Modern versus Traditional logic


Davy

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The basic problem with Aristotle's approach was existential import. Here is an example: statements of type A: all S are P imply statements of type I: some S are P. Now consider the type A: assertion all Unicorns are white. This must be true. Otherwise its contradictory, the type O: statement some Unicorns are not white, is true. If that were the case, then there would exist something that is (1) a Unicorn and (2) non- white, which would mean there exists a Unicorn. O.K. So the statement all Unicorns are White must be true. But this implies the type I statement some Unicorns are white. (see any reference to the traditional square of opposition for this). Well, from this it follows there is something that is (1) a Unicorn and (2) is white. Which means there is something that is a Unicorn. The insistence on existential import leads to nonsensical proofs for the existence of God as well as Unicorns. George Boole cured this by introducing categorical logic in which the rule of existential import was relaxed. This simplified the square of opposition considerably. Goodbye to alternation, subalternation. The only links left in the square are for contradictories: A and O—and I and E.

Oh, Kant rare, Bob. The basic problem with Aristotle’s approach was not existential import, but the ambiguity and equivocation that result from the failure of his critics and supporters alike to clarify the mode of existence in their statements. It is unreasonable to hold Aristotle’s doctrine of truth-value in immediate inference responsible for the shortcomings of those making ambiguous statements about imaginary creatures like Unicorns that do not really exist.

All we have to do to rectify this situation—and to indicate a simple way to obviate a large part of the supposed “improvements” foisted on us by modern logic—is to make one small clarifying change in the original A statement (and the corresponding I and O statements), a change which erases the ambiguity: All Unicorns are real creatures that are white. This says that all Unicorns are real creatures (that are white), which is clearly false.

Now consider the alleged “problem” with the A statement’s contradictory, the O statement, which becomes: Some Unicorns are not real creatures that are white. This says some Unicorns are not real creatures (that are white), and that is certainly true. With the ambiguity removed, we now see that Aristotle’s doctrine of opposite truth-value for contradictories holds after all, and the two propositions both are clear in their meaning and their truth-value.

On to the I statement, which is implied by the A statement, and which now becomes: Some Unicorns are real creatures that are white. This says some Unicorns are real creatures (that are white) which, again, is clearly false, as it should be, as the implication of a false A statement. Aristotle’s doctrine of an I statement having the same truth value as a true A statement is upheld, and the two propositions both are clear in their meaning and their truth-value.

As we can see, none of this implies that there is “something that is a unicorn.” Some of the statements say there is, some of them say there isn’t—and the former are false, while the latter are true.

I figured all this out 20 years ago, while studying through the existential import and immediate inference sections of David Kelley’s logic textbook. I don’t think he understands this issue very well, at least not in any of the first three editions. (We’re still waiting for the 4th edition, which supposedly was completed over 18 months ago.) But to be fair, the other major college logic texts (Copi and Hurley) have done an even worse job. Copi’s exercises on this subject are tantamount to child abuse, in my opinion.

Aristotle’s Square of Opposition and immediate inference alive and well. All of the links in Aristotle’s square are robustly intact, not just the ones for contradictories.

Boole’s alleged “cure” is worse than the supposed “disease,” which is remedied by simple verbal clarification. All we have to remember to do is to say explicitly what we mean, when there is any question about the existence of what we’re talking about. Maybe Aristotle didn’t say this in so many words, but that’s no excuse for the Boolean-Russellian travesty.

REB

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Modern logic made computer design possible. Without Boolean functions there would be no reasonable way of designing complex computer circuits.

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The traditional square of opposition can be rescued by a very simple ploy. Assume there is some S. And all the traditional arrows follow. The Boolean square of opposition relaxes this assumption. S is allowed to be empty. We need empty sets. Why. So the algebra of sets can from a complete lattice and all the boolean functions can be constructed. Without that we would have no algebra to construction arbitrarily complicated computer circuits.

Many of the people who think Boolean algebra is a scam also believe we could have gotten transistors without the quantum theory of solids. So even in the technological realm there are flat-earthers.

Ba'al Chatzaf

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The traditional square of opposition can be rescued by a very simple ploy. Assume there is some S. And all the traditional arrows follow. The Boolean square of opposition relaxes this assumption. S is allowed to be empty. We need empty sets. Why. So the algebra of sets can from a complete lattice and all the boolean functions can be constructed. Without that we would have no algebra to construction arbitrarily complicated computer circuits.

Many of the people who think Boolean algebra is a scam also believe we could have gotten transistors without the quantum theory of solids. So even in the technological realm there are flat-earthers.

Ba'al Chatzaf

It's not a ploy. It's a very simple, ethically mandatory requirement of proper communication: say exactly what you mean. If you fail to do this regarding things that do not exist, paradoxes happen and inference breaks down.

We already know this to be true from informal logic. The fallacy of equivocation is a prime example. Asking people not to commit this fallacy is not a "ploy." Again, it's just a request for responsible, ethical communication.

I believe that people who fail to honor this request (and standing obligation) are very suspicious characters, and that everything flowing from their weasely tactics should be examined very carefully and rubbed in their faces when necessary, starting with Bertrand Russell and his henchmen in the academic logic rackets. Garbage in, garbage out.

Even when they stumble over facts and useful information with their flawed methodology, it is likely by accident and is probably in need of a sounder theoretical and factual grounding. Mere deductive logic is not sufficient to establish fact, even if a true conclusion follows validly from false premises.

All cows are two-legged creatures.

All two-legged creatures are mammals.

Therefore, all cows are mammals. (computers and transistors)

This is also true in the physical sciences. Quantum mechanics, for instance, is true in its mathematics, and very useful in its technological applications, but most interpretatiions of it are hideously false in their ontologies.

REB

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...Subject-predicate or as Sommers puts it term-functor-logic is keyed to reasoning in natural language mode. Its main defect is the inability to deal with relationals. In categorical logic of the traditional scholastic form one cannot make this argument: a horse is an animal. sven is the owner of a horse therefore sven is the owner of an animal. That is because the assertion x is an R of y (R a relation) does not have a categorical subject-predicate structure with the usual quantity-quality designators...

There is no such defect in subject-predicate logic. If there were, traditional logic couldn't even do "All men are mortal. Socrates is a man. Therefore all men are mortal."

Consider some typical "relational" propositions...

In "Sven is the owner of an animal"--which I would express less ambiguously as "Sven is an owner of an animal"--what is designated by "Sven" is also designated by "an owner of an animal." Sven is the same thing in reality as one of the members of the class "owner of an animal."

In "Chicago is south of Milwaukee"--which I would express as "Chicago is a city that is south of Milwaukee"--what is designated by "Chicago" is also designated by "a city south of Milwaukee." Chicago is the same thing in reality as one of the members of the class "city south of Milwaukee."

In "Socrates is the teacher of Plato," what is designated by "Socrates" is also designated by "the teacher of Plato." Socrates is the same thing in reality as the teacher of Plato. (For the purpose of the example, we are considering "the teacher of Plato" to be a single-member class. No doubt, he had other teachers, but Socrates was perhaps Plato's only philosophy teacher.)

Doing standard traditional logic with such propositions is completely unproblemmatic.

And actually, ~all~ categorical subject-predicate propositions are also "relational" propositions," and all relational propositions can (and should) be rewritten as categorical subject-predicate propositions, where the predicate's class is made explicit.

In "Socrates is a man", what is designated by "Socrates" is also designated by "man." Socrates is the same thing in reality as one of the members of the class "man."

The key, though, is to bear in mind that all of the various ~other~ relationships (including being-owner-of, being-a-city-south-of, being-the-teacher-of, being-a-member-of-the-class-of) reduce to the fundamental Identity-relationship of being-the-same-thing-in-reality-as.

"Ayn Rand is the author of Atlas Shrugged"--relational? Sure. Being-the-author-of-Atlas-Shrugged is certainly a relation, and there is a person who stands in that relation. But that relation, like all relations, defines a class of things--in this case, the single-member class "author of Atlas Shrugged," because one and only one person stands in the relation that defines that class.

So, "Ayn Rand is the author of Atlas Shrugged" most fundamentally should be understood as saying: "Ayn Rand IS THE SAME THING IN REALITY AS the author of Atlas Shrugged." All of propositional logic can be dealt with in this way.

"Is the same thing in reality" is what "is" means, when you strip away all the ambiguities. (Are you listening, Bill Clinton?)

It even works when you state the foregoing as a subject-predicate categorical proposition: "Is the same thing in reality" is the same thing in reality as the meaning of "is." Or, compressing the relationship to a simple copula: "Is the same thing in reality" is the meaning of "is." (But I repeat myself....)

To me, this insight is the bridge between what is valid of traditional "what" logic and modern "relational" logic.

"Chicago is (south of Milwaukee)" can be converted to "Chicago (is south of) Milwaukee" and back again, with no loss of meaning or perspective or understanding. So can "Chicago is (the same thing in reality as one of the cities south of Milwaukee) and "Chicago (is the same thing in reality as) one of the cities south of Milwaukee."

No, we don't have to use the stilted, clunky, verbose formulations involving "is the same thing in reality as," but it's very important to bear in mind, and to be able to apply it when necessary, in cases of confusion, ambiguity, and the like.

Also, using propositions to express the identity or "what" of something is a more fundamental cognitive function than using propositions to embody and display more specific existential relations. To that extent, modern logic is formulated in terms of non-essentials and is thus more prone to error in deal with fallacies, paradoxes, immediate inference, etc. that involve the issue of truth-value in relation to existential import.

REB

Prove the Pythagorean theorem using only the categorical logic os syllogisms.

That should be an amusing exercise for you.

See Euclid Book I and redo Book I using only the 19 valid categorical syllogisms that Aristotle and his successor at the Lyceum identified. Aristotle did not do syllogisms in the 4 th figure. 4 more valid forms were added to the 15 that Aristotle identified.

It is time to put up or be somewhat more still.

Ba'al Chatzaf

I'm not going to do all of Euclid Book I at this time, nor even Book I, Proposition 17. It's not necessary to support my case, and it would be nice to see some other old man sweat a bit, to demonstrate that he can take it as well as dish it out. The following categoricals-only proof of Book I, Proposition 2 should be sufficient. (Take out your pencils, paper, straight-edges, and compasses, if you want to follow along.)

First, let me say that when I took high school geometry, I don’t recall using any logical processes other than straightforward deduction from axioms, definitions, and previously established results. What few conditional statements the proofs may have contained could easily be rephrased as categoricals, and that is how I will proceed here.

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 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.

Now, the assignment is to prove that the construction has been successful and, therefore, that segment AL = segment BC. Here is the proof:

  1. BC = BG (all radii of the same circle are equal)
  2. DL = DG (all radii of the same circle are equal)
  3. DA = DB (Proposition 1)
  4. AL = BG (from 2 and 3: all equal segments diminished by the same amount are equal)
  5. AL = BC (from 1 and 4: all segments equal to the same segment are equal)

As you can see, despite Euclid’s requirement of constructing 4 circles and 2 (or 3) line segments, the process for proving his proposition is relatively simple, requiring only two deductions. So, I did not have to stand on my head or get out a “buggy whip” in order to coerce my brain into laying out the proof.

Also, the deductions are both categorical syllogisms, using categorical propositions, and the premises (in parenthesis) are all categorical as well. (Radii of the same circle are equal. Equal segments diminished by the same amount are equal. Segments equal to the same segment are equal.) Basic Aristotelian deductive logic applies nicely to Euclidean geometry.

And now for the qualifications and objections…

First of all, Aristotle would be quick to admit that more is needed in order to establish Euclid's propositions than deductive logic and categorical propositions and syllogisms. 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.

Nowadays, we call this “reverse engineering.” How do I get this result? What if I did this? What result would I get? Once we find the right logical pathway, we can cast the result in categorical terms, rather than hypothetical or conditional. But this is not difficult, and it is not new, and it is not controversial.

Secondly, though, where are the categorical syllogisms? The five-step proof given above was highly compressed, with some of the premises in parenthesis, some of the conclusions also serving as premises, etc. In total, there were five syllogisms, all in the “Barbara” form:

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 need repeat all these steps explicitly. Nor should we, except for pedagogical purposes, such as training young math students to “show their work.”

But once the method of proof has been mastered, mental shortcuts and notational economies and abbreviations are normally and reasonably permitted, and this does not somehow make the arguments non-categorical or otherwise fishy. If we hadn’t been granted such boons, high school geometry would surely have taken four years instead of just one!

Another pedagogical purpose, of course, is showing good-faith skeptics that one has “dotted the i” and provided an airtight categorical proof of, for instance, Euclid’s proposition I-2. For instance, in a single step of the above proof, say #5, we 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.

So, making the implicit explicit can be illuminating, especially for those who do not see the connections in their full detail or who need to ingrain them in their thinking. But it is work, and if you want to see an old man sweat, well, you came to the right place!

And no, we’re not done yet. Now we have to address the question of whether the propositions themselves were in categorical form: All S are P, No S are P, Some S are P, Some S are not P. These are the officially authorized forms for categorical syllogisms.

Take, for instance, the conclusion of the first syllogism in the above proof: “BC and BG are equal.” Now, what does this mean? Is it like saying: “Plato and Aristotle are men”? This is a condensed way of saying: “Plato is a man,” and: “Aristotle is a man.” If so, then it is saying: “BC is equal,” and “BG is equal.” But this is nonsense.

Surely it is instead saying something more like: “Plato and Aristotle are friends,” which more clearly expressed is: “Plato and Aristotle are men who are friends to one another.” And indeed, when we restate the conclusion of the first syllogism as: “BC and BG are line segments that are equal in length to one another,” we see that this is precisely what it means.

What we learn from this is that it is vitally important to say exactly what we mean—or at least to know exactly what we mean, and to be able to spell it out more explicitly, if doubt arises. But there should be no controversy about the ability of Aristotelian logic to deal with binary relations like “are friends of one another” or “are line segments equal in length to one another,” and there should be no serious objection to abbreviating these expressions as “are friends” or “are equal.”

A further quibble involves the fact that, for instance, line segments BC and BG are individuals (individual line segments), not categories—and the statement about them is like saying: “Bill and Mary are equal,” rather than: “Men and women are equal.”

This is true enough, but it does not rule out such propositions for use in categorical syllogisms. If it did, then the Socrates syllogism would be out the window. However, traditional practice for centuries has been to reword “Socrates is mortal” as: “All Socrates is mortal,” treating Socrates as belonging to a single-member class, and there is good justification for it, justification that goes deeper epistemologically than the logical formalisms of “All S are P,” etc.

What we are saying in “Socrates is mortal,” or more precisely: “Socrates is a mortal being,” is that what is referred to by “Socrates” is the same thing in reality as one of the things referred to by “a mortal being.” This is exactly the same cognitive process with the same epistemic justification as in saying “All men are mortal (beings),” which is to say that the things referred to by “All men” are the same things in reality as some of the things referred to by “mortal beings.”

This fits in very well with Ayn Rand’s unit-perspective model, viewing a single-member group as being like a “mental file folder” (Rand’s analogy) containing one item along with a lot of data about it, similar to a “dossier.” The same is true for two-member groups, such as the one containing line segments BC and BG in the Euclid I-2 proof. (Whether or not we want to label these single- or double-member groups as “concepts,” they function in the same way.)

So, expressing the first syllogism in this way, and spelling out the relationship more clearly, will be a bit less pretty than simply saying: “BC and BG are equal,” but here it is: “All BC and BG [of which there are two in the present construction] are line segments that are equal in length to one another.” And fleshing out the entire syllogism leading to that conclusion looks like this:

All line segments BC and BG [of which there are two 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 [of which there are two in the present construction] are line segments that are equal in length to one another.

This is a straightforward “Barbara” syllogism, as are the other four I used above in my categoricals-only proof of Euclid I-2. All categoricals, all the time.

Quid erat gohumpastumpum.

REB

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...Subject-predicate or as Sommers puts it term-functor-logic is keyed to reasoning in natural language mode. Its main defect is the inability to deal with relationals. In categorical logic of the traditional scholastic form one cannot make this argument: a horse is an animal. sven is the owner of a horse therefore sven is the owner of an animal. That is because the assertion x is an R of y (R a relation) does not have a categorical subject-predicate structure with the usual quantity-quality designators...

There is no such defect in subject-predicate logic. If there were, traditional logic couldn't even do "All men are mortal. Socrates is a man. Therefore all men are mortal."

Consider some typical "relational" propositions...

In "Sven is the owner of an animal"--which I would express less ambiguously as "Sven is an owner of an animal"--what is designated by "Sven" is also designated by "an owner of an animal." Sven is the same thing in reality as one of the members of the class "owner of an animal."

In "Chicago is south of Milwaukee"--which I would express as "Chicago is a city that is south of Milwaukee"--what is designated by "Chicago" is also designated by "a city south of Milwaukee." Chicago is the same thing in reality as one of the members of the class "city south of Milwaukee."

In "Socrates is the teacher of Plato," what is designated by "Socrates" is also designated by "the teacher of Plato." Socrates is the same thing in reality as the teacher of Plato. (For the purpose of the example, we are considering "the teacher of Plato" to be a single-member class. No doubt, he had other teachers, but Socrates was perhaps Plato's only philosophy teacher.)

Doing standard traditional logic with such propositions is completely unproblemmatic.

And actually, ~all~ categorical subject-predicate propositions are also "relational" propositions," and all relational propositions can (and should) be rewritten as categorical subject-predicate propositions, where the predicate's class is made explicit.

In "Socrates is a man", what is designated by "Socrates" is also designated by "man." Socrates is the same thing in reality as one of the members of the class "man."

The key, though, is to bear in mind that all of the various ~other~ relationships (including being-owner-of, being-a-city-south-of, being-the-teacher-of, being-a-member-of-the-class-of) reduce to the fundamental Identity-relationship of being-the-same-thing-in-reality-as.

"Ayn Rand is the author of Atlas Shrugged"--relational? Sure. Being-the-author-of-Atlas-Shrugged is certainly a relation, and there is a person who stands in that relation. But that relation, like all relations, defines a class of things--in this case, the single-member class "author of Atlas Shrugged," because one and only one person stands in the relation that defines that class.

So, "Ayn Rand is the author of Atlas Shrugged" most fundamentally should be understood as saying: "Ayn Rand IS THE SAME THING IN REALITY AS the author of Atlas Shrugged." All of propositional logic can be dealt with in this way.

"Is the same thing in reality" is what "is" means, when you strip away all the ambiguities. (Are you listening, Bill Clinton?)

It even works when you state the foregoing as a subject-predicate categorical proposition: "Is the same thing in reality" is the same thing in reality as the meaning of "is." Or, compressing the relationship to a simple copula: "Is the same thing in reality" is the meaning of "is." (But I repeat myself....)

To me, this insight is the bridge between what is valid of traditional "what" logic and modern "relational" logic.

"Chicago is (south of Milwaukee)" can be converted to "Chicago (is south of) Milwaukee" and back again, with no loss of meaning or perspective or understanding. So can "Chicago is (the same thing in reality as one of the cities south of Milwaukee) and "Chicago (is the same thing in reality as) one of the cities south of Milwaukee."

No, we don't have to use the stilted, clunky, verbose formulations involving "is the same thing in reality as," but it's very important to bear in mind, and to be able to apply it when necessary, in cases of confusion, ambiguity, and the like.

Also, using propositions to express the identity or "what" of something is a more fundamental cognitive function than using propositions to embody and display more specific existential relations. To that extent, modern logic is formulated in terms of non-essentials and is thus more prone to error in deal with fallacies, paradoxes, immediate inference, etc. that involve the issue of truth-value in relation to existential import.

REB

Prove the Pythagorean theorem using only the categorical logic os syllogisms.

That should be an amusing exercise for you.

See Euclid Book I and redo Book I using only the 19 valid categorical syllogisms that Aristotle and his successor at the Lyceum identified. Aristotle did not do syllogisms in the 4 th figure. 4 more valid forms were added to the 15 that Aristotle identified.

It is time to put up or be somewhat more still.

Ba'al Chatzaf

I'm not going to do all of Euclid Book I at this time, nor even Book I, Proposition 17. It's not necessary to support my case, and it would be nice to see some other old man sweat a bit, to demonstrate that he can take it as well as dish it out. The following categoricals-only proof of Book I, Proposition 2 should be sufficient. (Take out your pencils, paper, straight-edges, and compasses, if you want to follow along.)

First, let me say that when I took high school geometry, I don’t recall using any logical processes other than straightforward deduction from axioms, definitions, and previously established results. What few conditional statements the proofs may have contained could easily be rephrased as categoricals, and that is how I will proceed here.

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 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.

Now, the assignment is to prove that the construction has been successful and, therefore, that segment AL = segment BC. Here is the proof:

  1. BC = BG (all radii of the same circle are equal)
  2. DL = DG (all radii of the same circle are equal)
  3. DA = DB (Proposition 1)
  4. AL = BG (from 2 and 3: all equal segments diminished by the same amount are equal)
  5. AL = BC (from 1 and 4: all segments equal to the same segment are equal)

As you can see, despite Euclid’s requirement of constructing 4 circles and 2 (or 3) line segments, the process for proving his proposition is relatively simple, requiring only two deductions. So, I did not have to stand on my head or get out a “buggy whip” in order to coerce my brain into laying out the proof.

Also, the deductions are both categorical syllogisms, using categorical propositions, and the premises (in parenthesis) are all categorical as well. (Radii of the same circle are equal. Equal segments diminished by the same amount are equal. Segments equal to the same segment are equal.) Basic Aristotelian deductive logic applies nicely to Euclidean geometry.

And now for the qualifications and objections…

First of all, Aristotle would be quick to admit that more is needed in order to establish Euclid's propositions than deductive logic and categorical propositions and syllogisms. 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.

Nowadays, we call this “reverse engineering.” How do I get this result? What if I did this? What result would I get? Once we find the right logical pathway, we can cast the result in categorical terms, rather than hypothetical or conditional. But this is not difficult, and it is not new, and it is not controversial.

Secondly, though, where are the categorical syllogisms? The five-step proof given above was highly compressed, with some of the premises in parenthesis, some of the conclusions also serving as premises, etc. In total, there were five syllogisms, all in the “Barbara” form:

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 need repeat all these steps explicitly. Nor should we, except for pedagogical purposes, such as training young math students to “show their work.”

But once the method of proof has been mastered, mental shortcuts and notational economies and abbreviations are normally and reasonably permitted, and this does not somehow make the arguments non-categorical or otherwise fishy. If we hadn’t been granted such boons, high school geometry would surely have taken four years instead of just one!

Another pedagogical purpose, of course, is showing good-faith skeptics that one has “dotted the i” and provided an airtight categorical proof of, for instance, Euclid’s proposition I-2. For instance, in a single step of the above proof, say #5, we 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.

So, making the implicit explicit can be illuminating, especially for those who do not see the connections in their full detail or who need to ingrain them in their thinking. But it is work, and if you want to see an old man sweat, well, you came to the right place!

And no, we’re not done yet. Now we have to address the question of whether the propositions themselves were in categorical form: All S are P, No S are P, Some S are P, Some S are not P. These are the officially authorized forms for categorical syllogisms.

Take, for instance, the conclusion of the first syllogism in the above proof: “BC and BG are equal.” Now, what does this mean? Is it like saying: “Plato and Aristotle are men”? This is a condensed way of saying: “Plato is a man,” and: “Aristotle is a man.” If so, then it is saying: “BC is equal,” and “BG is equal.” But this is nonsense.

Surely it is instead saying something more like: “Plato and Aristotle are friends,” which more clearly expressed is: “Plato and Aristotle are men who are friends to one another.” And indeed, when we restate the conclusion of the first syllogism as: “BC and BG are line segments that are equal in length to one another,” we see that this is precisely what it means.

What we learn from this is that it is vitally important to say exactly what we mean—or at least to know exactly what we mean, and to be able to spell it out more explicitly, if doubt arises. But there should be no controversy about the ability of Aristotelian logic to deal with binary relations like “are friends of one another” or “are line segments equal in length to one another,” and there should be no serious objection to abbreviating these expressions as “are friends” or “are equal.”

A further quibble involves the fact that, for instance, line segments BC and BG are individuals (individual line segments), not categories—and the statement about them is like saying: “Bill and Mary are equal,” rather than: “Men and women are equal.”

This is true enough, but it does not rule out such propositions for use in categorical syllogisms. If it did, then the Socrates syllogism would be out the window. However, traditional practice for centuries has been to reword “Socrates is mortal” as: “All Socrates is mortal,” treating Socrates as belonging to a single-member class, and there is good justification for it, justification that goes deeper epistemologically than the logical formalisms of “All S are P,” etc.

What we are saying in “Socrates is mortal,” or more precisely: “Socrates is a mortal being,” is that what is referred to by “Socrates” is the same thing in reality as one of the things referred to by “a mortal being.” This is exactly the same cognitive process with the same epistemic justification as in saying “All men are mortal (beings),” which is to say that the things referred to by “All men” are the same things in reality as some of the things referred to by “mortal beings.”

This fits in very well with Ayn Rand’s unit-perspective model, viewing a single-member group as being like a “mental file folder” (Rand’s analogy) containing one item along with a lot of data about it, similar to a “dossier.” The same is true for two-member groups, such as the one containing line segments BC and BG in the Euclid I-2 proof. (Whether or not we want to label these single- or double-member groups as “concepts,” they function in the same way.)

So, expressing the first syllogism in this way, and spelling out the relationship more clearly, will be a bit less pretty than simply saying: “BC and BG are equal,” but here it is: “All BC and BG [of which there are two in the present construction] are line segments that are equal in length to one another.” And fleshing out the entire syllogism leading to that conclusion looks like this:

All line segments BC and BG [of which there are two 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 [of which there are two in the present construction] are line segments that are equal in length to one another.

This is a straightforward “Barbara” syllogism, as are the other four I used above in my categoricals-only proof of Euclid I-2. All categoricals, all the time.

Quid erat gohumpastumpum.

REB

Equality is a binary predicate. The Aristotelian categorical statements do not include binary predicates. You also begged the question on the transitivity of equality and symmetry. If A = B and C = B then A = C requires symmetry and transitivity. That is handled by Euclid in his set of Common Notions. Symmetry and Transitivity are assumed.

Here is something you cannot prove using the categorical syllogism. All horses are mammals. Therefore, The tail of a horse is the tail of a mammal.

This does not fit the immediate categorical inferences studied by Aristotle in his Organon (that is the set of books: Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, On Sophistical Refutations).

Try it.

It is unfortunate the most of the writings of Cryssipus did not survive to this day. He wrote 300 books on logic, but the only trace we have of him today are in the writings of his opponents and a few original fragments. See Kneale and Kneale "The Development of Logic". He is the Stoic philosopher who formulated -conditional logic- which is an alternative logic to categorical logic, the logic of terms. It essentially goes like this: if the first, then the second. The first is true hence the second is true. We know it by its latin name modus ponens. Conditional logic bears a similar relation to categorical term logic as Mac does to Windows.

Conditional logic is the logic used by mathematicians and is capable of handling relations of any order or "arity". Binary, Ternary, Quadrenary ... relations. Aristotle based is development on categorical sentence forms. There are only four: All S is P. No S is P. Some S is P and some S is not P.

That is a straight jacket for mathematical development. One could never develop three and found dimensional geometry using just the logic of terms. That is why Fred Sommers had to do major plastic surgery on term logic to break it out of the straight jacket of monadic predicates. The classical form that Aristotle and the later Medieval Scholars developed was unequal to the task.

If you go back to the writings classical times and read the Codices of Archimedes the great mathematician of ancient times, second only to Eudoxus, you will not find a single categorical syllogism. Not one. Mathematicians have used conditional logic for 2300 years. The most famous compendium of this approach to mathematical proof is still "The Elements of Euclid", which has been in continuous use for 2300 years. That record is exceeded only by the Bible. Mathematical thinking and Aristotelian logic parted company after the establishment of the Stoic School, the Stoa Perkile around 300 b.c.e.

Ba'al Chatzaf

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Equality is a binary predicate. The Aristotelian categorical statements do not include binary predicates. You also begged the question on the transitivity of equality and symmetry. If A = B and C = B then A = C requires symmetry and transitivity. That is handled by Euclid in his set of Common Notions. Symmetry and Transitivity are assumed.

It is not that difficult to incorporate binary predicates into Aristotelian categorical statements. All you do is embed the relationship in the category expressed in the predicate.

unary: John is a man.

binary: John and Mary are siblings (of one another).

All this requires is that we form the category "people who are siblings of one another." John and Mary are members of this category. "John and Mary are two [members of the group of] people who are siblings of one another." "John and Mary are siblings (of one another).

Surely you're not excluding the possibility of this being a legitimate predicate, are you?

I'm not an expert on ancient Greek grammar, but it seems to me that Aristotle would not have found this controversial. Nor would the Scholastics.

The same would be true for the category "line segments that are equal in length to one another."

Some Objectivists look askance at such (to them) ad hoc categories. They're comfortable with concepts like "people" but not "people who are siblings of one another." But as Rand said, we form concepts (and categories) not willy-nilly, but by necessity.

As for transitivity of equality and symmetry, I didn't "beg" anything. I gratefully accepted what was given to me by Euclid. When you asked me to prove Book I, Proposition 2, you did not say that I also had to prove every assumption that Euclid stated at the outset and used in his proof. It's a double standard to deny me the same assumptions you would use without question. Also, it has nothing to do with conditions vs. categoricals.

What I ~did~ do was take any proposition he or you or someone might be more comfortable phrasing as an "if...then" or "since..." and rephrase it as a categorical. "If two quantities are both equal to the same third quantity, then they are equal to one another"-->"[All] two quantities that are both equal to the same third quantity are equal to one another."

REB

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Here is something you cannot prove using the categorical syllogism. All horses are mammals. Therefore, The tail of a horse is the tail of a mammal.

This does not fit the immediate categorical inferences studied by Aristotle in his Organon (that is the set of books: Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, On Sophistical Refutations).Try it.

Syllogisms work, but not with the premise you offer. In fact, your premise is reached by induction and then deduction, just like the conclusion you ask me to prove. The syllogisms are quite parallel...

1. A thing is the same thing in reality as that thing considered as a member of a wider category to which it belongs. A horse is a thing. Therefore, a horse is the same thing in reality as that horse considered as a mammal (i.e., as a member of the wider category of mammals, to which it belongs). I.e., a horse is a mammal.

2. An attribute of a thing is the same thing in reality as that attribute of that thing considered as a member of a wider category to which it belongs. The tail of a horse is an attribute of a thing. Therefore, the tail of a horse is the same thing in reality as that tail of that horse considered as a mammal (i.e., as a member of the wider category of mammals, to which it belongs). I.e., the tail of a horse is the tail of a mammal.

REB

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Have you ever wondered why none of the mathematicians (either classical or modern) use the logic of categorical syllogisms? Not one. They all use the conditional logic developed by the Stoic logicians. If... then... logic.

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Have you ever wondered why none of the mathematicians (either classical or modern) use the logic of categorical syllogisms? Not one. They all use the conditional logic developed by the Stoic logicians. If...then...logic.

Not at all. Conditional propositions and conditional syllogisms are the language of exploration, of ~looking for connections~. I would be surprised if mathematicians did ~not~ use conditionals in their thought processes and speculations. Conditionals are the "training wheels" of conceptual thought, the scaffolding of trial-and-error.

Once the connections are found and validated, however, it is our ~option~ to leave it in conditional form or to put it in categorical form.

Leaving an argument in conditional form, when there is really no unfulfilled premise (like "if it rains tomorrow"), shows the points where the speculative process took place during the construction of the proof. That's certainly useful sometimes for pedagogical purposes.

But it really sounds odd to say: "If x is a horse, then x is a mammal," rather than: "All horses are mammals," or more simply: "Horses are mammals." When I hear the former locution, I want to ask: "You mean you don't already know that horses are mammals? Did you skip induction class?"

Similarly, it sounds odd to say: "If x is a circle, then its radii are equal in length," rather than: "The radii of a circle are equal in length." Once you know a fact, you know it, and you don't have to keep standing on your head, or relying on your conditional woobie, when you state it.

Euclid does use appear to use conditional talk here and there. He says things like: "Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to triangle FBC." But this is equivalent to saying: "AB is equal to FB and BD is equal to BC. Therefore triangle ABD must be congruent to triangle FBC." If "since" is conditional, then so is "therefore."

Either way of speaking is just a way of making explicit the causal relationship between premises and conclusion, i.e., the fact that the premises are the reason for the conclusion. "Because (by reason of the fact that) AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to triangle FBC."

Just because this is the fully spelled out meaning of a categorical syllogism does not somehow make it non-categorical. It just means that when it is appropriate, we can strip away all the extraneous conditional or causal verbiage and lay out the bare structure of the argument, understanding that "x and y, therefore z."

I personally don't care how mathematicians find it most felicitous to express the steps of their reasoning, but when the rubber meets the road, their conclusions are in the form "S is P," not "If S, then P." And if they are sure enough about their conclusions to express them as categoricals, what's wrong with doing the same with their premises? (Again, we're not talking about hypothetical arguments such as: If he went to the party, he got drunk. He went to the party. Therefore he got drunk.)

If x is a man, then x is mortal.

Socrates is a man.

Therefore, Socrates is mortal.

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

I understand why modern logicians prefer the former way of stating the first premise. It's the old Existential Import bugaboo. Perish forbid we should "imply" that something exists when it doesn't really exist, as in: "All sea monsters are reptiles." Much better to play it safe and say: "If x is a sea monster, then x is a reptile."

I've already shown why propositions about non-existent things go awry. It's because the mode of existence predicated of the subject term is not spelled out. The corrective is to state the full, explicit meaning of the proposition: "All sea monsters are real creatures that are reptiles." Yes, this is claiming (not implying) that sea monsters really exist. But all we have to do is simply declare it false, not wring our hands over its committing a fallacy.

Logicians had the option of manning up and doing this for all universal propositions, and instead they wimped out with their "if x, then y" evasions. They were like the teacher who punishes the whole class for the difficulties caused by one student. It is so easy to finesse these situations with minor, common sense modifications of traditional logic. Instead, the moderns tore apart Aristotle's Square of Opposition and immediate inference and left us with hypothetical universal propositions.

No thank you.

REB

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You see almost no sign of categorical logic (as defined by Aristotle and his successors through until the Renaissance) in mathematical publications. Why do you suppose that is? I will tell. Categorical Logic a la Aristotle is a straight jacket on logical thinking. Conditional logic is much more natural, much more flexible and much easier to subject to metamathematical validation. If we were stuck with categorical syllogisms only the likes of the Incompleteness Theorems would never have been discovered. If we were stuck only with categorical logic, we would not even have Euclid's Elements.

I still leave to you as an exercise. Prove the Pythagorean theorem using ONLY categorical syllogisms. I bet you can't.

Ba'al Chatzaf

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You see almost no sign of categorical logic (as defined by Aristotle and his successors through until the Renaissance) in mathematical publications. Why do you suppose that is? I will tell. Categorical Logic a la Aristotle is a straight jacket on logical thinking. Conditional logic is much more natural, much more flexible and much easier to subject to metamathematical validation. If we were stuck with categorical syllogisms only the likes of the Incompleteness Theorems would never have been discovered. If we were stuck only with categorical logic, we would not even have Euclid's Elements.

I still leave to you as an exercise. Prove the Pythagorean theorem using ONLY categorical syllogisms. I bet you can't.

Ba'al Chatzaf

Why should I go to the effort? After several years of discussion on this, you still haven't even conceded that I have proved Euclid Book 1, Proposition 2. You have uttered no "congrats, didn't think you could do it," just a steady string of "yes-buts." I've met all your challenges, and you have hopped around like a flea on a dog to the next skeptical barrage of flak. I honestly don't think you're discussing in good faith.

As for conditionals, I will use them in working through a thought process just like the next guy. But when I summarize my proof and state my premises and conclusions, I want them stated firmly and unconditionally. Einstein didn't say "If something is e, then golly gee, it must be mc2.

REB

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You see almost no sign of categorical logic (as defined by Aristotle and his successors through until the Renaissance) in mathematical publications. Why do you suppose that is? I will tell. Categorical Logic a la Aristotle is a straight jacket on logical thinking. Conditional logic is much more natural, much more flexible and much easier to subject to metamathematical validation. If we were stuck with categorical syllogisms only the likes of the Incompleteness Theorems would never have been discovered. If we were stuck only with categorical logic, we would not even have Euclid's Elements.

I still leave to you as an exercise. Prove the Pythagorean theorem using ONLY categorical syllogisms. I bet you can't.

Ba'al Chatzaf

Why should I go to the effort? After several years of discussion on this, you still haven't even conceded that I have proved Euclid Book 1, Proposition 2. You have uttered no "congrats, didn't think you could do it," just a steady string of "yes-buts." I've met all your challenges, and you have hopped around like a flea on a dog to the next skeptical barrage of flak. I honestly don't think you're discussing in good faith.

As for conditionals, I will use them in working through a thought process just like the next guy. But when I summarize my proof and state my premises and conclusions, I want them stated firmly and unconditionally. Einstein didn't say "If something is e, then golly gee, it must be mc2.

REB

Your "proof" is nonsense.

1. It since the postulates are not in categorical form, one cannot infer a conclusion from them by categorical syllogistic logic.

2. you did not even invoke the postulates -as written- by Euclid.

You get a D - . l

Ba'al Chatzaf

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  • 1 year later...

There is a careful and informative online paper pertaining to the discussion in this thread and in another.* Its title is “Why is the Angle Inscribed in the Semicircle a Right Angle? An Examination of the Aristotelian Logic of the Middle Term” (2009). The author is John Brungardt. The work referred to in his opening quotation from Leibniz, the work that attempted to put the first six books of Euclid’s Elements into syllogistic form is Analyseis Geometricae Sex Librorum Euclidis (1566) by Christian Herlinus and Konrad Dasypodius.

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There is a careful and informative online paper pertaining to the discussion in this thread and in another.* Its title is “Why is the Angle Inscribed in the Semicircle a Right Angle? An Examination of the Aristotelian Logic of the Middle Term” (2009). The author is John Brungardt. The work referred to in his opening quotation from Leibniz, the work that attempted to put the first six books of Euclid’s Elements into syllogistic form is Analyseis Geometricae Sex Librorum Euclidis (1566) by Christian Herlinus and Konrad Dasypodius.

Did he succeed in transcribing Euclid's conditional logic proofs to proofs using the 19 valid syllogisms?

Here is the key paragraph in the paper you referred to:

"Why are mathematical theorems as expressed in symbols much less cumbersome than
their corresponding syllogisms? Alexander of Aphrodisias comments, “Syllogisms do not exist in
the words they contain but in what the words mean.”56 When the mind sees the relationships in
mathematical objects, it grasps the ratio, logos, or account of those objects. This is because the
intellect as a power is proportioned to its object, namely the forms of things (even mathematical
things). It expresses these objects through definitions, whether of subjects or predicates. These
mathematical forms and the order of causation among them are ‘speakable’, for the
apprehension of the intellect is a principle of speech. What symbolic expression hides, the
syllogism reveals: the mind’s discursive act as it grasps the cause as cause and as necessary of why
this attribute belongs to this subject."
-------------------------------------------------------------
Syllogistic proofs are clumsy to the n-th degree. The symbolic conditional logic is constructed in such a way that the formalism does most of the "thinking" for the mathematician constructing the proof. If mathematicians had not adapted formal conditional logic (in the guise of the quantified predicate calculus) we would not have calculus today. the theory of real and complex variable based on point set topology simply would not exist.
The bottom line is the Aristotle's syllogistic logic cannot handle multi-termed relations in a general way.
Binary Relations taken singly have to be beaten into submission so they appear as one term predicates
Ternary Relations? Forget about it.
Now about the theorem you references talks about: It is called Thales Theorem in the literature. Here is a reference to three honest to god real mathematical proofs of the theorem:
Every one of them is short, everyone of them is clear. every one of them is correct and none of them, not one, is burdened down with philosophical baggage.
Another difficulty with the "Philosophical Approach" is that it does not generalize or extend readily to manifolds other than planes (and even that requires that that one little theorem be beaten bloody by the philosophers). In a word, we never would have gotten to non-euclidean geometry. That means no Riemann Geometry, no General Theory of Relativity and eventually no GPS.

Ba'al Chatzaf

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Bob, I gather that philosophers have been interested not only in understanding the relation of Euclid’s proofs to syllogistic proof, but to proof using modern relational logic, such as in De Morgan, Peirce, and Russell. The motive seems to be in part the same as with Alexander of Aphrodisias, though his logical vista for logical inference was limited to the syllogistic. That sort of interest of philosophers, including logicians—that sort of interest as on show in the Russell-Whitehead tome—is philosophy about mathematics and logic. Philosophy of mathematics is not mathematics, at least not the sort of mathematics we hold dear. That said, however ponderous and boring such projects from the (highly technical) philosophers, one should not conclude the issues they pursue are worthless simply because what they are doing is not the project that is mathematics itself.

A similar story goes for the von Neumann tome on probability and quantum mechanics. That he contributes nothing new or exciting to QM is only to say he was pursuing theory about QM, not doing QM.

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A similar story goes for the von Neumann tome on probability and quantum mechanics. That he contributes nothing new or exciting to QM is only to say he was pursuing theory about QM, not doing QM.

On top of which von Neuman made a blunder which was pointed out by J.S. Bell.

"Ironically, the theory of hidden variables was not seriously considered as an alternative to the Copenhagen Interpretation because of an error by mathematician John von Neumann; now referred to as von Neumann's silly mistake. Von Neumann's work, published in 1932, seemed to give proof that no hidden variable theory could ever describe quantum mechanics. Although von Neumann's mistake was revealed in 1935 by Grete Hermann, the mistake was ignored. It was not until 1966 that von Neumann's mistake was seen and taken seriously. John Bell rediscovered the mistake and finally proved that hidden variables could describe quantum events if non-locality was included. Non-locality was proved to rule the quantum world through the experiments done by Alain Aspect which tested the EPR thought experiment. The profound discovery of Aspect is that any interpretation of quantum reality must include non-locality. "If you want to believe there is a real world out there, you cannot do without non-locality; if you want to believe that no form of communication takes place faster than the speed of light, you cannot have a real world, independent of the observer [Gribben, p.159, 1995]."

and see also:

http://books.google.com/books?id=j3OnfgmkCn4C&pg=PA67&lpg=PA67&dq=von+Neumann%27s+blunder&source=bl&ots=ir3_zKT57R&sig=ZLGddTqyNSAxRhNomm-Xp_1ic5g&hl=en&sa=X&ei=AU9eVLOvGoS1yATiuILgAg&ved=0CDcQ6AEwBQ#v=onepage&q=von%20Neumann's%20blunder&f=false

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