**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. 🙂

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 foodclipart.com

**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 Ω — as follows:

The Sample Space Ω =

Ω = { a1^{rw}, a2^{rw}, a3^{rw}, a4^{rw}, a5^{rw}, a6^{rw}, a7^{rw}, a8^{rw}, a9^{yw}, a10^{yw}, a11^{yw}, a12^{yw}, a13^{yw}, a14^{yw}, a15^{yw}, a16^{yw} }

where a1…a*n* 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* = { a1^{rw}, a2^{rw}, a3^{rw}, a4^{rw}, a5^{rw}, a6^{rw}, a7^{rw}, a8^{rw} }

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

*F* = { a1^{rw}, a2^{rw}, a3^{rw}, a4^{rw}, a5^{rw}, a6^{rw}, a7^{rw}, a8^{rw},a9^{yw}, a10^{yw}, a11^{yw}, a12^{yw}}

And of course the intersection of *E* and *F*, *E* ∩ *F*, the set of apples that are *both* red *and* wormy =

{ a1^{rw}, a2^{rw}, a3^{rw}, a4^{rw}, a5^{rw}, a6^{rw}, a7^{rw}, a8^{rw}}

I will be assuming that each apple in Ω 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* **∩** *F* ) **/** P(E)

P( *E* **∩** *F* ) = |*E* **∩** *F| ***/** **|**Ω**|*** = *8**/**16 = 1**/**2

P(E) = **|***E***| / |**Ω**| = **8**/**16 = 1**/**2

So:

P( *E* **∩** *F* ) **/** P(E) = 1/2 **/** 1/2 = 1

So:

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* **∩** *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* **∩** *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.