A challenge to Aristotle's categorical term logic


BaalChatzaf

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see.

Ba'al Chatzaf

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see.

Ba'al Chatzaf

Propositional logic and predicate logic have their weaknesses, too. But I don't see you dismissing them for their weaknesses.

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see.

Ba'al Chatzaf

Propositional logic and predicate logic have their weaknesses, too. But I don't see you dismissing them for their weaknesses.

First Order logic with Natural Deduction cover 99 % of mathematical reasoning. Second Order logic covers the rest.

What weakness?

Ba'al Chatzaf

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What weakness?

For starters, The Logic of Natural Language.

This book is available from Amazon.com for the modest sum of:

Hardcover -- -- $124.95

Paperback -- -- $238.32

Don't wait -- order your copy today! :-)

REB

P.S. -- and while you wait for it to arrive, ponder the logic of a paperback that costs nearly twice as much as the hardcover version.

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see.

Ba'al Chatzaf

Ba'al, you may not remember it, but you issued this challenge nearly 4 years ago in the Metaphysics section. You then said you would accept a similar proof of Prop 2, which I provided. You then asked how it was categorical, and I responded and showed the proof as using strictly categorical propositions and syllogisms all the way. Here is the interchange from November 2007:

[ba'al 2007]Take in hand a universally accepted translation of Euclid's Elements. I would recommend the Heath translation which includes all kinds of reference notes and analysis of the proofs. Look at Prop 47, Book I. This is the famous Pythagoras Theorem for right angle triangles. The sum of the squares on the legs of a right triangle equals the square on the hypotenuse. O.K. Are you with me? Now using the list of valid categorical syllogisms given in the base line reference (the Wiki article) produce a proof of Prop 47 using only categorical syllogisms. Tough you say? You bet. But I am an easy fellow. Try proving the following: Prop 2 using only valid categorical syllogisms. Prop 2 states (using the Heath translation): To place at a given point (as an extremity) a straight line equal to a given straight line. The proof given by Euclid (as translated by Heath) does not use a single valid categorical syllogism. Not one. I am challenging you to replace Euclid's proof for this simple proposition with a proof that uses only valid categorical syllogisms (a la Aristotle as given in Prior Analytics). [underscoring added by REB]

Now Ba'al hedges:

[ba'al 2007] I do not deny that anyone capable of doing Aristotelean syllogistic has the wits necessary to prove Euclid's theorems.

So what's the point of the challenge???

Ba'al went on:

[ba'al 2007] But I also say, valid categorical syllogisms were not the way Euclid's theorems were proven. Hence the above challenges. Syllogistic logic, therefore, is NOT NECESSARY to do Euclid's geometry. And in light of the problem of the existence of least upper bounds, the syllogistic logic of Aristotle is NOT SUFFICIENT to do the necessary mathematics required by physics (basically, integral calculus). I would go even further to say that casting Euclidean proofs in the form of valid categorical syllogisms is not worth the effort. Logic based on the hypothetical conditional (which is now expressed as propositional calculus) and beefed up with quantifiers to get first order logic is good for many of the proofs in modern mathematics and physics and is a hell of a lot easier (Google <Natural Deduction>). Or put in a less charitable way, the Aristotelian Syllogistic is as necessary as buggy whips to get about.

The "buggy whips" comment is gratuitous and unfounded, as I showed in my response (see the underscored):

[REB 2007] Sorry, but none of this makes any sense to me. I aced Euclidean geometry, and I don't recall using any logical processes other than straightforward deduction from axioms, definitions, and previously established results.

You offer Euclid's Proposition 2 as a challenge. OK.

Given and having constructed the following:

1. Given segment BC and outside point A.

2. Connect A and B, making segment AB.

3. Make circle 1 on B with radius AB.

4. Make circle 2 on A with radius AB.

5. Make make circle 3 on B with radius BC.

6. Extend chord DB from intersection D of circles 1 and 2, through B to point G on circle 3.

7. Make circle 4 on D with radius DG.

8. Extend chord AD from D to A to point L on circle 4.

===================================

Prove that segment AL = segment BC:

1. BC = BG (radii of the same circle are equal)

2. DL = DG (radii of the same circle are equal)

3. DA = DB (Proposition 1)

4. AL = BG (from 2 and 3: equal segments diminished by the same amount are equal)

5. AL = BC (from 1 and 4: segments equal to the same segment are equal)

Despite the need for constructing 4 circles and 2 (or 3) line segments, the deductive process is relatively simple, requiring only two deductions. So, I did not have to stand on my head or get out the "buggy whip" in order to lay out the proof.

And the deductions are both categorical syllogisms, with the categorical premises being "All equal segments diminished by the same amount are equal" and "All segments equal to the same segment are equal". So, as I said before, Euclidean geometry is carried out by basic Aristotelian deductive logic.

Now, Aristotle would be the first to admit that more is needed in order to establish Euclid's propositions than deductive logic. One must discover or intuit the deductive pathway, perhaps by "thinking backward" from the intended conclusion, to figure out what is needed in order to establish the conclusion. [REB 2011: E.g., thinking hypothetically, "What if I did this? What result would I get?" Then casting the result in categorical terms, rather than hypothetical, which is not that difficult!] But this is nothing new or controversial.

If I missed anything in the 8 preliminary steps or the 5 proof steps, I'm sure you'll let me know. :-)

P.S. -- I wouldn't mind tackling Proposition 47 sometime when I'm not so busy. But I expect that it will be just somewhat "busier" and more long-winded than the proof for Proposition 2, not different in essence. Since you offered Proposition 2 as an acceptable stand-in, I will take your challenge as having been answered, in principle. [underscoring added]

Ba'al's response to this was twofold: to challenge me to point out the categorical syllogisms I used (they were implicit), and to say that even if, or even though, the proof can be done that way, it is logey, "constipated," equating categorical statements and arguments to driving a horse and buggy, while conditional statements and arguments are presumably more like driving a DeLorean <g>:

[ba'al 2007]Point out where you used categorical syllogisms. Starting with BaRBaRa and going to the others. My point is, that categorical syllogisms are rarely used in a mathematical context. And for good reason. The categorical style is constipated. Mathematics runs on conditional modes, not categorical modes. Different tune, different trope although they are all music.

I replied, making the syllogisms explicit this time:

[REB 2007] Simple! Barbara, all the way...

BC and BG are radii of the same circle.

Radii of the same circle are equal.

1. So, BC and BG are equal.

DL and DG are radii of the same circle.

Radii of the same circle are equal.

2. So, DL and DG are equal.

DA and DB are sides of equilaterial triangle (ADB).

Sides of an equilaterial triangle are equal. (Proposition 1.)

3. So, DA and DB are equal.

Equal segments diminished by the same amount are equal.

AL and BG are equal segments (DL and BG) diminished by equal amounts (DA and DB). (2 and 3)

4. So, AL and BG are equal.

Segments equal to the same segment are equal.

BC and AL are equal to the same segment (BG). (1 and 4)

5. So, AL and BC are equal.

Naturally, once we automatize the various definitions and propositions and the ways in which they are deductively related, we don't repeat all these steps explicitly. The deductions are abbreviated. For instance, in a single step of a proof, say #5, you refer to steps 1 and 4 and draw on the principle that segments equal to the same segment are equal. Just because this is abbreviated, however, does NOT mean that you are NOT using deductive logic, with categorical syllogisms.

Also, I must say that it does NOT appear "constipated" to me, to illustrate the categorical deductive structure of a Euclidean proof. Making the implicit explicit is illuminating, not constipating. To me, anyway. I guess I'm only in charge of ~my~ mental bowel habits. :-)

So, here we are, nearly 4 years later, and Ba'al has yet to reply and acknowledge that I gave a simple, clean, categorical syllogism proof of Proposition 2, Book I of Euclid's Elements.

Now he wants the same kind of proof of the Pythagorean Theorem (Proposition 47). For what purpose? This seems like a bait and switch. Four years ago, Ba'al said a proof of Proposition 2 would suffice to make his point, and I showed that it did no such thing. His response? <crickets chirping>

I mean, if someone wants to see an old man sweat, doing something another old man ought to be able to do himself, well, OK. But I've already provided the example to counter his denigration of categorical logic. Do I have to do all the work here?

REB

P.S. -- Also, for the record, I'm more than a little suspicious that categorical logic is not sufficient for using integral calculus, as Ba'al claimed on the other thread. But even if proving Euclid's Proposition 47 is as beastly as Ba'al seems to think it is, I will definitely tackle that one first, before trying to bring calculus into the real world. (Hypotheticals and conditionals are so weasel-ish, don't you know. :-)

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see. Ba'al Chatzaf
Ba'al, you may not remember it, but you issued this challenge nearly 4 years ago in the Metaphysics section. You then said you would accept a similar proof of Prop 2, which I provided. You then asked how it was categorical, and I responded and showed the proof as using strictly categorical propositions and syllogisms all the way. Here is the interchange from November 2007:
[ba'al 2007]Take in hand a universally accepted translation of Euclid's Elements. I would recommend the Heath translation which includes all kinds of reference notes and analysis of the proofs. Look at Prop 47, Book I. This is the famous Pythagoras Theorem for right angle triangles. The sum of the squares on the legs of a right triangle equals the square on the hypotenuse. O.K. Are you with me? Now using the list of valid categorical syllogisms given in the base line reference (the Wiki article) produce a proof of Prop 47 using only categorical syllogisms. Tough you say? You bet. But I am an easy fellow. Try proving the following: Prop 2 using only valid categorical syllogisms. Prop 2 states (using the Heath translation): To place at a given point (as an extremity) a straight line equal to a given straight line. The proof given by Euclid (as translated by Heath) does not use a single valid categorical syllogism. Not one. I am challenging you to replace Euclid's proof for this simple proposition with a proof that uses only valid categorical syllogisms (a la Aristotle as given in Prior Analytics). [underscoring added by REB]
Now Ba'al hedges:
[ba'al 2007] I do not deny that anyone capable of doing Aristotelean syllogistic has the wits necessary to prove Euclid's theorems.
So what's the point of the challenge??? Ba'al went on:
[ba'al 2007] But I also say, valid categorical syllogisms were not the way Euclid's theorems were proven. Hence the above challenges. Syllogistic logic, therefore, is NOT NECESSARY to do Euclid's geometry. And in light of the problem of the existence of least upper bounds, the syllogistic logic of Aristotle is NOT SUFFICIENT to do the necessary mathematics required by physics (basically, integral calculus). I would go even further to say that casting Euclidean proofs in the form of valid categorical syllogisms is not worth the effort. Logic based on the hypothetical conditional (which is now expressed as propositional calculus) and beefed up with quantifiers to get first order logic is good for many of the proofs in modern mathematics and physics and is a hell of a lot easier (Google <Natural Deduction>). Or put in a less charitable way, the Aristotelian Syllogistic is as necessary as buggy whips to get about.
The "buggy whips" comment is gratuitous and unfounded, as I showed in my response (see the underscored):
[REB 2007] Sorry, but none of this makes any sense to me. I aced Euclidean geometry, and I don't recall using any logical processes other than straightforward deduction from axioms, definitions, and previously established results. You offer Euclid's Proposition 2 as a challenge. OK. Given and having constructed the following: 1. Given segment BC and outside point A. 2. Connect A and B, making segment AB. 3. Make circle 1 on B with radius AB. 4. Make circle 2 on A with radius AB. 5. Make make circle 3 on B with radius BC. 6. Extend chord DB from intersection D of circles 1 and 2, through B to point G on circle 3. 7. Make circle 4 on D with radius DG. 8. Extend chord AD from D to A to point L on circle 4. =================================== Prove that segment AL = segment BC: 1. BC = BG (radii of the same circle are equal) 2. DL = DG (radii of the same circle are equal) 3. DA = DB (Proposition 1) 4. AL = BG (from 2 and 3: equal segments diminished by the same amount are equal) 5. AL = BC (from 1 and 4: segments equal to the same segment are equal) Despite the need for constructing 4 circles and 2 (or 3) line segments, the deductive process is relatively simple, requiring only two deductions. So, I did not have to stand on my head or get out the "buggy whip" in order to lay out the proof. And the deductions are both categorical syllogisms, with the categorical premises being "All equal segments diminished by the same amount are equal" and "All segments equal to the same segment are equal". So, as I said before, Euclidean geometry is carried out by basic Aristotelian deductive logic. Now, Aristotle would be the first to admit that more is needed in order to establish Euclid's propositions than deductive logic. One must discover or intuit the deductive pathway, perhaps by "thinking backward" from the intended conclusion, to figure out what is needed in order to establish the conclusion. [REB 2011: E.g., thinking hypothetically, "What if I did this? What result would I get?" Then casting the result in categorical terms, rather than hypothetical, which is not that difficult!] But this is nothing new or controversial. If I missed anything in the 8 preliminary steps or the 5 proof steps, I'm sure you'll let me know. :-) P.S. -- I wouldn't mind tackling Proposition 47 sometime when I'm not so busy. But I expect that it will be just somewhat "busier" and more long-winded than the proof for Proposition 2, not different in essence. Since you offered Proposition 2 as an acceptable stand-in, I will take your challenge as having been answered, in principle. [underscoring added]
Ba'al's response to this was twofold: to challenge me to point out the categorical syllogisms I used (they were implicit), and to say that even if, or even though, the proof can be done that way, it is logey, "constipated," equating categorical statements and arguments to driving a horse and buggy, while conditional statements and arguments are presumably more like driving a DeLorean <g>:
[ba'al 2007]Point out where you used categorical syllogisms. Starting with BaRBaRa and going to the others. My point is, that categorical syllogisms are rarely used in a mathematical context. And for good reason. The categorical style is constipated. Mathematics runs on conditional modes, not categorical modes. Different tune, different trope although they are all music.
I replied, making the syllogisms explicit this time:
[REB 2007] Simple! Barbara, all the way... BC and BG are radii of the same circle. Radii of the same circle are equal. 1. So, BC and BG are equal. DL and DG are radii of the same circle. Radii of the same circle are equal. 2. So, DL and DG are equal. DA and DB are sides of equilaterial triangle (ADB). Sides of an equilaterial triangle are equal. (Proposition 1.) 3. So, DA and DB are equal. Equal segments diminished by the same amount are equal. AL and BG are equal segments (DL and BG) diminished by equal amounts (DA and DB). (2 and 3) 4. So, AL and BG are equal. Segments equal to the same segment are equal. BC and AL are equal to the same segment (BG). (1 and 4) 5. So, AL and BC are equal. Naturally, once we automatize the various definitions and propositions and the ways in which they are deductively related, we don't repeat all these steps explicitly. The deductions are abbreviated. For instance, in a single step of a proof, say #5, you refer to steps 1 and 4 and draw on the principle that segments equal to the same segment are equal. Just because this is abbreviated, however, does NOT mean that you are NOT using deductive logic, with categorical syllogisms. Also, I must say that it does NOT appear "constipated" to me, to illustrate the categorical deductive structure of a Euclidean proof. Making the implicit explicit is illuminating, not constipating. To me, anyway. I guess I'm only in charge of ~my~ mental bowel habits. :-)
So, here we are, nearly 4 years later, and Ba'al has yet to reply and acknowledge that I gave a simple, clean, categorical syllogism proof of Proposition 2, Book I of Euclid's Elements. Now he wants the same kind of proof of the Pythagorean Theorem (Proposition 47). For what purpose? This seems like a bait and switch. Four years ago, Ba'al said a proof of Proposition 2 would suffice to make his point, and I showed that it did no such thing. His response? <crickets chirping> I mean, if someone wants to see an old man sweat, doing something another old man ought to be able to do himself, well, OK. But I've already provided the example to counter his denigration of categorical logic. Do I have to do all the work here? REB P.S. -- Also, for the record, I'm more than a little suspicious that categorical logic is not sufficient for using integral calculus, as Ba'al claimed on the other thread. But even if proving Euclid's Proposition 47 is as beastly as Ba'al seems to think it is, I will definitely tackle that one first, before trying to bring calculus into the real world. (Hypotheticals and conditionals are so weasel-ish, don't you know. :-)

None of the statement you provided were in categorical form A, E, I, O.

Just an example sides AB and DE are equal. Does that mean AB is equal and DE is equal? That is nonsense. Aristotelian categorical term logic does not deal with binary relations at all. To say that AB is equal to DE is not a categorical

statement at all.

An A statement all P are S.

and E statement No P are S

and I statement some P are S

and O statement some P are not S.

These are the kind of statements that can appear in syllogisms.

You have not used categorical statements as Aristotle specified them.

Try again.

I will be kind. Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal.

You can use any of the 19 valid categorical syllogisms and you can even string them together as a sorites. See how you do.

Ba'al Chatzaf

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Reply to Bissell's #5. If you want something cheaper:

1. Something To Reckon With: The Logic of Terms Extensive coverage of Sommer's term functor logic.

2. The Old New Logic: Essays on the Philosophy of Fred Sommers

3. Philosophical Logic: An Introduction to Advanced Topics I haven't read this. It's far-ranging based on the Table of Contents, which can be viewed at Amazon.

Roger, I guess you could look at The Logic of Natural Language at Vanderbilt.

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see.

Ba'al Chatzaf

Do you find something lacking in the traditional proofs of the Pythagorean Theorem?

Shayne

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Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal.

Sommers' term logic can handle it.

But not Aristotle.

Ba'al Chatzaf

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Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal.

Sommers' term logic can handle it.

But not Aristotle.

So what? Euclid couldn't do calculus. I don't see you grumbling about how inadequate Euclid was.

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Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal.

Sommers' term logic can handle it.

But not Aristotle.

So what? Euclid couldn't do calculus. I don't see you grumbling about how inadequate Euclid was.

Doing calculus is a bit "further out" than deducing that if all Dogs are Mammals then the tail of a Dog is the tail of a Mammal.

Archimedes who was just 60 years after Euclid, wrote the first book on calculus a little less than 2000 years before Newton and Leibniz. Euclid was born just about the time Aristotle died. Euclid had access to the works of Eudoxus (about 400 b.c.e). Eudoxus did develop the method of exhaustion which was a preliminary step toward integral calculus. Archimedes finished what Eudoxus started. It is interesting to note that none of the Greek mathematical (geometric) masters used categorical term logic in their work.

Ba'al Chatzaf

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Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal.

Sommers' term logic can handle it.

But not Aristotle.

So what? Euclid couldn't do calculus. I don't see you grumbling about how inadequate Euclid was.

Arrrrgh. Sure, Sommers' term logic can handle it, but so can Aristotle's logic.

First of all, you have to reduce the five terms down to three, or you're not going to get anywhere! The minor premise, "All dogs are mammals," is the culprit here, and the solution is simple: substitute the needed premise for the one given -- in this case, what's needed is: "All attributes of dogs are attributes of mammals." The rest is a piece of cake.

(The substitution is based on an immediate inference derived from this corollary of the Law of Identity: "All attributes of entities are attributes of the entities that they are." A dog is the same thing in reality as that dog considered as a member of a wider class, such as "mammal," which is why we can say "All dogs are mammals." And for the substitution: the tail of a dog is the same thing in reality as the tail of that same dog considered as a member of the wider class "tail of a mammal," which is why we can say "All tails of dogs are tails of mammals."

So, here is the categorical argument:

All tails of dogs are tails of mammals.

T is the tail of a dog.

Therefore, T is the tail of a mammal.

Possible objection: the major premise and the conclusion above are not universal categorical statements, but singular statements. True enough, as in the famous Socrates syllogism, where "Socrates is a man. All men are mortal. Therefore, Socrates is mortal." But traditional Aristotelian logic, for reasons best supported in terms of Rand's unit-perspective model of concepts, holds that singular propositions can validly be treated as universal propositions. So, "Socrates is a man" can be taken as equivalent to "All Socrates is a man," and "T is the tail of a dog" as equivalent to "All T is the tail of a dog," and similarly for the conclusion above. On this interpretation, "Socrates" and "T" are viewed and treated as though they were classes with a single member. Rand would say they were like "file folders" containing a single object, along with much information about it--like the "dossier" of a suspect or person of interest.

This, by the way, indicates in part how I think Rand's unit-perspective model of concepts can be used (in extended form, to include individuals considered as single-member classes or single items in a "mental file folder") to resolve a number of thorny issues surrounding truth and falsity, existential import, propositions with non-existent subjects, etc. Had Rand's model existed in the 1800s, a lot of the bogus crap that Boole, Frege, and Russell were able to perpetrate would not have stood. As it is, the Augean stables of modern logic are going to be almost impossible to scoop out. By contrast, Hercules only had a 30-year accumulation of horse-crap to deal with.

REB

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Can anyone out there give us a proof of Prop 47, Book I of Euclid's Elements using Aristotelian categorical term logic. This I have to see. Ba'al Chatzaf
Ba'al, you may not remember it, but you issued this challenge nearly 4 years ago in the Metaphysics section. You then said you would accept a similar proof of Prop 2, which I provided. You then asked how it was categorical, and I responded and showed the proof as using strictly categorical propositions and syllogisms all the way. Here is the interchange from November 2007:
[ba'al 2007]Take in hand a universally accepted translation of Euclid's Elements. I would recommend the Heath translation which includes all kinds of reference notes and analysis of the proofs. Look at Prop 47, Book I. This is the famous Pythagoras Theorem for right angle triangles. The sum of the squares on the legs of a right triangle equals the square on the hypotenuse. O.K. Are you with me? Now using the list of valid categorical syllogisms given in the base line reference (the Wiki article) produce a proof of Prop 47 using only categorical syllogisms. Tough you say? You bet. But I am an easy fellow. Try proving the following: Prop 2 using only valid categorical syllogisms. Prop 2 states (using the Heath translation): To place at a given point (as an extremity) a straight line equal to a given straight line. The proof given by Euclid (as translated by Heath) does not use a single valid categorical syllogism. Not one. I am challenging you to replace Euclid's proof for this simple proposition with a proof that uses only valid categorical syllogisms (a la Aristotle as given in Prior Analytics). [underscoring added by REB]
Now Ba'al hedges:
[ba'al 2007] I do not deny that anyone capable of doing Aristotelean syllogistic has the wits necessary to prove Euclid's theorems.
So what's the point of the challenge??? Ba'al went on:
[ba'al 2007] But I also say, valid categorical syllogisms were not the way Euclid's theorems were proven. Hence the above challenges. Syllogistic logic, therefore, is NOT NECESSARY to do Euclid's geometry. And in light of the problem of the existence of least upper bounds, the syllogistic logic of Aristotle is NOT SUFFICIENT to do the necessary mathematics required by physics (basically, integral calculus). I would go even further to say that casting Euclidean proofs in the form of valid categorical syllogisms is not worth the effort. Logic based on the hypothetical conditional (which is now expressed as propositional calculus) and beefed up with quantifiers to get first order logic is good for many of the proofs in modern mathematics and physics and is a hell of a lot easier (Google <Natural Deduction>). Or put in a less charitable way, the Aristotelian Syllogistic is as necessary as buggy whips to get about.
The "buggy whips" comment is gratuitous and unfounded, as I showed in my response (see the underscored):
[REB 2007] Sorry, but none of this makes any sense to me. I aced Euclidean geometry, and I don't recall using any logical processes other than straightforward deduction from axioms, definitions, and previously established results. You offer Euclid's Proposition 2 as a challenge. OK. Given and having constructed the following: 1. Given segment BC and outside point A. 2. Connect A and B, making segment AB. 3. Make circle 1 on B with radius AB. 4. Make circle 2 on A with radius AB. 5. Make make circle 3 on B with radius BC. 6. Extend chord DB from intersection D of circles 1 and 2, through B to point G on circle 3. 7. Make circle 4 on D with radius DG. 8. Extend chord AD from D to A to point L on circle 4. =================================== Prove that segment AL = segment BC: 1. BC = BG (radii of the same circle are equal) 2. DL = DG (radii of the same circle are equal) 3. DA = DB (Proposition 1) 4. AL = BG (from 2 and 3: equal segments diminished by the same amount are equal) 5. AL = BC (from 1 and 4: segments equal to the same segment are equal) Despite the need for constructing 4 circles and 2 (or 3) line segments, the deductive process is relatively simple, requiring only two deductions. So, I did not have to stand on my head or get out the "buggy whip" in order to lay out the proof. And the deductions are both categorical syllogisms, with the categorical premises being "All equal segments diminished by the same amount are equal" and "All segments equal to the same segment are equal". So, as I said before, Euclidean geometry is carried out by basic Aristotelian deductive logic. Now, Aristotle would be the first to admit that more is needed in order to establish Euclid's propositions than deductive logic. One must discover or intuit the deductive pathway, perhaps by "thinking backward" from the intended conclusion, to figure out what is needed in order to establish the conclusion. [REB 2011: E.g., thinking hypothetically, "What if I did this? What result would I get?" Then casting the result in categorical terms, rather than hypothetical, which is not that difficult!] But this is nothing new or controversial. If I missed anything in the 8 preliminary steps or the 5 proof steps, I'm sure you'll let me know. :-) P.S. -- I wouldn't mind tackling Proposition 47 sometime when I'm not so busy. But I expect that it will be just somewhat "busier" and more long-winded than the proof for Proposition 2, not different in essence. Since you offered Proposition 2 as an acceptable stand-in, I will take your challenge as having been answered, in principle. [underscoring added]
Ba'al's response to this was twofold: to challenge me to point out the categorical syllogisms I used (they were implicit), and to say that even if, or even though, the proof can be done that way, it is logey, "constipated," equating categorical statements and arguments to driving a horse and buggy, while conditional statements and arguments are presumably more like driving a DeLorean <g>:
[ba'al 2007]Point out where you used categorical syllogisms. Starting with BaRBaRa and going to the others. My point is, that categorical syllogisms are rarely used in a mathematical context. And for good reason. The categorical style is constipated. Mathematics runs on conditional modes, not categorical modes. Different tune, different trope although they are all music.
I replied, making the syllogisms explicit this time:
[REB 2007] Simple! Barbara, all the way... BC and BG are radii of the same circle. Radii of the same circle are equal. 1. So, BC and BG are equal. DL and DG are radii of the same circle. Radii of the same circle are equal. 2. So, DL and DG are equal. DA and DB are sides of equilaterial triangle (ADB). Sides of an equilaterial triangle are equal. (Proposition 1.) 3. So, DA and DB are equal. Equal segments diminished by the same amount are equal. AL and BG are equal segments (DL and BG) diminished by equal amounts (DA and DB). (2 and 3) 4. So, AL and BG are equal. Segments equal to the same segment are equal. BC and AL are equal to the same segment (BG). (1 and 4) 5. So, AL and BC are equal. Naturally, once we automatize the various definitions and propositions and the ways in which they are deductively related, we don't repeat all these steps explicitly. The deductions are abbreviated. For instance, in a single step of a proof, say #5, you refer to steps 1 and 4 and draw on the principle that segments equal to the same segment are equal. Just because this is abbreviated, however, does NOT mean that you are NOT using deductive logic, with categorical syllogisms. Also, I must say that it does NOT appear "constipated" to me, to illustrate the categorical deductive structure of a Euclidean proof. Making the implicit explicit is illuminating, not constipating. To me, anyway. I guess I'm only in charge of ~my~ mental bowel habits. :-)
So, here we are, nearly 4 years later, and Ba'al has yet to reply and acknowledge that I gave a simple, clean, categorical syllogism proof of Proposition 2, Book I of Euclid's Elements. Now he wants the same kind of proof of the Pythagorean Theorem (Proposition 47). For what purpose? This seems like a bait and switch. Four years ago, Ba'al said a proof of Proposition 2 would suffice to make his point, and I showed that it did no such thing. His response? <crickets chirping> I mean, if someone wants to see an old man sweat, doing something another old man ought to be able to do himself, well, OK. But I've already provided the example to counter his denigration of categorical logic. Do I have to do all the work here? REB P.S. -- Also, for the record, I'm more than a little suspicious that categorical logic is not sufficient for using integral calculus, as Ba'al claimed on the other thread. But even if proving Euclid's Proposition 47 is as beastly as Ba'al seems to think it is, I will definitely tackle that one first, before trying to bring calculus into the real world. (Hypotheticals and conditionals are so weasel-ish, don't you know. :-)

None of the statement you provided were in categorical form A, E, I, O.

Just an example sides AB and DE are equal. Does that mean AB is equal and DE is equal? That is nonsense. Aristotelian categorical term logic does not deal with binary relations at all. To say that AB is equal to DE is not a categorical statement at all.

An A statement all P are S.

and E statement No P are S

and I statement some P are S

and O statement some P are not S.

These are the kind of statements that can appear in syllogisms.

You have not used categorical statements as Aristotle specified them.

Try again. [Tail of dog substitute challenge deleted...reb]

C'mon, Ba'al, you're doing a Complex Question here. Or, at least, muddling two distinct issues, both of which I'll be happy to address. Plus, sides AB and DE aren't even in the proof!

1. You say the premises in my syllogisms aren't categorical statements, and you remind me that I must express them either as A, E, I, or O propositions, or I will fail your test. Fine, I will follow traditional practice and reword the individual propositions and treat them AS THOUGH they are universal categorical propositions, in much the same way as logicians reword "Socrates is mortal" as "All Socrates is mortal," treating Socrates as belonging to a single-member class. (I referred in my previous post to how this fits in with Rand's unit-perspective model, viewing a single-member class as being like a "mental file folder" containing one item along with a lot of data about it, similar to a "dossier.") It will be uglier than simply saying "BC and BG are equal," but if you have to actually SEE the word "All" to be satisfied, then here ya go! "All BC and BG are equal."

2. But that's not enough. Noooooo. You claim that "BC and BG are equal" is not a categorical proposition, and that it cannot be coherently interpreted. Oh, Kant rare! You need it spelled out for you? Here! "All line segments BC and BG [in the present construction] are radii of the same circle. All radii of the same circle are line segments of the same length. So, All line segments BC and BG [in the present construction] are line segments of the same length."

As I said before: all categorical, all universal-ish, and clear as a bell. But now also wordy and logey as hell, because you wouldn't accept shorthand as complying with your challenge. <DEEP SIGH> Satisfied yet?

REB

P.S. -- I've jumped through enough hoops for you Ba'al. I've shown you how it's done. Now, why don't ~you~ fry a few brain cells and do the categorical proposition proof of the Pythagorean Theorem that you've badgered us about? I know it ~can~ be done, so here's my offer to you: if ~you~ will be so kind as to do the categorical proposition proof of the Pythagorean Theorem, ~I~ will do a categorical proposition derivation of the integral calculus. (That should keep him busy for a while....snicker.)

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http://youtu.be/WqRf0LTOD3o

http://en.wikipedia....agorean_theorem

This one is really cool...

http://www.themathpage.com/abooki/propi-47.htm

Can someone explain to me again how children in the public school are not learning?

Edited by Selene
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As I said before: all categorical, all universal-ish, and clear as a bell. But now also wordy and logey as hell, because you wouldn't accept shorthand as complying with your challenge. <DEEP SIGH> Satisfied yet?

REB

Universal-ish???? If one of my students ever used that on an exam in a logic course I taught I would flunk him out instantly.

I am strict. You did not use categorical propositions. You did not use any of the 19 valid syllogisms. You lose. Aristotelian syllogistics might suffice for some verbal arguments but they are inadequate for mathematics. Also Aristotelian logic is totally inadequate for n-ary relations where n >= 2. Aristotle's logic is, in essence, the logic of monadic predicates. It simply does not work for higher order predicates. That is why the tail of a dog argument fails in syllogistic logic.

Mathematicians use conditional reasoning, not syllogistic reasoning. This form of logic is best captured by Natural Deduction after the formulation of Gentzen. See: http://en.wikipedia.org/wiki/Natural_deduction

Logic was not liberated from its Aristotelian manacles until the time of Boole, Frege and Peirce. (Leibniz tried to formulate a hypothetical type logic, but he did not quite manage).

Ba'al Chatzaf

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Archimedes who was just 60 years after Euclid, wrote the first book on calculus a little less than 2000 years before Newton and Leibniz. Euclid was born just about the time Aristotle died. Euclid had access to the works of Eudoxus (about 400 b.c.e). Eudoxus did develop the method of exhaustion which was a preliminary step toward integral calculus. Archimedes finished what Eudoxus started. It is interesting to note that none of the Greek mathematical (geometric) masters used categorical term logic in their work.

Thus you have even more reason to grumble about Euclid's inadequacy.

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Archimedes who was just 60 years after Euclid, wrote the first book on calculus a little less than 2000 years before Newton and Leibniz. Euclid was born just about the time Aristotle died. Euclid had access to the works of Eudoxus (about 400 b.c.e). Eudoxus did develop the method of exhaustion which was a preliminary step toward integral calculus. Archimedes finished what Eudoxus started. It is interesting to note that none of the Greek mathematical (geometric) masters used categorical term logic in their work.

Thus you have even more reason to grumble about Euclid's inadequacy.

Euclid was more of a compiler and an organizer than an original mathematician. Nevertheless the layout he created was a prototype for mathematical treatises for the next 2000 years. His trademark organization was not improved upon until the middle of the 19th century. Euclid "cleaned up" the various geometrical styles of the more original contributors. He gathered up the loose ends and created a fairly tight logical organization of the material. Euclid's achievement was only matched in the early 20 th centural by Bourbaki, a consortium of mathematicians.

Speculation: Was Euclid one person, or also a consortium over the years?

Ba'al Chatzaf

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As I said before: all categorical, all universal-ish, and clear as a bell. But now also wordy and logey as hell, because you wouldn't accept shorthand as complying with your challenge. <DEEP SIGH> Satisfied yet?

REB

Universal-ish???? If one of my students ever used that on an exam in a logic course I taught I would flunk him out instantly.

I am strict. You did not use categorical propositions. You did not use any of the 19 valid syllogisms. You lose. Aristotelian syllogistics might suffice for some verbal arguments but they are inadequate for mathematics. Also Aristotelian logic is totally inadequate for n-ary relations where n >= 2. Aristotle's logic is, in essence, the logic of monadic predicates. It simply does not work for higher order predicates. That is why the tail of a dog argument fails in syllogistic logic.

I'm sorry, your "strictness" is an overly puristic mangling of traditional Aristotelian procedure. Either that, or you are using some radically different definition of a categorical proposition than I am. You asked me to try to avoid conditional propositions and use categorical ones instead. Fine. That is what I have done.

Conditional propositions use "if...then." Categoricals simply connect subject with predicate via "is" or some other form of "to be." Categorical means unconditional, not "if....then," but simply "S is P."

Yes, the four standard forms of catprops are "All S is P," "No S is P," "Some S is P," and "Some S is not P." But there are also ~individual~ catprops, such as "Socrates is a man" or "Line segments BC and BG [in the present construction] are radii of the same circle." And these, BY STANDARD PROCEDURE, are justifiably treated as one of the four standard forms -- in this case, as a universal affirmative categorical proposition.

Yes, what I gave in the proof was originally an ~individual~ affirmative categorical proposition, not a universal affirmative categorical proposition. But by employing standard procedure, I TREATED IT AS a universal affirmative categorical proposition. That is why I called it "universal-ish." I'm sorry such linguistic playfulness offends your sensibilities as a teacher of logic. But if I were your student, and you tried to "flunk [me] out immediately" for that, I would summon the ghost of Aristotle to kick your anal-retentive a**.

You can deny that this treatment is ~justified~ (and you have already clearly shown that you are opaque to the justification I've offered), but you cannot deny that this is standard procedure in traditional Aristotelian logic.

So, by standard procedure, the stock Socrates syllogism, "[ALL]Socrates is a man. All men are mortal. Therefore, [ALL] Socrates is a mortal" is a straightforward BARBARA syllogism. By the same token, so is each of the syllogisms I used in my proof of Proposition 2, Book I of Euclid's Elements: "[ALL] Line segments BC and BG [in the present construction] are radii of the same circle. All radii of the same circle are line segments of the same length. So, [ALL] Line segments BC and BG [in the present construction] are line segments of the same length."

Again, you may not like to admit it, but that ~is~ a BARBARA categorical syllogism. You may reject it, but you are wrong.

[.....]

Logic was not liberated from its Aristotelian manacles until the time of Boole, Frege and Peirce. (Leibniz tried to formulate a hypothetical type logic, but he did not quite manage).

Logic was not placed into its Kantian manacles until the time of Boole, Frege, and Peirce. Kant (with the help of Hume) fashioned the manacles by claiming (in his Critique of Pure Reason that existence is not a predicate, a manifest absurdity which destroys knowledge as the mental grasp of facts of reality. But it was not until decades later that those wishing to detach logic from reality figured out how to use it to deny existential import to universal categorical propositions, claiming that "Some" connects a proposition to reality, but "All" does not.

Only if you view things as having no graspable, essential nature which is true of all things of that kind -- basically, a skeptic-empiricist view -- can you plausibly claim that ~some~ things of that kind having that nature is ~observable~ and thus validly assertable, while ~all~ things of that kind having that nature is ~unobservable~ and thus ~not~ validly assertable. We've been over this ground before. Hume's ontological error of dismissing real essences and his epistemic error of dismissing induction (other than enumerative induction) are the likely root of this bifurcation of the existential import of universal ["All"] and particular ["Some"] categorical propositions.

Hume's noxious philosophy is a lot like poison ivy. You really don't want to get it on you, or you will suffer. Unlike poison ivy, though, it's not contagious...unless your teachers intimidate you (with threats of bad grades or flunking) into touching it.

REB

P.S. -- BTW, ALL deductive arguments are "verbal" arguments, including mathematics, so because Aristotelian logic suffices ALL "verbal" arguments, it also suffices for mathematical arguments. This is because, more fundamentally, all mathematical equations can be recast verbally as categorical "S is P" propositions, from "2 + 2 = 4" on up to the most complex equation you can imagine. There might not be any practical advantage in doing so, but the point is that either mathematics is, in principle, ontologically groundable in this way, or it is, at best, a sometimes inadvertently useful activity of establishing logical connections between symbols that are a congeries of floating abstractions.

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  • 4 weeks later...

Here's another challenge for Aristotle's logic:

Everything has a cause (premise 1)

If God exists, Then something isn't caused (namely, God) (premise 2)

Therefore, God doesn't exist (conclusion).

The problem is that premise 2 has a quantified term in the consequent of the hypothetical.

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P.S. -- BTW, ALL deductive arguments are "verbal" arguments, including mathematics, so because Aristotelian logic suffices ALL "verbal" arguments, it also suffices for mathematical arguments. This is because, more fundamentally, all mathematical equations can be recast verbally as categorical "S is P" propositions, from "2 + 2 = 4" on up to the most complex equation you can imagine. There might not be any practical advantage in doing so, but the point is that either mathematics is, in principle, ontologically groundable in this way, or it is, at best, a sometimes inadvertently useful activity of establishing logical connections between symbols that are a congeries of floating abstractions.

Prove the Heine Borel compactness theorem using Aristotelian Term Logic. This I have to see! Also prove the Goedel Incompleteness Theorem using Aristotelian Term Logic., If you can do that I will believe in Miracles.

No claims, no assertions now. Show proof!

Ba'al Chatzaf

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P.S. -- BTW, ALL deductive arguments are "verbal" arguments, including mathematics, so because Aristotelian logic suffices ALL "verbal" arguments, it also suffices for mathematical arguments. This is because, more fundamentally, all mathematical equations can be recast verbally as categorical "S is P" propositions, from "2 + 2 = 4" on up to the most complex equation you can imagine. There might not be any practical advantage in doing so, but the point is that either mathematics is, in principle, ontologically groundable in this way, or it is, at best, a sometimes inadvertently useful activity of establishing logical connections between symbols that are a congeries of floating abstractions.

Prove the Heine Borel compactness theorem using Aristotelian Term Logic. This I have to see! Also prove the Goedel Incompleteness Theorem using Aristotelian Term Logic., If you can do that I will believe in Miracles.

No claims, no assertions now. Show proof!

Ba'al Chatzaf

No claims, no assertions -- and no more proofs. I've jumped through enough hoops for you, Ba'al. In principle, since any hypothetical proposition can be expressed as a categorical proposition, any mathematical proof stated with hypotheticals can be restated with categoricals.

The conclusion of the Heine Borel compactness theorem as a hypothetical (actually, a biconditional) is: "A subset of a metric space is compact if and only if it is complete and totally bounded." Expressed as a categorical, it is: "A compact metric space subset is a complete and totally bounded metric space subset -- and a complete and totally bounded metric space subset is a compact metric space subset."

It might be fun, in a soda-crackers-in-the-desert dry sort of way, to wade through the process of converting all the conditionals to categoricals, but why should I? You know it can be done. No one does it, for the obvious reason that it's more verbally compact to say it as a conditional or biconditional than a categorical.

Handling the Heine Borel theory's conclusion (and its various premises) in this way is, again in principle, no different from re-expressing: "A living being is a human being if and only if it is a rational being" as: "A human being is a rational being, and a rational being is a human being." (Until some other rational living beings are discovered or created, and we have to change the contextually correct definition of "human being.")

But as I said, you keep raising the bar, even while refusing to acknowledge that I have met your previous challenges. Why should I humor you? It's a lot of fricking work, and I have other things to do, especially since I have no reason to believe that you will treat any additional demonstrations by me any differently than you have treated the ones I've met so far. (I'm sure if you ~didn't~ think I had met them, you'd still be arguing with me over them, instead of repeatedly redrawing the line in the sand.)

Perhaps you're trying to relive your youthful days of talking chicks out of their clothes by betting them they didn't have a blouse on under their coat, then not a bra on under their blouse, etc. :-)

REB

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Here's another challenge for Aristotle's logic:

Everything has a cause (premise 1)

If God exists, Then something isn't caused (namely, God) (premise 2)

Therefore, God doesn't exist (conclusion).

The problem is that premise 2 has a quantified term in the consequent of the hypothetical.

Huh? Why not say it straight out: "God is uncaused," or "No God is a thing that is caused."

Now, here's a ~real~ syllogism:

All things that exist are things that are caused.

No God is a thing that is caused.

Therefore, no God is a thing that exists.

All P is R

No S is R

Therefore, No S is P

Looks pretty good. The Venn diagrams confirm it (as the reader may see for himself). No deity to see here, move along. :-)

If both premises are true, then the conclusion is true. I.e., all syllogisms with valid logic and true premises are syllogisms with true conclusions.

Looking in particular at the first premise, all the evidence we have supports it, and none contradicts it. But even if somehow the first premise ~were~ false, theists could not take heart, because the conclusion would then be GIGO, as all good computer programmers know. :-)

In other words, the truth status of the conclusion of a valid syllogism with one or more false premises is independent of the truth value of the premises. (All cows are fish, all fish are mammals, therefore all cows are mammals -- but All cows are fish, all fish are crustaceans, therefore all cows are crustaceans.)

REB

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