stated it, do they think the assertion is wrong? If so, why? ¬†Does the troll think the assertion is unclear?)) rather than allowing the troll to rely on abusive innuendo.

The Problem

What Is Relevance Anyhow?

The Relevance-Making Relation Is Not The Causal Relation

The example that at least initially makes treating the relevance-making relation in terms of Dretske’s notion of informational content attractive: Dretske’s Doorbell Example.

This seems to run aground on the tautology IF p THEN p. The revisions needed to accommodate this tautology.

The ‘peanut is under which shell’ example. Will this example end up making Relevant Logic at least as weird and bizarre as Classical Logic by making the truth of implication statements relative to what one knows?

The measles and wormy red apples example.


No post of mine can do without an homage to Plato’s SYMPOSIUM. Here the homage will take the form of Channing Tatum.


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

June 10, 2017: ¬†Made some minor changes. ¬†Removed a joke I think wasn’t ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬†working.

June 14, 2017:  Added quote at the top.