In other words, once you remove logic from reality, there can still be a contradiction in the rules?

And Aristotle should be ashamed because he didn't know the new rules because he was dead centuries before someone came up with them?

Where the hell did reality go?

I've used the following example from God's Little Acre by Erskine Caldwell before, but it is such a great metaphor for this case, it bears repeating.

On 8/2/2010 at 8:47 PM, Michael Stuart Kelly said:

In Georgia, Ty Ty had been digging up his entire farm for years looking for gold that never appeared. But he kept one piece of land marked off in his mind as sacred. He refused to dig on it since he said he gave it to God out of gratitude. He called it "God's little acre." Eventually as the story went on, Ty Ty had dug up everywhere. Then someone told him that for sure the gold was in God's little acre. And after all these years they deserved to dig it up, but if they did that, the preacher would get the gold since God was not in need of it. Here is Ty Ty's response about moving the digging operation for the umpteenth time. Pluto is a fat country-bumpkin sheriff.

Erskine Caldwell said:

"All right, boys," Ty Ty agreed. "I'll move it again, but I ain't aiming to do away with God's little acre altogether. It's His and I can't take it away from Him after twenty-seven years. That wouldn't be right. But there ain't nothing wrong with shifting it a little, if need be. It would be a heathen shame to strike the lode on it, to be sure, the first thing, and I reckon I'd better shift it so we won't be bothered."

"Why don't you put it over here where the house and barn are, Pa?" Griselda suggested. "There's nothing under this house, and you can't be digging under it, anyway."

"I'd never thought of doing that, Griselda," Ty Ty said, "but it sure sounds fine to me. I reckon I'll shift it over here. Now, I'm pretty much glad to get that off my mind."

Pluto turned his head and looked at Ty Ty.

"You haven't shifted it already, have you Ty Ty?" he asked.

"Shifted it already" Why sure. This is God's little acre we're sitting on right now. I moved it from over yonder to right here."

"You're the quickest man of action I've ever heard about," Pluto said, shaking his head. "And that's a fact."

Michael. Apparently you did not grasp the error built into the classical Square of Opposition. The truthful assertion all unicorns have one horn in the middle of their foreheads does NOT imply unicorns exists. I provided a proof for the fact All S is P does NOT imply there exists some S. This is an error the medieval logicians committed building on Aristotle's constricted view of logic. There is more to logic than categorical propositions as ALL logicians since Boole and Frege have shown. Aristotle came up with version 1.0 of logic. Logic was freed of its verbal chains and allowed to expand its scope since the middle of the 19 th century.

We no longer think that the earth is the center of the solar system and we no long believe that logic is only about 4 types of categorical assertions.

The truthful assertion all unicorns have one horn in the middle of their foreheads does NOT imply unicorns exists. I provided a proof for the fact All S is P does NOT imply there exists some S.

Bob,

Of course unicorns exist--in our heads. The little word game does not permit a unicorn to suddenly become a giraffe with one horn in the middle of its forehead. So even as an imaginary creature, a unicorn does have identity. Therefore it exists, even though it exists only in the imagination. It is an existent with a specific nature.

You can either tie logic to direct observation (including observation of the imagination) and axiomatic concepts like existence, identity and consciousness, or you can keep moving Aristotle's little acre in your head among abstract symbols and pretending it connects to reality solely within the mental game where an abstraction refers to something except when it doesn't.

So long as this process is divorced from observation outside of logic, all it does is provide a new rule for the abstraction game, but not a new rule that connects the game to reality--and not to Aristotle, for that matter. He observed. And observed and observed and observed...

Of course unicorns exist--in our heads. The little word game does not permit a unicorn to suddenly become a giraffe with one horn in the middle of its forehead. So even as an imaginary creature, a unicorn does have identity. Therefore it exists, even though it exists only in the imagination. It is an existent with a specific nature.

You can either tie logic to direct observation (including observation of the imagination) and axiomatic concepts like existence, identity and consciousness, or you can keep moving Aristotle's little acre in your head among abstract symbols and pretending it connects to reality solely within the mental game where an abstraction refers to something except when it doesn't.

So long as this process is divorced from observation outside of logic, all it does is provide a new rule for the abstraction game, but not a new rule that connects the game to reality--and not to Aristotle, for that matter. He observed. And observed and observed and observed...

Michael

All round-squares are geometrical objects. Does this imply round-squares exist???? No. The inference All S is P implies Some S is P is just plain wrong. It is not a valid inference. And there are empty sets. For example: The set of all living Americans born prior to 1800. Not a single one. Classical logic was not only constricted to categorical propositions, it was unable to deal with empty sets which are indeed sets. Without empty sets we could not deal with set differences thus without empty sets we could not have an algebra of sets.

Aristotle founded logic 1.0. We are now up to logic 8.0.

Round-squares do not exist, so nothing can be predicated of them.

The quoted proposition isn't the same as the unicorn example. It is an equivocation fallacy, to equivocate a mental existant (the unicorn) with what is presented as an existant--these are different things.

In the Irwin translation, Aristotle said in Categories that to predicate, it had to be predicated of something (said of something), this something is the primary substance. I would use MSK's post previous to this one, if I were to type it out myself. I would use it to conclude here.

Round-squares do not exist, so nothing can be predicated of them.

The quoted proposition isn't the same as the unicorn example. It is an equivocation fallacy, to equivocate a mental existant (the unicorn) with what is presented as an existant--these are different things.

In the Irwin translation, Aristotle said in Categories that to predicate, it had to be predicated of something (said of something), this something is the primary substance. I would use MSK's post previous to this one, if I were to type it out myself. I would use it to conclude here.

4

All round squares are arbitrary constructs. They exist in the mind as arbitrary constructs, as products of the imagination. The imagination can construct all kinds of contradictory things that cannot exist in mind-independent reality.

A: All round squares are real geometrical objects. False.

I: Some round squares are real geometrical objects. False.

E: No round squares are real geometrical objects. True.

O: Some round squares are not real geometrical objects. True.

A: All round squares are arbitrary constructs. True.

I: Some round squares are arbitrary constructs. True.

E: No round squares are arbitrary constructs. False.

O: Some round squares are not arbitrary constructs. False.

This is Logic 2.0B. It follows the dictum of Aquinas that only wholes can be legitimately predicated of wholes, and that once you restrict predication in this way, Aristotle's square of opposition and the laws of immediate inference are preserved intact. You can handle propositions about the existent, the imaginary, and the non-existent with equal ease and validity. Truth values flow quickly and clearly from this approach, like low-hanging fruit. Which may be why most academics and die-hard computer gurus have resisted it for so long. (Just guessing.)

By contrast, Logic 2.0A (Bob's preference) is based on (supposed) Boolean logic as promoted by Russell et al. It is shot through with errors, due to failure to realize that predicating parts of wholes will lead to logical paradoxes and a drastically reduced ability of immediate inference to function properly.

I wrote about this extensively in my December 2014 essay in The Journal of Ayn Rand Studies. I invite you to check it out, if you are interested in delving more deeply than is permitted by the repartee format of OL.

Round-squares do not exist, so nothing can be predicated of them.

The quoted proposition isn't the same as the unicorn example. It is an equivocation fallacy, to equivocate a mental existant (the unicorn) with what is presented as an existant--these are different things.

In the Irwin translation, Aristotle said in Categories that to predicate, it had to be predicated of something (said of something), this something is the primary substance. I would use MSK's post previous to this one, if I were to type it out myself. I would use it to conclude here.

There are no unicorns and I have a notion of what a round square is. Using the taxicab metric all circles come out square.

Existential import using Aristotle rules produces a contradiction as I showed. It is a bogus notion and we are well rid of it.

All round squares are arbitrary constructs. They exist in the mind as arbitrary constructs, as products of the imagination. The imagination can construct all kinds of contradictory things that cannot exist in mind-independent reality.

A: All round squares are real geometrical objects. False.

I: Some round squares are real geometrical objects. False.

E: No round squares are real geometrical objects. True.

O: Some round squares are not real geometrical objects. True.

A: All round squares are arbitrary constructs. True.

I: Some round squares are arbitrary constructs. True.

E: No round squares are arbitrary constructs. False.

O: Some round squares are not arbitrary constructs. False.

This is Logic 2.0B. It follows the dictum of Aquinas that only wholes can be legitimately predicated of wholes, and that once you restrict predication in this way, Aristotle's square of opposition and the laws of immediate inference are preserved intact. You can handle propositions about the existent, the imaginary, and the non-existent with equal ease and validity. Truth values flow quickly and clearly from this approach, like low-hanging fruit. Which may be why most academics and die-hard computer gurus have resisted it for so long. (Just guessing.)

By contrast, Logic 2.0A (Bob's preference) is based on (supposed) Boolean logic as promoted by Russell et al. It is shot through with errors, due to failure to realize that predicating parts of wholes will lead to logical paradoxes and a drastically reduced ability of immediate inference to function properly.

I wrote about this extensively in my December 2014 essay in The Journal of Ayn Rand Studies. I invite you to check it out, if you are interested in delving more deeply than is permitted by the repartee format of OL.

REB

In my own thinking, I separate out an arbitrary from the imaginary, and I consider an arbitrary to be something presented as fact-based, but not fact-based. An imaginary thing isn't presented as fact-based, is meant to be imaginary, and a mental existant. So "All round-squares are geometrical objects" would be an arbitrary because All round-squares (non-real) are geometrical objects (a real category with real objects)--meaning the non-real subject, in my own thinking, would automatically disqualify anything being predicated of it, to the point where a copula isn't possible. An example of an imaginary would be "All unicorns are white", which is fine, it's not asserting it belongs to a category with real existants.

In my own thinking, I wouldn't give "All round-squares are geometrical objects" the status of a proposition, as this proposition could not be arrived at by induction. I would call it invalid, reject it, and wouldn't assign a truth value to it (or its I, E, O forms). The reason this is so is because of the axioms, the statement didn't arrive from them. Would this compare require logic? Perhaps a quick, semi-automatized method of inductive difference--comparing the statement to one's context of knowledge. In my own thinking, this is what I do, and I don't assign the statement the status of possible truth or falsity, I hold it up as real/unreal, without a truth value. I don't allow it to enter the laws of logic, nor do I even consider the statement to have identity. I ask the question, "is this real or unreal", rather than, "what identity does this statement have". Only if a statement passes the sniff test to the context of my knowledge does it begin to be considered.

I am interested in what you've written and would like to read it. These are my thoughts, some a product of reading, some a product of my own thinking, and I'm sure some things that you've heard before.

There are no unicorns and I have a notion of what a round square is. Using the taxicab metric all circles come out square.

Existential import using Aristotle rules produces a contradiction as I showed. It is a bogus notion and we are well rid of it.

I watched the video, and found several errors in it. She is omitting some of the rules of the square of opposition, namely how super and sub implication works. This is from Lionel Ruby, Logic an Introduction, page 252:

If the superimplicant is true, then the subimplicant must be true.

If the superimplicant is false, then the subimplicant may be true or false.

If the subimplicant is true, then the superimplicant may be true or false.

If the subimplicant is false, then the superimplicant must be false.

1:45 - She makes the statement, "some politicians are not honest, so no politicians are honest". This is invalid, when saying "some politicians are not honest" she automatically moves to "no politicians are honest" as a true statement--from "some politicians are not honest" you can't logically say anything about the other politicians, because the rest of them might be dishonest, the rest of them might be honest.

4:35 - She actually demonstrates amphiboly here, but says she's doing something different. Holding two red and one green and staying at the A form, she says, "All blocks are red is false," but actually demonstrates, "All blocks are not red", which is amphiboly. She moves to the E form, as amphiboly can be taken to mean a negation of "All blocks are red", or it can mean "Some blocks are not red", which the latter she obviously should have moved to.

8:35 - She directs her hands that if someone was to know one of the propositions on the square of opposition, that you can move around the square to know the other propositions, but this is invalid, she misses the rules of super and sub implication.

8:53 - She invalidly moves from a sub implication to its super as if it is true, but it cannot be determined (again, she omits some of the rules of the square of opposition).

14 hours ago, BaalChatzaf said:

there are empty sets. For example: The set of all living Americans born prior to 1800. Not a single one. Classical logic was not only constricted to categorical propositions, it was unable to deal with empty sets which are indeed sets.

I consider that statement to be an example of an arbitrary, it's being asserted that there is "the set of all living Americans born prior to 1800", but it is not possible to create that class. "There are zero living Americans born prior to 1800" is a true statement, and classical logic handles this fine.

Aristotle spoke of the "not thing" in one of his refutations to Plato, that we need to be concerned with what something is, not what something not is (I'm paraphrasing heavily here), if we were to keep track of what things aren't, there would be infinite possibilities. I mentioned before about Categories, about the primary substance, that to predicate, it had to be predicated of something (said of something), that this something is the primary substance. What I'm getting at here, is when I studied Aristotle, he is talking about things that exist. In On Interpretation, Aristotle talks about possibility and necessity--about "possible to be", "possible not to be", "not possible to be", "not possible not to be". So why come up with things that are not possible, things that not possible to be, and insist it breaks his logic? Throw out the not possible to be, throw out the arbitrary--his logic is meant to handle things that exist, it was derived from reality where things do exist. Speak about those things and classical logic is fine.

there are empty sets. For example: The set of all living Americans born prior to 1800. Not a single one. Classical logic was not only constricted to categorical propositions, it was unable to deal with empty sets which are indeed sets.

I consider that statement to be an example of an arbitrary, it's being asserted that there is "the set of all living Americans born prior to 1800", but it is not possible to create that class. "There are zero living Americans born prior to 1800" is a true statement, and classical logic handles this fine.

Aristotle spoke of the "not thing" in one of his refutations to Plato, that we need to be concerned with what something is, not what something not is (I'm paraphrasing heavily here), as if we were to keep track of what things aren't, there would be infinite possibilities. I mentioned before about Categories, about the primary substance, that to predicate, it had to be predicated of something (said of something), that this something is the primary substance. What I'm getting at here, is when I studied Aristotle, he is talking about things that exist. In On Interpretation, Aristotle talks about possibility and necessity--about "possible to be", "possible not to be", "not possible to be", "not possible not to be". So why come up with things that are not possible, things that not possible to be, and insist it breaks his logic? Throw out the not possible to be, throw out arbitrary--his logic is meant to handle things that don't exist, it was derived from reality where things do exist. Speak about those things and classical logic is just fine.

I think you can speak about anything whatever, and "classical logic is just fine." You just have to clearly spell out what you are asserting, whenever there is any doubt. Otherwise, the ambiguities leave an opening wide enough for Bertrand Russell to drive a Mack truck through it!

We don't have to "keep track of what things aren't," but that is not an impediment to judging that things which don't exist don't exist! In performing the conceptual inventory of "real geometric figures" in order to assess truth/falsity of "A round-square is a real geometric figure," all you have to do is look in your "real geometric figure" file folder and note that there aren't any round-squares there! You can't see what isn't there, so certainly you're under no obligation to keep track of what doesn't exist!

I like your example. Certainly, we cannot now form the class of actual people denoted by "living Americans born before 1800," but it's equally true that 200 years ago, someone could have! So, we need to build that present/past distinction into any assertion we make about that class - just like we need to build the imaginary/actual or arbitrary/real distinction into some of the other assertions we've been considering.

With that in mind, here's how I'd deal with it. "All living Americans born prior to 1800 are people who are now (in 2016) very old." True or false? (I could have said: "...are actual people who are very old.")

Well, do the Bertie Russell inventory. Look in the class of "people who are now (in 2016) very old." Do you see any living Americans born prior to 1800? Nope. The assertion is false.

Now, negate: "Some living Americans born prior to 1800 ARE NOT people who are now (in 2016) very old." True or false? Are there any living Americans born prior to 1800 in the class of "people who are now (in 2016) very old? Again, no. So, this statement is true, as the negation of a false assertion must be.

You really don't have to decide whether the subject refers to a product of cognition (valid concept) or a product of imagination - and you don't have to decide whether the product of imagination was cobbled together from compatible or incompatible elements.

You just have to be clear on whether you are asserting that the subject refers to something real or imaginary, and then (if necessary for clarity) say so in a fuller expression of the assertion. And then perform the conceptual inventory and on that basis judge whether the "is" or "is not" is correct (and the statement true) or incorrect (and the statement false).

I think you can speak about anything whatever, and "classical logic is just fine." You just have to clearly spell out what you are asserting, whenever there is any doubt. Otherwise, the ambiguities leave an opening wide enough for Bertrand Russell to drive a Mack truck through it!

We don't have to "keep track of what things aren't," but that is not an impediment to judging that things which don't exist don't exist! In performing the conceptual inventory of "real geometric figures" in order to assess truth/falsity of "A round-square is a real geometric figure," all you have to do is look in your "real geometric figure" file folder and note that there aren't any round-squares there! You can't see what isn't there, so certainly you're under no obligation to keep track of what doesn't exist!

I like your example. Certainly, we cannot now form the class of actual people denoted by "living Americans born before 1800," but it's equally true that 200 years ago, someone could have! So, we need to build that present/past distinction into any assertion we make about that class - just like we need to build the imaginary/actual or arbitrary/real distinction into some of the other assertions we've been considering.

With that in mind, here's how I'd deal with it. "All living Americans born prior to 1800 are people who are now (in 2016) very old." True or false? (I could have said: "...are actual people who are very old.")

Well, do the Bertie Russell inventory. Look in the class of "people who are now (in 2016) very old." Do you see any living Americans born prior to 1800? Nope. The assertion is false.

Now, negate: "Some living Americans born prior to 1800 ARE NOT people who are now (in 2016) very old." True or false? Are there any living Americans born prior to 1800 in the class of "people who are now (in 2016) very old? Again, no. So, this statement is true, as the negation of a false assertion must be.

You really don't have to decide whether the subject refers to a product of cognition (valid concept) or a product of imagination - and you don't have to decide whether the product of imagination was cobbled together from compatible or incompatible elements.

You just have to be clear on whether you are asserting that the subject refers to something real or imaginary, and then (if necessary for clarity) say so in a fuller expression of the assertion. And then perform the conceptual inventory and on that basis judge whether the "is" or "is not" is correct (and the statement true) or incorrect (and the statement false).

REB

How about the set of all three headed Americans born before 1800 and still alive. Not a single one. It is the empty set.

Bob, perhaps the most common solution, in Aristotle’s behalf, to the seeming existential import you illustrated in the root post is to “introduce into the assertoric syllogistic a presupposition to the effect that the plurality associated with every term has at least one member” (42). I have no problem in general with extralogical assumptions that constrain the syllogistic (assertoric or modal); it depends on what the extralogical presupposition is. I concur with Rand’s extralogical presupposition that logic depends on the axiom “Existence exists.” Branden’s Vision and Peikoff’s OPAR draw out constraints from that dependency as to the character of what we are doing with the normative discipline of deduction. But I’ll leave what they say to another time and for now to the interested reader to dig out. To the question of the present proposed solution to the problem of existential import, I don’t think the dependence of logic on the axiom “Existence exists” entails the constraint that the plurality associated with every term must have at least one existing member (at some time or other). Leaving the issue of Objectivist (or my own) extralogical presuppositions aside, the drawback of this proposed solution is that the assertoric syllogistic “would not be the universally applicable system of formal logic that it is often thought to be” (43).

I’m quoting from Marko Malink’s Aristotle’s Modal Syllogistic (2013 Harvard).

A second sort of solution allows into the syllogistic terms whose associated plurality has no existing members (at any time). They just cannot serve as terms that are subjects in any universal propositions that are true. Inferring “B belongs to some A” from the false “B belongs to all A” is simply invalid. This strategy has been adopted by Moody and several others (and Roger?), but Malink observes that it is incompatible with Aristotle when he seem to say “B does not belong to some B” cannot be true (APr. 64b7–13, in tandem with 64a4–7and 64a23–30), whereas on this second strategy in behalf of Aristotle, “B does not belong to some A” must be true (since “B belongs to all A” is false) where A has no members. Then it would be true that “B does not belong to some B” where B has no members, in contradiction of Aristotle’s apparent position.

A third sort of solution, for which there is some evidence in APr., is to replace material implication with what is called connexive implication,^{§2} though Malink doubts this view is truly Aristotle’s. Rather than any of those three proposals, in Aristotle’s behalf, Malink’s own is to reject the idea that the plurality associated with a term in the syllogistic is to be identified with the set of individuals falling under the term. He proposes to identify the plurality associated with a term in the syllogistic with the set of items of which the term is predicated universally: {A1, A2, . . . such that the term B satisfies “B belongs to all A1, all A2, . . . .”}. Then there is at least one member in the plurality associated with each term, for it is the case for every B that “B belongs to all B.” (44, 68) I’m still getting a grip on this.

How about the set of all three headed Americans born before 1800 and still alive. Not a single one. It is the empty set.

What fun!

All three-headed Americans born before 1800 are real, presently living people.

You have introduced two criteria that require mention in the predicate: being real (as opposed to imaginary) and being alive (as opposed to dead).

Compare subject and predicate. In the set of "real, presently living people," do you find a three-headed American born before 1800? No. Thus, the things referred to by the subject (three-headed Americans born before 1800) ARE NOT THE SAME as any of the things referred to by the predicate (real, presently living people). The proposition asserts that they ARE THE SAME, and that assertion of identity is therefore incorrect, and thus the proposition is false.

You cannot infer from a false A statement that its corresponding I statement is also false, so we'll leave the truth status of the I statement for now.

Moving on to the O statement: since it's the contradictory of the A, it looks like this: Some three-headed Americans born before 1800 ARE NOT real, presently living people. Bertrand Russell inventory: do you see any three-headed Americans in your file folder (class) of real, presently living people? Nope. Thus, the assertion of NON-IDENTITY (i.e., the denial of identity) between subject-referent and predicate-referent is CORRECT, and the proposition is true. Aristotelian truth-value of negation holds firmly.

OK, what about the E statement? No three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. This is the assertion, and it is correct, so the E statement is true.

Now we can take up the I statement: since it's the contradictory of the E, it looks like this: Some three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. Thus, the things referred to by the subject ARE NOT THE SAME as the things referred to by the predicate. Since the proposition asserts they ARE, that assertion of identity is therefore incorrect, and the I proposition is false.

I've taken on your grotesque, Baroquely adorned challenge and shown it to be easily analyzed according to the Aristotelian truth-value in re negation, both for the A (false) <-->O (true) and the E (true) <-->I (false).

What this shows to me is that formal logic is nowhere near as clear or powerful at dealing with logical inference as is ordinary language in combination with Aristotle's laws of logic and common sense (or, as he would have called it, practical wisdom).

Not only this approach it easier to use, it's more useful when someone is trying to pull the wool over your eyes. Godel's slingshot would never have flown if he had spelled it out in ordinary language. It's built on a gross equivocation(s), covered over with a blizzard of alphabet-soup symbolization.

All three-headed Americans born before 1800 are real, presently living people.

You have introduced two criteria that require mention in the predicate: being real (as opposed to imaginary) and being alive (as opposed to dead).

Compare subject and predicate. In the set of "real, presently living people," do you find a three-headed American born before 1800? No. Thus, the things referred to by the subject (three-headed Americans born before 1800) ARE NOT THE SAME as any of the things referred to by the predicate (real, presently living people). The proposition asserts that they ARE THE SAME, and that assertion of identity is therefore incorrect, and thus the proposition is false.

You cannot infer from a false A statement that its corresponding I statement is also false, so we'll leave the truth status of the I statement for now.

Moving on to the O statement: since it's the contradictory of the A, it looks like this: Some three-headed Americans born before 1800 ARE NOT real, presently living people. Bertrand Russell inventory: do you see any three-headed Americans in your file folder (class) of real, presently living people? Nope. Thus, the assertion of NON-IDENTITY (i.e., the denial of identity) between subject-referent and predicate-referent is CORRECT, and the proposition is true. Aristotelian truth-value of negation holds firmly.

OK, what about the E statement? No three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. This is the assertion, and it is correct, so the E statement is true.

Now we can take up the I statement: since it's the contradictory of the E, it looks like this: Some three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. Thus, the things referred to by the subject ARE NOT THE SAME as the things referred to by the predicate. Since the proposition asserts they ARE, that assertion of identity is therefore incorrect, and the I proposition is false.

I've taken on your grotesque, Baroquely adorned challenge and shown it to be easily analyzed according to the Aristotelian truth-value in re negation, both for the A (false) <-->O (true) and the E (true) <-->I (false).

What this shows to me is that formal logic is nowhere near as clear or powerful at dealing with logical inference as is ordinary language in combination with Aristotle's laws of logic and common sense (or, as he would have called it, practical wisdom).

Not only this approach it easier to use, it's more useful when someone is trying to pull the wool over your eyes. Godel's slingshot would never have flown if he had spelled it out in ordinary language. It's built on a gross equivocation(s), covered over with a blizzard of alphabet-soup symbolization.

REB

Boolean Logic made computers possible. What has cramped, confined, constipated logic of categorical propositions done for science and mathematics? Damned little.

Try this with categorical logic.: all lions are animals. Therefore the tail of a lion is the tail of an animal. It cannot be done with categorical propositions.

Bob, perhaps the most common solution, in Aristotle’s behalf, to the seeming existential import you illustrated in the root post is to “introduce into the assertoric syllogistic a presupposition to the effect that the plurality associated with every term has at least one member” (42). I have no problem in general with extralogical assumptions that constrain the syllogistic (assertoric or modal); it depends on what the extralogical presupposition is. I concur with Rand’s extralogical presupposition that logic depends on the axiom “Existence exists.” Branden’s Vision and Peikoff’s OPAR draw out constraints from that dependency as to the character of what we are doing with the normative discipline of deduction. But I’ll leave what they say to another time and for now to the interested reader to dig out. To the question of the present proposed solution to the problem of existential import, I don’t think the dependence of logic on the axiom “Existence exists” entails the constraint that the plurality associated with every term must have at least one existing member (at some time or other). Leaving the issue of Objectivist (or my own) extralogical presuppositions aside, the drawback of this proposed solution is that the assertoric syllogistic “would not be the universally applicable system of formal logic that it is often thought to be” (43).

I’m quoting from Marko Malink’s Aristotle’s Modal Syllogistic (2013 Harvard).

3

Stephen, if by "at least one existing member," you mean existing in mind-independent reality, then surely this is too restrictive. We can and do form and reason with assertions about products of the imagination as well as arbitrary constructs:

Harry Potter is a creation of J. K. Rowling. All creations of J. K. Rowling are fictional characters. Therefore, Harry Potter is a fictional character.

A round-square is an arbitrary construct. All arbitrary constructs are things to which Leonard Peikoff is opposed. Therefore, a round-square is something to which Leonard Peikoff is opposed.

I think if you loosen the criterion to "at least one member existing mentally and/or extra-mentally" (or something like that), you would have something very close to how I extend/adapt Aristotle's rules of inference.

10 hours ago, Guyau said:

A second sort of solution allows into the syllogistic terms whose associated plurality has no existing members (at any time). They just cannot serve as terms that are subjects in any universal propositions that are true. Inferring “B belongs to some A” from the false “B belongs to all A” is simply invalid. This strategy has been adopted by Moody and several others (and Roger?), but Malink observes that it is incompatible with Aristotle when he seem to say “B does not belong to some B” cannot be true (APr. 64b7–13, in tandem with 64a4–7and 64a23–30), whereas on this second strategy in behalf of Aristotle, “B does not belong to some A” must be true (since “B belongs to all A” is false) where A has no members. Then it would be true that “B does not belong to some B” where B has no members, in contradiction of Aristotle’s apparent position.

4

I see all kinds of problems with this, Stephen. First, it is not clear whether "existing" means strictly existing in extra-mental reality or (additionally or alternatively) in imagination. But since you are contrasting this with the first solution, I assume that you mean the more restrictive sense of "existing."

On that assumption, this suggested solution is is very problematic in that it excludes universal propositions like "No sea serpents are real reptiles." This is a universal proposition, and its subject term does not have any existing members, in the sense of existing in the world external to someone's imagination - yet, it is true!

Negation of this true E proposition gives the I proposition: "Some sea serpents are real reptiles." This is a particular proposition, and it is false, as it should be, both since the corresponding E proposition is true and since it is not a correct identification of reality. Further, since this I proposition is false, the correspoinding A proposition: "All sea serpents are real reptiles," should also be false, and indeed it is. (Note, that if we had started with the false A proposition, we could not validly infer the false I proposition from it, but we can validly infer both the false I proposition and its corresponding false A proposition from the true E proposition.)

We have gone most of the way "around the horn," so to speak, starting with a true negative universal, inferring a false affirmative particular, then inferring a false affirmative universal. From this, we can (and we could have, at the start, as Stephen notes), further infer a true negative particular, the O proposition: "Some sea serpents are not real reptiles."

Now, what about the above-mentioned apparent conflict with Aristotle? It is, I believe, a chimera or misinterpretation. Since we customarily express a proposition's subject as A and its predicate as B, it can be confusing to state a proposition in the form "Some B belongs to A," rather than "All A are B." These forms are equivalent, and Aristotle uses them so in Prior Analytics. What has to be clarified, however, is that "belongs to" does not indicate class membership, but being possessed as an attribute. So, to say "B belongs to some A" in the present example is to say not that “real reptiles (B) belong to (i.e., are members of) the category of sea serpents (A(,” but instead that "real reptileness (B) belongs to (i.e., is an attribute of) some sea serpents (A)," which is the same as saying "some sea serpents (A) are real reptiles (B)." Bness belongs as an attribute to some As = Some As are Bs.

In the present context, there is an additional problem. Malink is saying that, according to Aristotle, it is false to say, “B does not belong to some B,” while Moody’s approach implies that it is true to say that “B does not belong to B,” when there are no members of B. I don’t see how this follows. First of all, as we are assigning the letters to subject and predicate, in the present example B denotes “real reptile,” and there certainly are members of this class, real ones, no less! Or, if you prefer, it denotes “real reptileness,” and there are real instances of this attribute. So, the supposed problem with Moody’s approach doesn’t even apply here. (It would only apply if B were denoting “sea serpents.” But B is, as per Aristotle, the term denoting the predicate, whether it is expressed B belongs to A or A is B.)

The supposed conflict between Moody and Aristotle also revolves on the ambiguity in the term “belong.” It is nonsense to say that “whiteness belongs to whiteness.” Whiteness belongs to white objects. (And it is nonsense to say that “whiteness is white.” White objects are white.) Yet, in Malink’s interpretation, Aristotle is saying just that: it is false to say that “whiteness does not belong to some whiteness.” This is equivalent to saying that it is false to say that “whiteness is not whiteness.” Well, duh, that’s just the Law of Identity in negative form: it is false to say that a thing is not itself. But what of the supposed problematic implication from Moody’s approach that it is true to say that “whiteness does not belong to some whiteness”? This is equivalent to saying that it is true to say that “whiteness is not whiteness,” if there are no instances of whiteness. Well, I guess that would be correct, too. If there are no instances of whiteness, then there is no whiteness to be itself! So, there really is no conflict with Aristotle. But it doesn’t help out with truth analysis of propositions either!

More importantly, we can still assess the proposition about that to which real reptileness is predicated. Are the referents of the subject, “sea serpents,” something to which real reptileness belongs? Examine the category of “sea serpent,” which is an imaginary creature. Do any of its members possess the attribute of real reptileness? No. This is equivalent to my preferred method of examining the category of “real reptile” and asking if any of its members are the same as what is referred to by the subject. I hope this makes clear how awkward and unwieldy attribute-belonging analysis is compared to match-mismatch analysis of subject and predicate referents.

This is my way of doing logic with the non-existent, and I do not think it is a case of this supposed second solution. I certainly have not (to my knowledge) tried to infer false particulars from false universals. (If you have seen any cases where I have tried to do so, Stephen, please point me to them!) Nor do I think that my framework says that you can do so. (Again, if you think otherwise, please explain.)

10 hours ago, Guyau said:

A third sort of solution, for which there is some evidence in APr., is to replace material implication with what is called connexive implication,^{§2} though Malink doubts this view is truly Aristotle’s. Rather than any of those three proposals, in Aristotle’s behalf, Malink’s own is to reject the idea that the plurality associated with a term in the syllogistic is to be identified with the set of individuals falling under the term. He proposes to identify the plurality associated with a term in the syllogistic with the set of items of which the term is predicated universally: {A1, A2, . . . such that the term B satisfies “B belongs to all A1, all A2, . . . .”}. Then there is at least one member in the plurality associated with each term, for it is the case for every B that “B belongs to all B.” (44, 68) I’m still getting a grip on this.

9

Let me know when you figure it out, Stephen. It doesn’t make a whole lot of sense to me. What little I can figure out is that, for the same reason as above, it sounds backwards and/or trivial. B is the predicate, not the subject, yet the formulation is trying to predicate the predicate of itself. All I can see that this means is that for every instance of whiteness, for instance, it is the case that “whiteness belongs to all whiteness.” Or equivalently, for every instance of whiteness, “all whiteness is whiteness.” Yup, very true, but not very edifying.

I know I’m not getting some of this, Stephen, so please clarify it with an example or two, when you can. Thanks.

Try this with categorical logic.: all lions are animals. Therefore the tail of a lion is the tail of an animal. It cannot be done with categorical propositions.

Baal, are you wanting to infer that a lion is an animal, because it has a tail?

Baal, are you wanting to infer that a lion is an animal, because it has a tail?

Do you have any concept at all of what constitutes a valid argument form? Do you know what a categorical syllogism is? Do you have any notion of how logic works?

Try this with categorical logic.: all lions are animals. Therefore the tail of a lion is the tail of an animal. It cannot be done with categorical propositions.

35 minutes ago, BaalChatzaf said:

Do you have any concept at all of what constitutes a valid argument form? Do you know what a categorical syllogism is? Do you have any notion of how logic works?

Try this with categorical logic.: all lions are animals. Therefore the tail of a lion is the tail of an animal. It cannot be done with categorical propositions.

1

Sure, but not with the premise you have given me! You know as well as I do that syllogistic can't even get off the ground, until you have three terms, no more and no less. So, to do a syllogism on this mess you've given me (how dare you!), we need to somehow reduce it down to three terms from the five you've used (lion, animal, tail, tail of a lion, tail of an animal).

And I'm here to tell you what you already know: it can't be done. The reason is that there is a huge relational structure baked into the challenge, one that the premise is worthless for dealing with.

However, there are premises that will syllogistically generate the conclusion you want. Here's the pattern:

A tail of a lion, considered as a species (lion), is also a tail of the same lion, considered as a genus (animal).

A tail of a lion, considered as an individual, is a tail of the same lion, considered as a species (lion).

Therefore, a tail of a lion, considered as an individual, is a tail of the same lion, considered as a genus (animal).

This avoids the whole cumbersome, rickety apparatus of modern logic and depends on only one very useful general premise: a thing that stands as a term of a certain relation, when that thing is considered under one aspect, will also stand as a term of the same relation, when it is considered under another aspect.

Or, more specifically, this corollary: a part of a whole, considered as a species, is also a part of the same whole, considered as a genus - and this corollary: a part of a whole, considered as an individual, is also a part of the same whole, considered as a species.

I credit this Aristotelian approach to Henry B. Veatch, Intentional Logic: A Logic Based on Philosophical Realism, Yale University Press, 1952. His subsequent book, Two Logics, is a very helpful, systematic comparison of Aristotelian and modern logic. You will have to take out a second mortgage to buy these books second hand - or check them out from a university library and photocopy them. But they're worth every penny of it.

Sure, but not with the premise you have given me! You know as well as I do that syllogistic can't even get off the ground, until you have three terms, no more and no less. So, to do a syllogism on this mess you've given me (how dare you!), we need to somehow reduce it down to three terms from the five you've used (lion, animal, tail, tail of a lion, tail of an animal).

And I'm here to tell you what you already know: it can't be done. The reason is that there is a huge relational structure baked into the challenge, one that the premise is worthless for dealing with.

However, there are premises that will syllogistically generate the conclusion you want. Here's the pattern:

A tail of a lion, considered as a species (lion), is also a tail of the same lion, considered as a genus (animal).

A tail of a lion, considered as an individual, is a tail of the same lion, considered as a species (lion).

Therefore, a tail of a lion, considered as an individual, is a tail of the same lion, considered as a genus (animal).

This avoids the whole cumbersome, rickety apparatus of modern logic and depends on only one very useful general premise: a thing that stands as a term of a certain relation, when that thing is considered under one aspect, will also stand as a term of the same relation, when it is considered under another aspect.

Or, more specifically, this corollary: a part of a whole, considered as a species, is also a part of the same whole, considered as a genus - and this corollary: a part of a whole, considered as an individual, is also a part of the same whole, considered as a species.

I credit this Aristotelian approach to Henry B. Veatch, Intentional Logic: A Logic Based on Philosophical Realism, Yale University Press, 1952. His subsequent book, Two Logics, is a very helpful, systematic comparison of Aristotelian and modern logic. You will have to take out a second mortgage to buy these books second hand - or check them out from a university library and photocopy them. But they're worth every penny of it.

REB

Rickety logic can be used to design computer circuits. Syllogistic cannot. And you made my point (thank you). Categorical Syllogism cannot handle simple binary relations (such as x is the tail of y) nor can categorical syllogisms be used to prove most mathematical theorems. Challenge: formulate Pythagoras Theorem as an enthymeme (chain of syllogistic arguments using categorical propositions) and then prove it using the rules of categorical syllogistic arguments. Your "considered as" arguments are not valid categorical arguments. Which is the point. Aristotle's categorical argument forms are restricted, constricted and constipated. Which is why the Stoics invented conditional arguments which have stood the test of time and are not the primary argument forms in mathematics and the physical sciences. After Frege extended conditional argumentation to quantification his approach soon prevailed in logic (even with the difficulties with the set paradoxes). The Zermelo-Frankel formulation of set theory avoids the paradoxes and is the logical basis of most mathematical theories. Aristotle's logic is not wrong, it is too restrictive. It cannot handle general n-adic relations.

Once Boole and Frege mathematized logic, traditional categorical logic fell to the wayside, in spite of Fred Sommer's valiant attempts to revive it. Fred Sommer's Term Logic is not lot capable of formulating is own metalogic.

Rickety logic can be used to design computer circuits. Syllogistic cannot. And you made my point (thank you). Categorical Syllogism cannot handle simple binary relations (such as x is the tail of y) nor can categorical syllogisms be used to prove most mathematical theorems. Challenge: formulate Pythagoras Theorem as an enthymeme (chain of syllogistic arguments using categorical propositions) and then prove it using the rules of categorical syllogistic arguments. Your "considered as" arguments are not valid categorical arguments. Which is the point. Aristotle's categorical argument forms are restricted, constricted and constipated. Which is why the Stoics invented conditional arguments which have stood the test of time and are not the primary argument forms in mathematics and the physical sciences. After Frege extended conditional argumentation to quantification his approach soon prevailed in logic (even with the difficulties with the set paradoxes). The Zermelo-Frankel formulation of set theory avoids the paradoxes and is the logical basis of most mathematical theories. Aristotle's logic is not wrong, it is too restrictive. It cannot handle general n-adic relations.

Once Boole and Frege mathematized logic, traditional categorical logic fell to the wayside, in spite of Fred Sommer's valiant attempts to revive it. Fred Sommer's Term Logic is not lot capable of formulating is own metalogic.

Your "considered as" arguments are not valid categorical arguments.

Of course, they are. "Considered as" is just another way of saying "viewed from the perspective of." All identity ("is") statements - except instances of A is A - are assertions that a given thing viewed from two different perspectives is the same thing, i.e., is itself. That's why propositions work, and that's why syllogistic works. This is affirmed by various realist schools of thought, from Thomists like Aquinas and, more recently, Henry Veatch, to Ontological Atomists like Butchvarov (whose Being Qua Being has an extremely valuable and clarifying discussion of formal "a is a" identity and material identity, which is involved in non-tautological propositions).

When I put things in parenthesis, they are just for clarification, so that no shyster can come along and run with ambiguity and claim the argument is invalid. In other words, I'm making sure the reader knows what I am and am not trying to argue.

For instance, expanding the copula "is" to read "is (the same thing as)" is not a violation of logic but a clarification of it. It is highlighting the fact that the function of a categorical proposition is not to attribute the predicate to the subject, nor to assign the subject membership in the predicate class, but to assert that what is referred to by the subject is the same thing as what is referred to by the predicate.

And that is also why I, and all standard logic texts, and Aquinas, insist that propositions be put in "standard form," where you are literally saying that a thing (viewed from one perspective) is the same thing as that thing (viewed from another perspective). Again, that is how propositions and syllogisms work. Without the Law of Identity (and its corollaries) standing at least in the background, as the court of last resort, none of our utterances are intelligible.

Here is an example: Venus considered as the morning star is (the same thing, viewed from a different perspective, as) Venus considered as the evening star.

Two syllogisms, identity throughout:

Syllogism 1. Phosphorus is (the same thing as) Venus considered as the morning star.

Venus considered as the morning star is (the same thing, viewed from a different perspective, as) Venus considered as the evening star.

Therefore, Phosphorus is (the same thing, viewed from a different perspective, as) Venus considered as the evening star.

Syllogism 2. (from conclusion of 1.) Phosphorus is (the same thing, viewed from a different perspective, as) Venus considered as the evening star.

Venus considered as the evening star is (the same thing as) Hesperus.

Therefore, Phosphorus is (the same thing, viewed from a different perspective, as) Hesperus.

2 hours ago, BaalChatzaf said:

And you made my point (thank you). Categorical Syllogism cannot handle simple binary relations (such as x is the tail of y) nor can categorical syllogisms be used to prove most mathematical theorems.

Really? Hmmmm...

A tail of a lion, considered as a species (lion), is also a tail of the same lion, considered as a genus (animal).

A tail of a lion, considered as an individual, is a tail of the same lion, considered as a species (lion).

Therefore, a tail of a lion, considered as an individual, is a tail of the same lion, considered as a genus (animal).

Seems like all you're complaining about is that I didn't use X and Y in my argument.

X, a tail of a lion, Y, considered as a species (lion), is also X, a tail of the same lion, Y, considered as a genus (animal).

X, a tail of a lion, Y, considered as an individual, is X, a tail of the same lion, Y, considered as a species (lion).

Therefore, X, a tail of a lion, Y, considered as an individual, is X, a tail of the same lion, Y, considered as a genus (animal).

To me, that's just needless alphabet-soup, when the original version makes the case clearly and in depth. The original lays out the relations between parts and wholes and between individuals, species, and genera - and incorporates them into a categorical syllogism composed of three categorical propositions. By spelling out the relational complexities in ordinary language, it avoids paradox and fallacy that often results from excessive symbolization (e.g., the Goedel Slingshot argument). It doesn't get more Aristotelian than that.

Supposedly such "gotcha" cases as this reveal the weak underbelly of Aristotelian logic, but it seems to me that they instead reveal its power and adaptability. Ironically, your rehashing of the modernist sophistries is making me more confident in the power of perspicuously applied Aristotelian logic than before.

2 hours ago, BaalChatzaf said:

Challenge: formulate Pythagoras Theorem as an enthymeme (chain of syllogistic arguments using categorical propositions) and then prove it using the rules of categorical syllogistic arguments.

We've already been through this twice in the past. I met your challenge both times, and you refused to accept it both times. That's enough for the book I'm doing. (I won't quote you unless you want me to.)

2 hours ago, BaalChatzaf said:

the Stoics invented conditional arguments which have stood the test of time and are not the primary argument forms in mathematics and the physical sciences.

If they are NOT the primary argument forms in mathematics and the physical sciences, why are YOU championing them?

Categorical Syllogism cannot handle simple binary relations (such as x is the tail of y)

Aristotle's logic is not wrong, it is too restrictive. It cannot handle general n-adic relations.

Once Boole and Frege mathematized logic, traditional categorical logic fell to the wayside, in spite of Fred Sommer's valiant attempts to revive it. Fred Sommer's Term Logic is not lot capable of formulating is own metalogic.

Fred Sommers' Term Logic handles relations.

The Logic of Natural Language (Sommers), page 185:

+ a_{1} +R^{123} + a_{2} - a_{3}

_{An example proposition on page 187 is: "Some sailor read some (particular) poem to every girl."}

_{The minus sign here means “every” and “not some”. At other times it means "not".}

Something to Reckon With, by George Englebretsen inspired by Fred Sommers' work, page 116: “a complex term of the form +A+A is equivalent to +A.”

So I will apply this to your horse-tail-animal example. That's hopefully correctly; I'm not steeped in this stuff and read the books years ago.

- H + T Every horse has a tail.

+ A + T Some animals have tails.

Therefore:

- - H + A + T + T and

- - H + A + T Every horse is an animal with a tail.

Take another proposition:

Some animals have no tail. + A – T

Combining this with - H + T yields – H + A. Every horse is an animal.

P.S. Please ignore the two minus signs to the far left. I don't know how to erase them.

Yup. Induction regarding part-whole relations is a little trickier than induction in general, but it works. What I borrowed from Henry B. Veatch and adapted immediately following is:

13 hours ago, Roger Bissell said:

one very useful general premise: a thing that stands as a term of a certain relation, when that thing is considered under one aspect, will also stand as a term of the same relation, when it is considered under another aspect.

Or, more specifically, this corollary: a part of a whole, considered as a species, is also a part of the same whole, considered as a genus - and this corollary: a part of a whole, considered as an individual, is also a part of the same whole, considered as a species.

These inductive premises (paraphrasing my most recent comment):

41 minutes ago, Roger Bissell said:

...lay out the relations between parts and wholes and between individuals, species, and genera - and incorporate them into a categorical syllogism composed of three categorical propositions. By spelling out the relational complexities in ordinary language, they avoid paradox and fallacy that often results from excessive symbolization, and they reveal the power and adaptability of perspicuously applied Aristotelian logic

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## Michael Stuart Kelly

In other words, once you remove logic from reality, there can still be a contradiction in the rules?

And Aristotle should be ashamed because he didn't know the new rules because he was dead centuries before someone came up with them?

Where the hell did reality go?

I've used the following example from

God's Little Acreby Erskine Caldwell before, but it is such a great metaphor for this case, it bears repeating.Bob keeps moving Aristotle's little acre.

Michael

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## BaalChatzaf

AuthorMichael. Apparently you did not grasp the error built into the classical Square of Opposition. The truthful assertion all unicorns have one horn in the middle of their foreheads does NOT imply unicorns exists. I provided a proof for the fact All S is P does NOT imply there exists some S. This is an error the medieval logicians committed building on Aristotle's constricted view of logic. There is more to logic than categorical propositions as ALL logicians since Boole and Frege have shown. Aristotle came up with version 1.0 of logic. Logic was freed of its verbal chains and allowed to expand its scope since the middle of the 19 th century.

We no longer think that the earth is the center of the solar system and we no long believe that logic is only about 4 types of categorical assertions.

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## Michael Stuart Kelly

Bob,

Of course unicorns exist--in our heads. The little word game does not permit a unicorn to suddenly become a giraffe with one horn in the middle of its forehead. So even as an imaginary creature, a unicorn does have identity. Therefore it exists, even though it exists only in the imagination. It is an existent with a specific nature.

You can either tie logic to direct observation (including observation of the imagination) and axiomatic concepts like existence, identity and consciousness, or you can keep moving Aristotle's little acre in your head among abstract symbols and pretending it connects to reality solely within the mental game where an abstraction refers to something except when it doesn't.

So long as this process is divorced from

observation outside of logic, all it does is provide a new rule for the abstraction game, but not a new rule that connects the game to reality--and not to Aristotle, for that matter.He observed. And observed and observed and observed...Michael

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## BaalChatzaf

AuthorAll round-squares are geometrical objects. Does this imply round-squares exist???? No. The inference All S is P implies Some S is P is just plain wrong. It is not a valid inference. And there are empty sets. For example: The set of all living Americans born prior to 1800. Not a single one. Classical logic was not only constricted to categorical propositions, it was unable to deal with empty sets which are indeed sets. Without empty sets we could not deal with set differences thus without empty sets we could not have an algebra of sets.

Aristotle founded logic 1.0. We are now up to logic 8.0.

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## KorbenDallas

Round-squares do not exist, so nothing can be predicated of them.

The quoted proposition isn't the same as the unicorn example. It is an equivocation fallacy, to equivocate a mental existant (the unicorn) with what is presented as an existant--these are different things.

In the Irwin translation, Aristotle said in Categories that to predicate, it had to be predicated

ofsomething (said of something), this something is the primary substance. I would use MSK's post previous to this one, if I were to type it out myself. I would use it to conclude here.## Link to comment

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## Roger Bissell

All round squares are arbitrary constructs. They exist in the mind as arbitrary constructs, as products of the imagination. The imagination can construct all kinds of contradictory things that cannot exist in mind-independent reality.

A: All round squares are real geometrical objects. False.

I: Some round squares are real geometrical objects. False.

E: No round squares are real geometrical objects. True.

O: Some round squares are not real geometrical objects. True.

A: All round squares are arbitrary constructs. True.

I: Some round squares are arbitrary constructs. True.

E: No round squares are arbitrary constructs. False.

O: Some round squares are not arbitrary constructs. False.

This is Logic 2.0B. It follows the dictum of Aquinas that only wholes can be legitimately predicated of wholes, and that once you restrict predication in this way, Aristotle's square of opposition and the laws of immediate inference are preserved intact. You can handle propositions about the existent, the imaginary, and the non-existent with equal ease and validity. Truth values flow quickly and clearly from this approach, like low-hanging fruit. Which may be why most academics and die-hard computer gurus have resisted it for so long. (Just guessing.)

By contrast, Logic 2.0A (Bob's preference) is based on (supposed) Boolean logic as promoted by Russell et al. It is shot through with errors, due to failure to realize that predicating parts of wholes will lead to logical paradoxes and a drastically reduced ability of immediate inference to function properly.

I wrote about this extensively in my December 2014 essay in

The Journal of Ayn Rand Studies. I invite you to check it out, if you are interested in delving more deeply than is permitted by the repartee format of OL.REB

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## BaalChatzaf

AuthorThere are no unicorns and I have a notion of what a round square is. Using the taxicab metric all circles come out square.

Existential import using Aristotle rules produces a contradiction as I showed. It is a bogus notion and we are well rid of it.

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## KorbenDallas

In my own thinking, I separate out an arbitrary from the imaginary, and I consider an arbitrary to be something presented as fact-based, but not fact-based. An imaginary thing isn't presented as fact-based, is meant to be imaginary, and a mental existant. So "All round-squares are geometrical objects" would be an arbitrary because All round-squares (non-real) are geometrical objects (a real category with real objects)--meaning the non-real subject, in my own thinking, would automatically disqualify anything being predicated of it, to the point where a copula isn't possible. An example of an imaginary would be "All unicorns are white", which is fine, it's not asserting it belongs to a category with real existants.

In my own thinking, I wouldn't give "All round-squares are geometrical objects" the status of a proposition, as this proposition could not be arrived at by induction. I would call it invalid, reject it, and wouldn't assign a truth value to it (or its I, E, O forms). The reason this is so is because of the axioms, the statement didn't arrive from them. Would this compare require logic? Perhaps a quick, semi-automatized method of inductive difference--comparing the statement to one's context of knowledge. In my own thinking, this is what I do, and I don't assign the statement the status of possible truth or falsity, I hold it up as real/unreal, without a truth value. I don't allow it to enter the laws of logic, nor do I even consider the statement to have identity. I ask the question, "is this real or unreal", rather than, "what identity does this statement have". Only if a statement passes the sniff test to the context of my knowledge does it begin to be considered.

I am interested in what you've written and would like to read it. These are my thoughts, some a product of reading, some a product of my own thinking, and I'm sure some things that you've heard before.

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## KorbenDallas

I watched the video, and found several errors in it. She is omitting some of the rules of the square of opposition, namely how super and sub implication works. This is from Lionel Ruby, Logic an Introduction, page 252:

1:45 - She makes the statement, "some politicians are not honest, so no politicians are honest". This is invalid, when saying "some politicians are not honest" she automatically moves to "no politicians are honest" as a true statement--from "some politicians are not honest" you can't logically say anything about the other politicians, because the rest of them might be dishonest, the rest of them might be honest.

4:35 - She actually demonstrates amphiboly here, but says she's doing something different. Holding two red and one green and staying at the A form, she says, "All blocks are red is false," but actually demonstrates, "All blocks are not red", which is amphiboly. She moves to the E form, as amphiboly can be taken to mean a negation of "All blocks are red", or it can mean "Some blocks are not red", which the latter she obviously should have moved to.

8:35 - She directs her hands that if someone was to know one of the propositions on the square of opposition, that you can move around the square to know the other propositions, but this is invalid, she misses the rules of super and sub implication.

8:53 - She invalidly moves from a sub implication to its super as if it is true, but it cannot be determined (again, she omits some of the rules of the square of opposition).

I consider that statement to be an example of an arbitrary, it's being asserted that there is "the set of all living Americans born prior to 1800", but it is not possible to create that class. "There are zero living Americans born prior to 1800" is a true statement, and classical logic handles this fine.

Aristotle spoke of the "not thing" in one of his refutations to Plato, that we need to be concerned with what something is, not what something not is (I'm paraphrasing heavily here), if we were to keep track of what things aren't, there would be infinite possibilities. I mentioned before about Categories, about the primary substance, that to predicate, it had to be predicated

ofsomething (saidofsomething), that this something is the primary substance. What I'm getting at here, is when I studied Aristotle, he is talking aboutthings that exist.In On Interpretation, Aristotle talks about possibility and necessity--about "possible to be", "possible not to be", "not possible to be", "not possible not to be". So why come up with things that are not possible, things that not possible to be, and insist it breaks his logic? Throw out the not possible to be, throw out the arbitrary--his logic is meant to handle things that exist, it was derived from reality where thingsdoexist. Speak about those things and classical logic is fine.## Link to comment

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## Roger Bissell

I think you can speak about

anything whatever, and "classical logic is just fine." You just have to clearly spell out what you are asserting, whenever there is any doubt. Otherwise, the ambiguities leave an opening wide enough for Bertrand Russell to drive a Mack truck through it!We don't have to "keep track of what things aren't," but that is not an impediment to judging that things which don't exist

don't exist! In performing the conceptual inventory of "real geometric figures" in order to assess truth/falsity of "A round-square is a real geometric figure," all you have to do is look in your "real geometric figure" file folder and note that there aren't any round-squares there! You can't see what isn't there, so certainly you're under no obligation to keep track of what doesn't exist!I like your example. Certainly, we cannot now form the class of actual people denoted by "living Americans born before 1800," but it's equally true that 200 years ago, someone could have! So, we need to build that present/past distinction into any assertion we make about that class - just like we need to build the imaginary/actual or arbitrary/real distinction into some of the other assertions we've been considering.

With that in mind, here's how I'd deal with it. "All living Americans born prior to 1800 are people who are now (in 2016) very old." True or false? (I could have said: "...are actual people who are very old.")

Well, do the Bertie Russell inventory. Look in the class of "people who are now (in 2016) very old." Do you see any living Americans born prior to 1800? Nope. The assertion is false.

Now, negate: "Some living Americans born prior to 1800 ARE NOT people who are now (in 2016) very old." True or false? Are there any living Americans born prior to 1800 in the class of "people who are now (in 2016) very old? Again, no. So, this statement is true, as the negation of a false assertion

mustbe.You really don't have to decide whether the subject refers to a product of cognition (valid concept) or a product of imagination - and you don't have to decide whether the product of imagination was cobbled together from compatible or incompatible elements.

You just have to be clear on whether you are asserting that the subject refers to something real or imaginary, and then (if necessary for clarity) say so in a fuller expression of the assertion. And then perform the conceptual inventory and on that basis judge whether the "is" or "is not" is correct (and the statement true) or incorrect (and the statement false).

REB

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## BaalChatzaf

AuthorHow about the set of all three headed Americans born before 1800 and still alive. Not a single one. It is the empty set.

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## Guyau

Bob, perhaps the most common solution, in Aristotle’s behalf, to the seeming existential import you illustrated in the root post is to “introduce into the assertoric syllogistic a presupposition to the effect that the plurality associated with every term has at least one member” (42). I have no problem in general with extralogical assumptions that constrain the syllogistic (assertoric or modal); it depends on what the extralogical presupposition is. I concur with Rand’s extralogical presupposition that logic depends on the axiom “Existence exists.” Branden’s

Visionand Peikoff’sOPARdraw out constraints from that dependency as to the character of what we are doing with the normative discipline of deduction. But I’ll leave what they say to another time and for now to the interested reader to dig out. To the question of the present proposed solution to the problem of existential import, I don’t think the dependence of logic on the axiom “Existence exists” entails the constraint that the plurality associated with every term must have at least one existing member (at some time or other). Leaving the issue of Objectivist (or my own) extralogical presuppositions aside, the drawback of this proposed solution is that the assertoric syllogistic “would not be the universally applicable system of formal logic that it is often thought to be” (43).I’m quoting from Marko Malink’s

Aristotle’s Modal Syllogistic(2013 Harvard).A second sort of solution allows into the syllogistic terms whose associated plurality has no existing members (at any time). They just cannot serve as terms that are subjects in any universal propositions that are true. Inferring “B belongs to some A” from the false “B belongs to all A” is simply invalid. This strategy has been adopted by Moody and several others (and Roger?), but Malink observes that it is incompatible with Aristotle when he seem to say “B does not belong to some B” cannot be true (

APr.64b7–13, in tandem with 64a4–7and 64a23–30), whereas on this second strategy in behalf of Aristotle, “B does not belong to some A” must be true (since “B belongs to all A” is false) where A has no members. Then it would be true that “B does not belong to some B” where B has no members, in contradiction of Aristotle’s apparent position.A third sort of solution, for which there is some evidence in

APr., is to replace material implication with what is called connexive implication,^{§2}though Malink doubts this view is truly Aristotle’s. Rather than any of those three proposals, in Aristotle’s behalf, Malink’s own is to reject the idea that the plurality associated with a term in the syllogistic is to be identified with the set of individuals falling under the term. He proposes to identify the plurality associated with a term in the syllogistic with the set of items of which the term is predicated universally: {A1, A2, . . . such that the term B satisfies “B belongs to all A1, all A2, . . . .”}. Then there is at least one member in the plurality associated with each term, for it is the case for every B that “B belongs to all B.” (44, 68) I’m still getting a grip on this.## Link to comment

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## Roger Bissell

What fun!

All three-headed Americans born before 1800 are real, presently living people.You have introduced two criteria that require mention in the predicate: being real (as opposed to imaginary) and being alive (as opposed to dead).

Compare subject and predicate. In the set of "real, presently living people," do you find a three-headed American born before 1800? No. Thus, the things referred to by the subject (three-headed Americans born before 1800) ARE NOT THE SAME as any of the things referred to by the predicate (real, presently living people). The proposition asserts that they ARE THE SAME, and that assertion of identity is therefore incorrect, and thus the proposition is false.

You cannot infer from a false A statement that its corresponding I statement is also false, so we'll leave the truth status of the I statement for now.

Moving on to the O statement: since it's the contradictory of the A, it looks like this:

Some three-headed Americans born before 1800 ARE NOT real, presently living people. Bertrand Russell inventory: do you see any three-headed Americans in your file folder (class) of real, presently living people? Nope. Thus, the assertion of NON-IDENTITY (i.e., the denial of identity) between subject-referent and predicate-referent is CORRECT, and the proposition is true. Aristotelian truth-value of negation holds firmly.OK, what about the E statement?

No three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. This is the assertion, and it is correct, so the E statement is true.Now we can take up the I statement: since it's the contradictory of the E, it looks like this:

Some three-headed Americans born before 1800 ARE real, presently living people. Inventory: no three-headed Americans in the set of real, presently living people. Thus, the things referred to by the subject ARE NOT THE SAME as the things referred to by the predicate. Since the proposition asserts they ARE, that assertion of identity is therefore incorrect, and the I proposition is false.I've taken on your grotesque, Baroquely adorned challenge and shown it to be easily analyzed according to the Aristotelian truth-value in re negation, both for the A (false) <-->O (true) and the E (true) <-->I (false).

What this shows to me is that formal logic is nowhere near as clear or powerful at dealing with logical inference as is ordinary language in combination with Aristotle's laws of logic and common sense (or, as he would have called it, practical wisdom).

Not only this approach it easier to use, it's more useful when someone is trying to pull the wool over your eyes. Godel's slingshot would never have flown if he had spelled it out in ordinary language. It's built on a gross equivocation(s), covered over with a blizzard of alphabet-soup symbolization.

REB

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## BaalChatzaf

AuthorBoolean Logic made computers possible. What has cramped, confined, constipated logic of categorical propositions done for science and mathematics? Damned little.

Try this with categorical logic.: all lions are animals. Therefore the tail of a lion is the tail of an animal. It cannot be done with categorical propositions.

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## Roger Bissell

Stephen, if by "at least one existing member," you mean existing in

mind-independentreality, then surely this is too restrictive. We can and do form and reason with assertions about products of the imagination as well as arbitrary constructs:Harry Potter is a creation of J. K. Rowling. All creations of J. K. Rowling are fictional characters. Therefore, Harry Potter is a fictional character.

A round-square is an arbitrary construct. All arbitrary constructs are things to which Leonard Peikoff is opposed. Therefore, a round-square is something to which Leonard Peikoff is opposed.

I think if you loosen the criterion to "at least one member existing mentally and/or extra-mentally" (or something like that), you would have something very close to how I extend/adapt Aristotle's rules of inference.

I see all kinds of problems with this, Stephen. First, it is not clear whether "existing" means strictly existing in extra-mental reality or (additionally or alternatively) in imagination. But since you are contrasting this with the first solution, I assume that you mean the more restrictive sense of "existing."

On that assumption, this suggested solution is is very problematic in that it excludes universal propositions like "No sea serpents are real reptiles." This is a universal proposition, and its subject term does not have any existing members, in the sense of existing in the world external to someone's imagination - yet, it is true!

Negation of this true E proposition gives the I proposition: "Some sea serpents are real reptiles." This is a particular proposition, and it is false, as it should be, both since the corresponding E proposition is true and since it is not a correct identification of reality. Further, since this I proposition is false, the correspoinding A proposition: "All sea serpents are real reptiles," should also be false, and indeed it is. (Note, that if we had

startedwith the false A proposition, we could not validly infer the false I proposition from it, but wecanvalidly inferboththe false I proposition and its corresponding false A proposition from the true E proposition.)We have gone most of the way "around the horn," so to speak, starting with a true negative universal, inferring a false affirmative particular, then inferring a false affirmative universal. From this, we can (and we could have, at the start, as Stephen notes), further infer a true negative particular, the O proposition: "Some sea serpents are not real reptiles."

Now, what about the above-mentioned

apparentconflict with Aristotle? It is, I believe, a chimera or misinterpretation. Since we customarily express a proposition's subject as A and its predicate as B, it can be confusing to state a proposition in the form "Some B belongs to A," rather than "All A are B." These formsareequivalent, and Aristotle uses them so inPrior Analytics. What has to be clarified, however, is that "belongs to" does not indicate class membership, but being possessed as an attribute. So, to say "B belongs to some A" in the present example is to say not that “real reptiles (B) belong to (i.e., are members of) the category of sea serpents (A(,” but instead that "real reptileness (B) belongs to (i.e., is an attribute of) some sea serpents (A)," which is the same as saying "some sea serpents (A) are real reptiles (B)." Bness belongs as an attribute to some As = Some As are Bs.In the present context, there is an additional problem. Malink is saying that, according to Aristotle, it is false to say, “B does not belong to some B,” while Moody’s approach implies that it is true to say that “B does not belong to B,” when there are no members of B. I don’t see how this follows. First of all,

as we are assigning the letters to subject and predicate, in the present example B denotes “real reptile,” and there certainlyaremembers of this class,realones, no less! Or, if you prefer, it denotes “real reptileness,” and there are real instances of this attribute. So, the supposed problem with Moody’s approach doesn’t even apply here. (It would only apply if B were denoting “sea serpents.” But B is, as per Aristotle, the term denoting the predicate, whether it is expressed B belongs to A or A is B.)The supposed conflict between Moody and Aristotle also revolves on the ambiguity in the term “belong.” It is nonsense to say that “whiteness belongs to whiteness.” Whiteness belongs to white objects. (And it is nonsense to say that “whiteness is white.” White objects are white.) Yet, in Malink’s interpretation, Aristotle is saying just that: it is false to say that “whiteness does not belong to some whiteness.” This is equivalent to saying that it is false to say that “whiteness is not whiteness.” Well, duh, that’s just the Law of Identity in negative form: it is false to say that a thing is not itself. But what of the supposed problematic implication from Moody’s approach that it is

trueto say that “whiteness does not belong to some whiteness”? This is equivalent to saying that it istrueto say that “whiteness is not whiteness,” if there are no instances of whiteness. Well, I guess that would be correct, too. If there are no instances of whiteness, then there is no whiteness to be itself! So, there really is no conflict with Aristotle. But it doesn’t help out with truth analysis of propositions either!More importantly, we can still assess the proposition about that to which real reptileness is predicated. Are the referents of the subject, “sea serpents,” something to which real reptileness belongs? Examine the category of “sea serpent,” which is an imaginary creature. Do any of its members possess the attribute of real reptileness? No. This is equivalent to my preferred method of examining the category of “real reptile” and asking if any of its members are the same as what is referred to by the subject. I hope this makes clear how awkward and unwieldy attribute-belonging analysis is compared to match-mismatch analysis of subject and predicate referents.

This is

myway of doing logic with the non-existent, and I do not think it is a case of this supposed second solution. I certainly have not (to my knowledge) tried to infer false particulars from false universals. (If you have seen any cases where I have tried to do so, Stephen, please point me to them!) Nor do I think that my framework says that you can do so. (Again, if you think otherwise, please explain.)Let me know when you figure it out, Stephen. It doesn’t make a whole lot of sense to me. What little I can figure out is that, for the same reason as above, it sounds backwards and/or trivial. B is the predicate, not the subject, yet the formulation is trying to predicate the predicate of itself. All I can see that this means is that for every instance of whiteness, for instance, it is the case that “whiteness belongs to all whiteness.” Or equivalently, for every instance of whiteness, “all whiteness is whiteness.” Yup, very true, but not very edifying.

I know I’m not getting some of this, Stephen, so please clarify it with an example or two, when you can. Thanks.

REB

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## KorbenDallas

Baal, are you wanting to infer that a lion is an animal, because it has a tail?

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## BaalChatzaf

AuthorDo you have any concept at all of what constitutes a valid argument form? Do you know what a categorical syllogism is? Do you have any notion of how logic works?

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## KorbenDallas

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## Roger Bissell

Sure, but not with the premise you have given me! You know as well as I do that syllogistic can't even get off the ground, until you have three terms, no more and no less. So, to do a syllogism on this mess you've given me (how dare you!), we need to somehow reduce it down to three terms from the five you've used (lion, animal, tail, tail of a lion, tail of an animal).

And I'm here to tell you what you already know: it can't be done. The reason is that there is a huge relational structure baked into the challenge, one that the premise is worthless for dealing with.

However, there

arepremises that will syllogistically generate the conclusion you want. Here's the pattern:A tail of a lion, considered as a species (lion), is also a tail of the same lion, considered as a genus (animal).

A tail of a lion, considered as an individual, is a tail of the same lion, considered as a species (lion).

Therefore, a tail of a lion, considered as an individual, is a tail of the same lion, considered as a genus (animal).

This avoids the whole cumbersome, rickety apparatus of modern logic and depends on only one very useful general premise:

a thing that stands as a term of a certain relation, when that thing is considered under one aspect, will also stand as a term of the same relation, when it is considered under another aspect.Or, more specifically, this corollary:

a part of a whole, considered as a species, is also a part of the same whole, considered as a genus- and this corollary:a part of a whole, considered as an individual, is also a part of the same whole, considered as a species.I credit this Aristotelian approach to Henry B. Veatch,

Intentional Logic: A Logic Based on Philosophical Realism, Yale University Press, 1952. His subsequent book,Two Logics, is a very helpful, systematic comparison of Aristotelian and modern logic. You will have to take out a second mortgage to buy these books second hand - or check them out from a university library and photocopy them. But they're worth every penny of it.REB

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## BaalChatzaf

AuthorRickety logic can be used to design computer circuits. Syllogistic cannot. And you made my point (thank you). Categorical Syllogism cannot handle simple binary relations (such as x is the tail of y) nor can categorical syllogisms be used to prove most mathematical theorems. Challenge: formulate Pythagoras Theorem as an enthymeme (chain of syllogistic arguments using categorical propositions) and then prove it using the rules of categorical syllogistic arguments. Your "considered as" arguments are not valid categorical arguments. Which is the point. Aristotle's categorical argument forms are restricted, constricted and constipated. Which is why the Stoics invented conditional arguments which have stood the test of time and are not the primary argument forms in mathematics and the physical sciences. After Frege extended conditional argumentation to quantification his approach soon prevailed in logic (even with the difficulties with the set paradoxes). The Zermelo-Frankel formulation of set theory avoids the paradoxes and is the logical basis of most mathematical theories. Aristotle's logic is not wrong, it is too restrictive. It cannot handle general n-adic relations.

Once Boole and Frege mathematized logic, traditional categorical logic fell to the wayside, in spite of Fred Sommer's valiant attempts to revive it. Fred Sommer's Term Logic is not lot capable of formulating is own metalogic.

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## KorbenDallas

Induction handles your lion example easily.

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## Roger Bissell

Of course, they are. "Considered as" is just another way of saying "viewed from the perspective of." All identity ("is") statements - except instances of A is A - are assertions that a given thing viewed from two different perspectives is the

samething, i.e., is itself. That's why propositions work, and that's why syllogistic works. This is affirmed by various realist schools of thought, from Thomists like Aquinas and, more recently, Henry Veatch, to Ontological Atomists like Butchvarov (whoseBeing Qua Beinghas an extremely valuable and clarifying discussion of formal "a is a" identity and material identity, which is involved in non-tautological propositions).When I put things in parenthesis, they are just for clarification, so that no shyster can come along and run with ambiguity and claim the argument is invalid. In other words, I'm making sure the reader knows what I am

and am nottrying to argue.For instance, expanding the copula "is" to read "is (the same thing as)" is not a violation of logic but a clarification of it. It is highlighting the fact that the function of a categorical proposition is not to attribute the predicate to the subject, nor to assign the subject membership in the predicate class, but to assert that what is referred to by the subject

is the same thing aswhat is referred to by the predicate.And that is also why I, and all standard logic texts, and Aquinas, insist that propositions be put in "standard form," where you are literally saying that a thing (viewed from one perspective) is the same thing as that thing (viewed from another perspective). Again, that is how propositions and syllogisms work. Without the Law of Identity (and its corollaries) standing at least in the background, as the court of last resort, none of our utterances are intelligible.

Here is an example: Venus considered as the morning star is (the same thing, viewed from a different perspective, as) Venus considered as the evening star.

Two syllogisms, identity throughout:

Syllogism 1. Phosphorusis (the same thing as)Venus considered as the morning star.Venus considered as the morning star

is (the same thing, viewed from a different perspective, as)Venus considered as the evening star.Therefore, Phosphorus

is (the same thing, viewed from a different perspective, as)Venus considered as the evening star.Syllogism 2. (from conclusion of 1.) Phosphorus is(the same thing, viewed from a different perspective, as)Venus considered as the evening star.Venus considered as the evening star

is (the same thing as)Hesperus.Therefore, Phosphorus

is (the same thing, viewed from a different perspective, as)Hesperus.Really? Hmmmm...

A tail of a lion, considered as a species (lion), is also a tail of the same lion, considered as a genus (animal).

A tail of a lion, considered as an individual, is a tail of the same lion, considered as a species (lion).

Therefore, a tail of a lion, considered as an individual, is a tail of the same lion, considered as a genus (animal).

Seems like all you're complaining about is that I didn't use X and Y in my argument.

X, a tail of a lion, Y, considered as a species (lion), is also X, a tail of the same lion, Y, considered as a genus (animal).

X, a tail of a lion, Y, considered as an individual, is X, a tail of the same lion, Y, considered as a species (lion).

Therefore, X, a tail of a lion, Y, considered as an individual, is X, a tail of the same lion, Y, considered as a genus (animal).

To me, that's just needless alphabet-soup, when the original version makes the case clearly and in depth. The original lays out the relations between parts and wholes and between individuals, species, and genera - and incorporates them into a categorical syllogism composed of three categorical propositions. By spelling out the relational complexities in ordinary language, it avoids paradox and fallacy that often results from excessive symbolization (e.g., the Goedel Slingshot argument). It doesn't get more Aristotelian than that.

Supposedly such "gotcha" cases as this reveal the weak underbelly of Aristotelian logic, but it seems to me that they instead reveal its power and adaptability. Ironically, your rehashing of the modernist sophistries is making me

moreconfident in the power of perspicuously applied Aristotelian logic than before.We've already been through this twice in the past. I met your challenge both times, and you refused to accept it both times. That's enough for the book I'm doing. (I won't quote you unless you want me to.)

If they are NOT the primary argument forms in mathematics and the physical sciences, why are YOU championing them?

REB

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## merjet

Fred Sommers' Term Logic handles relations.

The Logic of Natural Language(Sommers), page 185:+ a

_{1}+R^{123}+ a_{2}- a_{3}_{An example proposition on page 187 is: "Some sailor read some (particular) poem to every girl."}_{The minus sign here means “every” and “not some”. At other times it means "not".}Something to Reckon With, by George Englebretsen inspired by Fred Sommers' work, page 116: “a complex term of the form +A+A is equivalent to +A.”So I will apply this to your horse-tail-animal example. That's hopefully correctly; I'm not steeped in this stuff and read the books years ago.

- H + T Every horse has a tail.

+ A + T Some animals have tails.

Therefore:

- - H + A + T + T and

- - H + A + T Every horse is an animal with a tail.

Take another proposition:

Some animals have no tail. + A – T

Combining this with - H + T yields – H + A. Every horse is an animal.

P.S. Please ignore the two minus signs to the far left. I don't know how to erase them.

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## Roger Bissell

Yup. Induction regarding part-whole relations is a little trickier than induction in general, but it works. What I borrowed from Henry B. Veatch and adapted immediately following is:

These inductive premises (paraphrasing my most recent comment):

REB

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