- #1

Clifford Engle Wirt

- 5

- 1

(Mentor note: link removed as not essential to the question.)

The problem is: what is relevance anyhow?

My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math, when I articulate an intuition, do people share or not share the same intuition?

Ω = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13y, a14y, a15y, a16y }

E is the event 'a red apple is picked up from the pile':

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

F is the event 'a wormy apple is picked up from the pile':

F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8r,a9yw, a10yw, a11yw, a12yw}

The probability that a wormy apple will be drawn from the pile is |F|/|Ω| = 12/16 = 3/4.

The conditional probability that the apple drawn from the pile will be wormy given that it is red is 1, as can be seen from the following:

P( F

Now the intersection E

E

and P( E

and P(E) = |E|/|Ω| = 8/16 = 1/2.

So P( E

But two distinct events are independent of one another if and only if

P(E

So in this case E and F are not independent events. Should the probability of F (a wormy apple is drawn) given E

This situation mirrors, I submit, two features of the following proposition, where 'this apple' refers to an apple drawn from the pile:

IF this apple is red, THEN it is wormy.

A necessary condition for the truth of this IF THEN proposition is that the conditional probability of the consequent be 1 given the antecedent. Check. Likewise, a necessary and sufficient condition for the relevance of the antecedent to the consequent is (so I claim) that the probability of the consequent increase given the antecedent. Check. (For why it should matter that the antecedent be relevant to the consequent, take a look at the examples provided in the link above.)

What grounds can be given for this (maybe dubious) claim that an increase in probability provides a necessary and sufficient condition for the relevance of the antecedent to the consequent? Let's see what happens when such an increase fails to occur. Again, given that I have absolutely zero natural talent in math, I would be very much interested in knowing if I got the following math right.

Suppose now that

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

E is as before. F is now identical with Ω --

P( F

P(F) is now 1, since all the apples are now wormy. P(E) is 1/2, as before.

and P( E

But again, two distinct events are independent of one another if and only if

P(E

1/2 = 1/2 * 1.

So E and F are now independent events.

Now once

IF this apple is red, THEN it is wormy

because now the possible redness of the apple is no longer

This (at least alleged) lack of relevance of the antecedent to the consequent mirrors the independence of E (the apple is red) and F (the apple is wormy) considered as events. E no longer has 'anything to do' with F because E and F are independent. I take this as an argument in favor of the claim that, as regards IF THEN propositions, a necessary condition for the relevance of the antecedent to the consequent consists in an increase, given the antecedent, in the probability of the consequent (which has to increase to 1) over and above what the probability is given just the original sample space. As for this being also a sufficient condition, I don't think it can be denied that if E increases the probability of F, E is relevant to F.

Again, if I have made a mistake in the math, I would be very interested to know. Also, I would be very interested to know if people do not share my intuitions, or have trouble deciding what to think because the intuitions are way off in left field.

The problem is: what is relevance anyhow?

My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math, when I articulate an intuition, do people share or not share the same intuition?

**A pile of apples:**The sample space Ω comprises a pile of 16 apples (identified by the numerals 1...16), 8 of which are red (indicated by the letter r), and 8 of which are yellow (indicated by the letter 'y'). All 8 of the red apples are, unfortunately, wormy (indicated by the letter 'w'). 4 of the yellow apples are also wormy, and 4 are, as of yet, not wormy (indicated by an absence of a 'w').Ω = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13y, a14y, a15y, a16y }

E is the event 'a red apple is picked up from the pile':

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

F is the event 'a wormy apple is picked up from the pile':

F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8r,a9yw, a10yw, a11yw, a12yw}

The probability that a wormy apple will be drawn from the pile is |F|/|Ω| = 12/16 = 3/4.

The conditional probability that the apple drawn from the pile will be wormy given that it is red is 1, as can be seen from the following:

P( F

**|**E ) = P( E**∩**F )**/**P(E)Now the intersection E

**∩**F is:E

**∩**F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw}and P( E

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

So P( E

**∩**F )**/**P(E) = (1/2)/(1/2) = 1. So P( F**|**E ) = 1.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. Should the probability of F (a wormy apple is drawn) given E

*increase*over the probability of F given just the draw from the pile, this would be a sufficient condition for F's having a dependency upon E. This probability does increase to 1 from 3/4. So F has a dependency upon E.This situation mirrors, I submit, two features of the following proposition, where 'this apple' refers to an apple drawn from the pile:

IF this apple is red, THEN it is wormy.

A necessary condition for the truth of this IF THEN proposition is that the conditional probability of the consequent be 1 given the antecedent. Check. Likewise, a necessary and sufficient condition for the relevance of the antecedent to the consequent is (so I claim) that the probability of the consequent increase given the antecedent. Check. (For why it should matter that the antecedent be relevant to the consequent, take a look at the examples provided in the link above.)

What grounds can be given for this (maybe dubious) claim that an increase in probability provides a necessary and sufficient condition for the relevance of the antecedent to the consequent? Let's see what happens when such an increase fails to occur. Again, given that I have absolutely zero natural talent in math, I would be very much interested in knowing if I got the following math right.

Suppose now that

*all*of the apples in the pile, red or yellow, have become wormy. (These things are known to happen.) The sample space now looks like this:Ω = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13yw, a14yw, a15yw, a16yw }

E is as before. F is now identical with Ω --

*all*the apples are now wormy. And now E no longer increases the probability of F. P(F) is 1; P(F | E) is also 1. E and F are now independent events, as can be seen from the following:P( F

**|**E ) = 1, as before.P(F) is now 1, since all the apples are now wormy. P(E) is 1/2, as before.

and P( E

**∩**F ) = | E**∩**F|/||Ω| = 8/16 = 1/2 as before.But again, two distinct events are independent of one another if and only if

P(E

**∩***F*) = P(E) * P(F)1/2 = 1/2 * 1.

So E and F are now independent events.

Now once

*the apples in the pile, red or yellow, are wormy, it seems (to me, at least) positively weird to say:***all**IF this apple is red, THEN it is wormy

because now the possible redness of the apple is no longer

*relevant*to its possible worminess. (The Relevant Logician would say that the proposition is false because of this; the Classical Logician would say the proposition may be a bit weird, but still true.) The possible redness of the apple no longer has anything to do with its possible worminess. -- Or rather, this is my strong intuition. If anyone has a different intuition regarding this, I would be keenly interested to know.This (at least alleged) lack of relevance of the antecedent to the consequent mirrors the independence of E (the apple is red) and F (the apple is wormy) considered as events. E no longer has 'anything to do' with F because E and F are independent. I take this as an argument in favor of the claim that, as regards IF THEN propositions, a necessary condition for the relevance of the antecedent to the consequent consists in an increase, given the antecedent, in the probability of the consequent (which has to increase to 1) over and above what the probability is given just the original sample space. As for this being also a sufficient condition, I don't think it can be denied that if E increases the probability of F, E is relevant to F.

Again, if I have made a mistake in the math, I would be very interested to know. Also, I would be very interested to know if people do not share my intuitions, or have trouble deciding what to think because the intuitions are way off in left field.

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