Tag Archives: Classical Logic

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.  🙂

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.

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:  the pile of apples in the orchard comprises 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.

The Sample Space Ω =

Ω = { 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.

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

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

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

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

And of course the intersection of E and F, E ∩ 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 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.)

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

Boxer_XML_OnlyComputerBookBoughtJustForTheCover_

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.

 

 

 

 

 

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Re-Igniting An Old Flame

A few weeks ago my interest in the French Philosopher Maurice Merleau-Ponty (1908-1961) suddenly got re-ignited upon finding out that a paper I published in a previous life (THE CONCEPT OF THE ECSTASIS, Journal Of The British Society For Phenomenology, 14(1):  79-90, 1983) actually got listed in the bibliography of Stephen Priest’s MERLEAU-PONTY:  THE ARGUMENTS OF THE PHILOSOPHERS.

The sudden explosion of this renewed interest is a bit like the result of throwing a lighted match on a bunch of rags soaked in gasoline.  In its heat, I’ve decided to start a new category of blog posts comprising an attempt to gain a deeper, fuller understanding of the topic of that paper.  What positions stated in the paper do I still hold?  What positions must I mark to market?  (<yes I am being ironic>Doubtlessly none — surely my paper is sacred text.</yes I am being ironic>) What can be stated more clearly, argued for more carefully?  Doing this kind of thing is what blogs are ideal for:

…you can work around the edges of an idea over days and weeks and months [and years] and really   come to understand it. It’s this process that blogging does better than pretty much any other medium.

Anil Dash On Blogging

 

The topic of my paper is, essentially:

The question concerning corporeity connects also with Merleau-Ponty’s reflections on space (l’espace) and the primacy of the dimension of depth (la profondeur) as implied in the notion of being in the world (être au monde; to echo Heidegger’s In-der-Welt-sein) and of one’s own body (le corps propre).

Wikipedia Article On Maurice Merleau-Ponty

 

So in the months and years to come I will be re-reading, working through, and blogging on Merleau-Ponty (THE PHENOMENOLOGY OF PERCEPTION, THE VISIBLE AND INVISIBLE, and other works) in order to really come to understand, truly get my head around, get a maximal grasp of this notion of ‘the primacy of the dimension of depth as implied in the notion of being in the world and of one’s own body.’  As part of this effort, I will be re-reading and blogging on George Berkeley’s works as well, which, partly as foil, partly in a kind of concurrence, shed light in an interesting way on Merleau-Ponty.

These efforts will fall under the category ‘Primacy Of The Dimension Of Depth.’

Of course, I am far from having finished the other two main categories I have been working on in this blog, to wit: ‘The Argument That Tagalog Lacks A Subject’ (a thread inspired largely by Paz Buenaventura Naylor’s article), and ‘Material Implication And Information Theory’ (inspired largely by Fred Dretske’s KNOWLEDGE AND THE FLOW OF INFORMATION and by Edwin D. Mares’ RELEVANT LOGIC).  I intend to continue working on these threads at the same time that I am re-igniting an old flame, my crush on Merleau-Ponty.

 

MerleauPontyArgumentsOfPhilosophers

 

If I bore anyone, tough.  You don’t have to read these incoherent/semi-incoherent ramblings.  I am writing largely in order to learn, to get as much clarity as I can in my own head regarding these topics.

Of course, it would be nice if someone else were interested in them, and, even better yet, had something useful and interesting to say about them, whether in disagreement or agreement with me.

It would also be nice if Ashton Kutcher gave me a call.

 

ashton_kutcher-4036

 

(No post even touching on philosophy would be completed without an homage to Plato’s SYMPOSIUM.)  I wonder if Alkibiades was as gorgeous.

 


My Attempt To Identify The IF-THEN Relation With The INFORMATION-THAT Relation Ignominiously Bites The Dust

Here is yet another challenge to the idea that ‘If p Then q’ is true when the occurrence of p is information that q.  Unfortunately, I think this challenge nails the matter. Consider Dretske’s shell game example.  The peanut is under shell #4.  So the following statement is true (given that my visual faculties are in sufficiently good working order, and that I am looking in the proper direction with my eyes open):

If I turn shell #4 over now (t0), I will see a peanut at time t1

(t1 being one millisecond or whatever later than t0.)  Is my turning shell #4 over at time t0 information that I see a peanut at t1? Certainly the situation largely fits Dretske’s definition of ‘information that’:

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

(k represents what the receiver already knows about the source.)  The conditional probability of my seeing the peanut at t1 is certainly 1 given my turning the shell over at t0 (and given the other conditions mentioned).  So the IF-THEN statement above certainly fits that part of the definition of informational content.

But is my turning the shell over at time t0 a signal that at time t1 that I see the peanut?  A signal is  “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F.”  (Dretske, p. 65.)  Does my turning the shell over now depend upon my seeing the peanut one millisecond in the future?  How can a present event depend upon a future event?  Clearly not.

A signal cannot occur before the event or thing or state of affairs the occurrence (existence, obtaining) of which it signals.  The smoke does not occur before the fire (or the smoldering).  The doorbell does not ring before the button is pushed.  The deer tracks in the snow do not appear before the deer show up.  Were the watchman in Aeschylus’ play AGAMEMNON in the ORESTEIA trilogy to light his fire before he spots Agamemnon’s ships, his fire would not be a signal informing Clytemnestra of the appearance of those ships on the scene:  Clytemnestra would be receiving false information.  Something cannot be announced before it occurs (exists, obtains).

“But the dark clouds signal the rain that is about to fall; the sports official signals the race that is about to start in one millisecond by firing the pistol into the air.”  Someone may object in this way to my (seemingly obvious) claim that a signal cannot occur before the thing it signals.  Yet, although we can doubtlessly “round up” the dark clouds and the firing of the pistol to the status of signals, they are not so in the very strictest sense of ‘signal’ that I intend to use here.  For the conditional probability that, given the dark clouds, rain will fall is perhaps only 99%, while the probability that the race actually will start given the firing of the pistol is perhaps only 99.9999999999% (the supernova that will hit us eventually may choose that exact millisecond to intervene by making its presence glaringly, searingly obvious, or a huge earthquake might strike at that very moment….).

A signal is  “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F” and therefore cannot occur before the occurrence (existence, obtaining) of s‘s coming to be F.   The examples I’ve just given are not signals because they occur after what they “signal”, and — surely not coincidentally — they do not depend upon what they “signal.”  Let me dwell a moment, perhaps a bit obsessively/compulsively, on this notion of dependence.  Let me say that an event, object, or state of affairs p depends upon an event, object, or state of affairs q when, given a condition c,  p would occur (exists, obtain) only because q occurs (exists, obtains).

Consider, for example, a doorbell whose wiring is defective in such a way that, 99% of the time when the button outside is getting depressed by someone or something, the doorbell rings.  But 1% of the time the doorbell does not ring when the button outside is getting depressed. (I state the example this way to make it mirror the fact that p does not follow from If p Then q; q.)  Also, there is no poltergeist inside the wiring that sometimes generates the ringing sound even when no one or nothing is pressing the button outside; likewise, there is never, ever any freak burst of electricity ultimately caused by a butterfly flapping its wings in the Amazon that generates a buttonless ringing sound.  Nor (somewhat more plausibly) is there any defect in the wiring that would ever cause a buttonless ringing sound to occur. Let c be the condition of the defective wiring as just described (including the absence of ring-generating poltergeists).  Given c (which I will call the non-poltergeist condition), the doorbell would ring only because the button outside is getting depressed (even though the button’s getting depressed does not necessarily result in the doorbell’s ringing)*.  Given c, the doorbell’s ringing depends upon someone or something’s depressing the button outside and is therefore a signal.  (A signal, moreover, carrying the information that someone or something is depressing the button outside, because the conditional probability of this is 1 given the doorbell’s ringing under condition c.  Another way to put this is to make the perhaps obvious/tautologous point that to be a signal is to carry information.)

Consider another example, one which is perhaps belongs more to the realm of probability than to causality.  One has turned over shells #1 and # 2 and verified that both are empty. They have information that the peanut is located in one of the four shells.  So c is now the condition that either the peanut is located under shell #3 or under shell #4.  Given c, shell #3 would be empty only because it is shell #4 that is covering the peanut.  It is, in fact, difficult to come up with any clear idea of anything else that could be the reason why shell #3 is empty.  Shell #3’s being empty therefore depends upon the peanut’s being located under shell #4, and the former would be a signal carrying information that the latter.  (Conversely, given that there is only 1 peanut at play in the game and given the rest of c, shell #4’s turning out to have the peanut would be a signal carrying information that shell #3 is empty.  Shell #4 would have the peanut only because shell #3 is empty. )

Now consider again the turning over shell #4 example given above as an instance of an event, object, or state of affairs that very definitely is not a signal carrying information.  It would be difficult to give any meaning to the assertion:

my turning shell #4 over at time toccurs only because I will see a peanut at time t1

Such an assertion would not, I submit, make any clear sense, since the dependency aka only because relationship is a vector traveling forward (to speak metaphorically) in time.

Also consider yet one more doorbell example:  suppose that the doorbell’s wiring is screwy in such a way that every now and then little bursts of electricity get generated which produce the ringing sound even when no one or no thing is depressing the button outside.  (Or, if you prefer, there is a poltergeist residing inside the wiring that every now and then gets agitated by a freak burst of air pressure inside the contraption that is ultimately caused by a butterfly flapping its wings in the Amazon….)  Nonetheless, the condition of the wiring is such that the doorbell always rings when the button is getting pushed.  100 percent of the time the doorbell rings when the button outside gets pushed, but 1% of the time the doorbell is ringing buttonlessly. (I state the example this way to make it mirror the fact that q does not follow from If q Then p; p.  And I am making it mirror this because, of course, the whole point of these interminable disquisitions is to dig into the nature of IF-THEN statements.)  Let me call this condition of the wiring c, as usual.  (In a moment I will be calling it the ‘poltergeist condition.>)  Given c, it would be difficult to give any sense to the following assertion:

My pressing the button outside occurs only because the doorbell is ringing.

Clearly, my pressing the button outside does not depend upon, and is not a signal for, the doorbell’s ringing.  Again, the pressing of the button does not depend upon the doorbell ringing because the dependency aka only because relationship is a vector traveling forward, not backward, in time.

“Feel free to come to the point when you finally have one,” my (possibly non-existent) reader may want to say.  Well, the point of all of the above is the following.  Given their respective condition c’s, each of the following IF-THEN statements is true:

1) If I turn shell #4 over now (t0), then I will see a peanut at time t1

2) If I press the button outside [given the poltergeist condition], then the doorbell will ring.

3) If shell #3 is empty, then the peanut is located under shell #4.

4) If the doorbell is ringing [given the non-poltergeist condition], then someone or something is depressing the button outside.

Although the antecedent p is a signal carrying the information that q in the last two examples, it is not such a signal in the first two examples.

These examples, I think, nail it:  IF-THEN statements cannot be identified with an information relation.  My attempt to identify the IF-THEN relation with the INFORMATION-THAT relation has ignominiously bitten the dust.  (Sob, sob.) Does this mean, then, that we are stuck after all with Classical Logic’s paradoxes of Material Implication, whereby both of the following statements are true?

If Cliff lives in Houston, Texas, then the earth has just one moon.

If Cliff lives in Orange County, California, then Paris, Texas is the capital of France.

(Please God, please God, please don’t let these statements be true.)  Well, maybe we aren’t forced to accept these horribly ugly statements as true after all.  For in each of the 4 numbered examples given above, the conditional probability of the consequent (given the antecedent plus the relevant condition c ((plus the relevant knowledge k))) remains 1.  It is just that in the first two examples the antecedent does not depend upon the consequent, and therefore is not a signal carrying the information that the consequent.  It is not a p only because q relationship.  Perhaps, then, we can identify the IF-THEN relation with a different (but similar) relation, which I will call ‘the conditional probability is 1‘ relation. If so, it would remain true that in examples 3 and 4 above, the antecedent p is a signal carrying information that q.  So whenever p does depend upon q in such a way as to be a signal for q the corresponding IF-THEN statements would, possibly, have the (at least to me) weird properties mentioned in a previous post:

Third, the informational relation is both intentional and relative, as described by Fred Dretske in his KNOWLEDGE AND THE FLOW OF INFORMATION.  Treating If p Then q as an information relation would make implication both intentional and relative.  The very same If p Then q statement would be true inside some frameworks and false inside others.  Rather than accept this, some would perhaps rather accept Classical Logic’s paradoxes of Material Implication.

(Sidenote:  Dretske’s measles example displays the intentional character of information.  By pure chance, all of Herman’s children happen to have the measles; moreover, one does not know this.  So when one discovers that a particular person is a child of Herman’s, they do not have information that this person has the measles.) Or are we truly stuck with this weirdness? Can we find a way to make implication non-relative and non-intentional even in those cases in which p happens to be a signal carrying the information that q?

Today’s homage to Plato’s SYMPOSIUM is this gorgeous young Asian Man: GorgeousAsianGuy

It is hard to understand how anyone can get any work done at all with Beauty like this walking the earth, but somehow we do. How sleek all those black, white, and gray tones are!

Post Updated on June 27, 2015 to make the temporal vector nature of the dependency/only because relation clearer. (Or, if my reader is particularly suspicious, they are free to think I made the update in order to cover up some totally obvious mistakes, not simply to make a somewhat muddy post slightly clearer.)


The Red And The Yellow: Pinning Down Some Intuitions

I want to pin down the following intuitions. (Maybe this is a bit like pinning down some bizarre insects in a collection done for a biology class.)  The intuitions have been … provoked, if that is the word…by my reading in Fred Dretske’s classic work Knowledge And The Flow Of Information, and are motivated by some claims I want to make (hopefully in a later post) regarding Relevant Logic (as opposed to Classical Logic).

Suppose that someone has thrown together a pile of apples in an orchard.  (I am picturing my maternal grandfather’s orchard in Iowa, near Council Bluffs.)  The pile comprises some red apples and some yellow apples.  By pure chance, all of the red apples happen to be wormy, while at least some of the yellow apples happen not to be wormy.  I pick up an apple from the pile.  The apple happens to be red.

Now I have the strong intuition that in this situation the following statement is true:

If the apple I have picked up from this pile is red, that apple is wormy.

The statement is true, at least for a particular stretch of time, because any apple I pick up from the pile will be wormy should it happen to be red.  During that stretch of time, the apple from that particular pile will be wormy if it is red. Maybe later someone will throw in a non-wormy red apple, in which case the If-Then statement above will cease to be true.  But for this moment, the statement is true.

Now let’s back up and change the example.  The pile still comprises both red and yellow apples, but now both the yellow apples and the red apples are all wormy.  In this case…well, perhaps saying I have the intuition is too strong…nonetheless, I am strongly tempted to claim that in this other situation the statement under discussion is false:

If the apple I have picked up from this pile is red, that apple is wormy.

Classical Logic thinks the statement is true because both the antecedent and the consequent are (in our thought experiment) true.  Nonetheless, I am at the moment of this writing willing (perhaps foolishly) to stick my head out and say the statement is false because, in this situation, the apple’s being red is no more relevant to its being wormy than any other accidental feature of the apple — say, its still having a leafed twig attached to it.  In this particular situation, neither the apple’s being red nor its still having a twig and leaf attached to it excludes the possibility that it is not wormy.  What does exclude that possibility (for a while, at least) is the apple’s coming from this particular pile.  So the above statement is no more true than this statement (suppose I happen to be driving down Highway 66 at the moment):

If I am driving down Highway 66, then the earth has just one moon.

Classical Logic thinks this statement is true because both the antecedent and the consequent are true; Relevant Logic thinks the statement is false because the fact that I happen to be driving down Highway 66 at the moment is not relevant to the earth’s having just one moon.

I would like to add that the statement remains false even though, 100 percent of the time when I drive down Highway 66, the earth still has just one moon.  (I hope to motivate this claim a bit later.)  In spite of this ‘100 percent of the time’, this reliability, my driving down Highway 66 is not a factor that excludes the possibility that the earth has more than one moon.

Today my homage to Plato’s Symposium will be the singer Von Smith. (According to Plato, one ascends a ladder that starts with the beauty of gorgeous young men, and leads up to the beauty of things like Classical and Relevant Logic.)

Von_Smith

How can anyone get anything done with beauty like this walking the earth?  Especially beauty like this walking the earth and singing.