A challenge to Aristotle's categorical term logic


BaalChatzaf

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Thanks Roger.

Actually, I just realised that another way of looking at it is to notice that "everything has a cause" is the negation of "something isn't caused", so if

A = everything has a cause

B = God exists

Then we have a standard pattern -

if B then ¬A

A

therefore, ¬B

Not sure about your assertion that any hypothetical proposition can be expressed as a categorical proposition though. As I understood it, a conditional proposition isn't quite the same as a hypothetical, or perhaps I'm thinking of a hypothetical as a so-called 'material' conditional.

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Show one piece of significant mathematics was was derived by term logic.

Ba'al Chatzaf

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Show one piece of significant mathematics was was derived by term logic.

Ba'al Chatzaf

Show that Newton's calculus was derived by modern predicate logic.

Abraham Robinson did it in the 1960's. Look up non-standard analysis. By means of predicate logic infinitessimals were put on a rigorous basis that was able to overcome the objections of Bishop Berkeley who was a sharp mathematician himself. If was Berkeley's objections to infinitessimals that embarrassed mathematicians into properly defining limits in the early 19th century.

ruveyn

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Show that Newton's calculus was derived by modern predicate logic.

Abraham Robinson did it in the 1960's. Look up non-standard analysis. By means of predicate logic infinitessimals were put on a rigorous basis that was able to overcome the objections of Bishop Berkeley who was a sharp mathematician himself. If was Berkeley's objections to infinitessimals that embarrassed mathematicians into properly defining limits in the early 19th century.

Newton did not. There is nothing on the Wikipedia pages for non-standard analysis or Abraham Robinson that he derived anything with modern predicate logic. That something is proven in another way and then expressed in modern predicate logic symbolism does not show that the proof was derived with modern predicate logic. I said derived, not consistent with or expressible in.
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The problem with the Predicate calculus is that it's not very user friendly because it's so far removed from natural language, and I'm pretty sure very few mathematicians use it to derive anything. In fact, you can't actually derive anything at all using the predicate calculus, all you can do is prove a conclusion that's already given. It's much more useful to be able to find what follows from a set of premises. Syllogistic can derive conclusions as well as prove by contradiction.

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The problem with the Predicate calculus is that it's not very user friendly because it's so far removed from natural language, and I'm pretty sure very few mathematicians use it to derive anything. In fact, you can't actually derive anything at all using the predicate calculus, all you can do is prove a conclusion that's already given. It's much more useful to be able to find what follows from a set of premises. Syllogistic can derive conclusions as well as prove by contradiction.

I agree with you, Davy. I think that logic can be sufficiently rigorous to avoid errors due to ambiguity fostered by category errors and the like, yet sufficiently close to natural language to avoid errors due to ambiguity fostered by abstruseness. Standard form with the same-category requirement for subject and predicate is a good first payment on the former; many (all?) of the paradoxes surrounding existential import can be dissolved by this requirement.

On the other end, grossities such as Goedel's "slingshot," which are easily revealed to be due to a covert ambiguity obscured by the modern symbolisms, would never have happened, if logicians hadn't gone off the deep end, playing with formalisms. (And that's the generous, charitable interpretation of what they have done.)

(I've discussed the "slingshot" elsewhere here on OL, to no avail. It purports to prove that every fact is every other fact, or: all true sentences stand for the same thing. It's really a breathtaking, scary argument, if you gloss over the equivocation that is hidden in how Goedel uses the quanitification symbols.)

REB

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Show one piece of significant mathematics was was derived by term logic.

Ba'al Chatzaf

I don't know if you would consider this "significant." It certainly doesn't compare with Newtonian calculus, &c. But about 20 years ago, I discovered (inductively) and validated (deductively) a new method for deriving Pythagorean triples -- the only one I am aware of, other than the original one the Greeks or Babylonians or Egyptians came up with several thousand years ago.

I did not use predicate logic for the validation. I just used some rather exhausting, multiple-column induction-by-inspection, along with the same form of math+ordinary language proof I was taught in plane geometry in my sophomore year of high school (1963-1964), and the quadratic formula, which I believe was discovered by the Babylonians even prior to the Pythagorean triple method. :-)

Here is a link to my abbreviated on-line essay, laying out the method I discovered and its validation:

http://rogerbissell.com/id11iiii.html

REB

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Interesting site you have there, Roger.

By the way, according to mathematician Stanley Burris, who has written a fair bit on Boole's (and other notable 19th Century logician's) logic, says that modern semantics was introduced by Charles Peirce in 1880, although in the Wiki entry on the Square of Opposition it's claimed that Boole argued for universal propositions to lack existential import. Boole's algebra is not the modern boolean algebra, but simply ordinary high school algebra restricted to values of 0 and 1. Burris says:

The literature from the late 1800s through the entire 20th century is filled with the wreckage of muffled circumlocutions and failed attempts to explain what Boole was doing. The universal error has been to assume that Boole was using modern semantics for simple class names...

Boole used Aristotelian Semantics!

For Boole the conversion by limitation argument 'All A is B', therefore 'Some B is A' was correct and proved in both his 1847 and 1854 texts.

Not only was his logic misunderstood, but his probability theory (which is covered in "The Laws of Thought") was thought to be wrong. According to David Miller (also mentioned in the Wiki entry on Boole), it's not only not wrong but useful for many problems which conventional probability cannot deal with.

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  • 2 years later...

On September 30, 2011, earlier in this thread, Ba'al Chatzaf wrote: "Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal. You can use any of the 19 valid categorical syllogisms and you can even string them together as a sorites. See how you do."


I replied the same day and thought I did pretty well. But Ba'al never responded, and my impression was that he was not impressed. No one else commented on my response either, so presumably they weren't impressed either.


Well, I'm back, and perhaps readers will find this reply more clarifying and more decisive.


First of all, it's instructive to note that Ba'al's example is a retread of an example used by Bertrand Russell in A History of Western Philosophy (1945): "A horse is an animal; therefore, the head of a horse is a head of an animal." (Of course, while Ba'al tried to make his version look like a syllogism, and I think he's right in trying to do so, Bertie's example doesn't even purport to be a syllogism, but instead an immediate inference.)


In any case, there is a conclusion, and we need to clarify the situation that justifies drawing that conclusion -- if possible, in terms of categorical propositions. So, what is that situation? All things that stand as terms of a certain relation, when those things are considered in one respect, will also stand as terms of the same relation when considered in another respect.


As a syllogism, it's like this: 1. All parts of wholes, considered as species, are also parts of the same wholes, considered as genesus. 2. All heads of horses are parts of wholes considered as species (viz., horses). 3. Therefore, all heads of horses are parts of the same wholes, considered as genuses (viz., animals).


You can convert these into plain language by dropping the universal quantifiers down to simple articles, like this: 1. A part of a whole, considered as a species, is also a part of the same whole, considered as a genus. 2. A head of a horse is part of a whole considered as a species (viz., horse). 3. Therefore, a head of a horse is part of the same whole considered as genus (viz., animal).


However, the meaning is obviously the same as before, and although they are no longer universal propositions in *technical* terms, logicians have traditionally treated them as though they *were* universal propositions, and there is really no reason not to, and every reason to do so.


The subject and predicate of each premise and the conclusion are both making a reference, and the question is whether they are both referring to the same thing in each case. If they are, then the syllogism is valid; and if the premises are true, then the conclusion is true.


The reader is invited to try out this approach on the famous example given by Leibniz: "Jesus is God; therefore the mother of Jesus is the mother of God." Peace be with you....REB

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On September 30, 2011, earlier in this thread, Ba'al Chatzaf wrote: "Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal. You can use any of the 19 valid categorical syllogisms and you can even string them together as a sorites. See how you do."

I replied the same day and thought I did pretty well. But Ba'al never responded, and my impression was that he was not impressed. No one else commented on my response either, so presumably they weren't impressed either.

Well, I'm back, and perhaps readers will find this reply more clarifying and more decisive.

First of all, it's instructive to note that Ba'al's example is a retread of an example used by Bertrand Russell in A History of Western Philosophy (1945): "A horse is an animal; therefore, the head of a horse is a head of an animal." (Of course, while Ba'al tried to make his version look like a syllogism, and I think he's right in trying to do so, Bertie's example doesn't even purport to be a syllogism, but instead an immediate inference.)

In any case, there is a conclusion, and we need to clarify the situation that justifies drawing that conclusion -- if possible, in terms of categorical propositions. So, what is that situation? All things that stand as terms of a certain relation, when those things are considered in one respect, will also stand as terms of the same relation when considered in another respect.

As a syllogism, it's like this: 1. All parts of wholes, considered as species, are also parts of the same wholes, considered as genesus. 2. All heads of horses are parts of wholes considered as species (viz., horses). 3. Therefore, all heads of horses are parts of the same wholes, considered as genuses (viz., animals).

You can convert these into plain language by dropping the universal quantifiers down to simple articles, like this: 1. A part of a whole, considered as a species, is also a part of the same whole, considered as a genus. 2. A head of a horse is part of a whole considered as a species (viz., horse). 3. Therefore, a head of a horse is part of the same whole considered as genus (viz., animal).

However, the meaning is obviously the same as before, and although they are no longer universal propositions in *technical* terms, logicians have traditionally treated them as though they *were* universal propositions, and there is really no reason not to, and every reason to do so.

The subject and predicate of each premise and the conclusion are both making a reference, and the question is whether they are both referring to the same thing in each case. If they are, then the syllogism is valid; and if the premises are true, then the conclusion is true.

The reader is invited to try out this approach on the famous example given by Leibniz: "Jesus is God; therefore the mother of Jesus is the mother of God." Peace be with you....REB

Where is the categorical proof using the 19 valid forms in a sorites. I don't see it in your posting.

The sad fact is there is no valid proof in categorical logic.

Categorical logic cannot handle relations well. It is bad on some binary relations and hopeless on n-ary relations where

n > 2.

That is why it was dumped for mathematical use from Boole and there thereafter.

Ba'al Chatzaf

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The sad fact is there is no valid proof in categorical [natural language predicate?] logic.

Unless you post in math expressions, we're stuck with discerning the validity of propositions and inferences by the meaning of terms.

Certainly true, the head of a horse is the head of an animal. It's not the head of a rock.

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I don't need a *chain* of syllogisms (one meaning of "sorites"), only a single syllogism, which I provided in categorical form in the sixth paragraph of my previous post.

The initial statements you gave were worthless for doing inference. I instead identified the situation they were *about* and constructed the syllogism from that.

It is all very Aristotelian and very categorical. If you disagree, then we are from such different definitional/conceptual frameworks that we will not be able to continue the discussion. Only you can answer that question...

I then translated it into ordinary language without the universal quantifier, which I know disqualifies it (in your opinion) from being categorical. Nonetheless, it *is* categorical -- no if's and's or but's, just straightforward assertion. Again, if you do not recognize and accept this as being categorical, we're really done here.

But the real problem with your challenge is asking for a syllogism based on those premises as stated. As I remarked several years ago, you're trapped in a framework that commits you to the Fourth Term Fallacy.

The person who invented the approach I used is an Aristotelian, and he's a hell of a lot smarter than Russell and his ilk.

And Boole was far more of an Aristotelian and less of a "Boolean" than most people realize or care to accept. If you want to read a couple of academic papers on this, I can provide links. But I fear that you will be as impervious to their thoughts as you are to mine.

And that's fine. To each, his own. It's certainly not stopping me from writing up and publishing my ideas. Just making me scratch my head a bit, wondering how the intellectual world could have been so lax as to allow Russell et al to gut Aristotelian logic.

REB

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Ba'al's misdirection in his preceding post reminds me of the old joke about the camel on the front of the Camel's cigarette package. The question was: "If you were this camel, and you wanted some relief from the sun, would you go to the palm tree in the oasis or the pyramid?" And the answer was: "Neither, you'd go around back to the hotel!" I essentially rejected the non-starter notion of fumbling around for some chain of syllogisms based on Ba'al's original premises, none of which could possibly work, and instead produced premises that accurately described the situation *and* would work together in a *single* syllogism to produce the desired conclusion.

I know this is "cheating," but like I give a big rat's patoot. I'm not Ba'al's student, so he can't give me an F. I give a flying F about his grading system, anyway.

Actually, it's not "cheating," it's rational thought and problem-solving, stepping outside the box, rather than being handcuffed to someone's false dilemma: "Use these statements I've provided to construct a valid categorical argument, or Aristotle's syllogistic is done for."

Of course, some logic teachers don't like that kind of student either. They want nice compliant ones that will (erroneously) accept that Aristotle's logic is of "limited usefulness" and get with the program of modern logic.

REB

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On September 30, 2011, earlier in this thread, Ba'al Chatzaf wrote: "Try proving this categorically: All Dogs are Mammal. T is the tail of a Dog therefore T is the tail of a Mammal. You can use any of the 19 valid categorical syllogisms and you can even string them together as a sorites. See how you do."

I replied the same day and thought I did pretty well. But Ba'al never responded, and my impression was that he was not impressed. No one else commented on my response either, so presumably they weren't impressed either.

Well, I'm back, and perhaps readers will find this reply more clarifying and more decisive.

First of all, it's instructive to note that Ba'al's example is a retread of an example used by Bertrand Russell in A History of Western Philosophy (1945): "A horse is an animal; therefore, the head of a horse is a head of an animal." (Of course, while Ba'al tried to make his version look like a syllogism, and I think he's right in trying to do so, Bertie's example doesn't even purport to be a syllogism, but instead an immediate inference.)

In any case, there is a conclusion, and we need to clarify the situation that justifies drawing that conclusion -- if possible, in terms of categorical propositions. So, what is that situation? All things that stand as terms of a certain relation, when those things are considered in one respect, will also stand as terms of the same relation when considered in another respect.

As a syllogism, it's like this: 1. All parts of wholes, considered as species, are also parts of the same wholes, considered as genesus. 2. All heads of horses are parts of wholes considered as species (viz., horses). 3. Therefore, all heads of horses are parts of the same wholes, considered as genuses (viz., animals).

You can convert these into plain language by dropping the universal quantifiers down to simple articles, like this: 1. A part of a whole, considered as a species, is also a part of the same whole, considered as a genus. 2. A head of a horse is part of a whole considered as a species (viz., horse). 3. Therefore, a head of a horse is part of the same whole considered as genus (viz., animal).

However, the meaning is obviously the same as before, and although they are no longer universal propositions in *technical* terms, logicians have traditionally treated them as though they *were* universal propositions, and there is really no reason not to, and every reason to do so.

The subject and predicate of each premise and the conclusion are both making a reference, and the question is whether they are both referring to the same thing in each case. If they are, then the syllogism is valid; and if the premises are true, then the conclusion is true.

The reader is invited to try out this approach on the famous example given by Leibniz: "Jesus is God; therefore the mother of Jesus is the mother of God." Peace be with you....REB

Try this. A lamp post is left of all the animals. A tiger is an animal, therefore the lamp post is left of the tiger.

The Sun is bigger than all animals. a tiger is an animal therefore the Sun is bigger than the tiger.

Can you prove this categorically.

The issue is not parts and wholes but binary relations.

Ba'al Chatzaf

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Bigger-smaller and left of-right of are reciprocal binary relations, just as are part-whole, individual-species, species-genus, many more, including affirmation-negation (is vs. is not). The discerning logician takes note of these facts when deciding how to transform premises so that deduction can proceed most efficiently (or even at all). The same general pattern of premise-formulation and reasoning I already used works for any challenge of this type that you can construct.

You should be able to handle these challenges yourself, working from the example I provided -- not asking me to take up my time to do them.

However, if you cannot figure out how to use Aristotelian principles to solve these challenges, let me know, and as a consolation for the F you will receive, I'll give you one of the solutions, so that you can try again on the other and maybe bring your overall grade up to a C. ;-)

REB

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. . .

As a syllogism, it's like this: 1. All parts of wholes, considered as species, are also parts of the same wholes, considered as genesus. 2. All heads of horses are parts of wholes considered as species (viz., horses). 3. Therefore, all heads of horses are parts of the same wholes, considered as genuses (viz., animals).

You can convert these into plain language by dropping the universal quantifiers down to simple articles, like this: 1. A part of a whole, considered as a species, is also a part of the same whole, considered as a genus. 2. A head of a horse is part of a whole considered as a species (viz., horse). 3. Therefore, a head of a horse is part of the same whole considered as genus (viz., animal).

. . .

Roger, I'm unsure what you mean by "same whole." Do you mean "same individual considered as a whole"? So your meaning is as follows?

1. All parts of individuals considered as wholes, and considered as species, are also parts of the same individuals considered as wholes, and considered as genuses. 2. All heads of horses are parts of individuals considered as wholes and considered as species (viz., horses). 3. Therefore, all heads of horses are parts of the same individuals considered as wholes and considered as genuses (viz., animals).

~~~~~~~~~~~~~~~~

Continuing from #35:

“. . . then the syllogism is valid; and if the premises are true then the conclusion is true.”

The premises of the syllogism or any other valid form of deductive inference would entail conclusions in a stronger way, and this is perhaps what you meant anyway:

“If the premises are true, then the conclusion cannot be false” or “if the premises are true, the conclusion must be true.”

I’m not sure Aristotle would allow the same necessity in the link of premises to conclusions where the species and genus stand in abstract membership relations to concretes as opposed to where they stand in concrete relations. Similarly with abstract versus concrete part-whole relations. As I recall, his logical theory is about the relations of species and genus in their abstract meaning. The example of horse and animal is a case of the abstract relation in that all individuals that are horses are animals, but it may also, and quite naturally, be read as a case of the biological concrete relations (in our modern understanding), which is a different matter. The example of head of a horse to a horse is an example of the abstract relation of part to whole, as in the color of the head being part of the color(s) of the whole horse, but it may be read also as a case of the biological concrete relation of head to horse. If we understand your premises and conclusions to be dealing in only the abstract sense of species-genus and in only the abstract sense of part-whole, perhaps Aristotle would go along with your picture. (I haven’t been able to do any digging on this for this post. Maybe you know already what Aristotle would think about your scheme, a scheme worth pondering, I’d say.)

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Bigger-smaller and left of-right of are reciprocal binary relations, just as are part-whole, individual-species, species-genus, many more, including affirmation-negation (is vs. is not). The discerning logician takes note of these facts when deciding how to transform premises so that deduction can proceed most efficiently (or even at all). The same general pattern of premise-formulation and reasoning I already used works for any challenge of this type that you can construct.

You should be able to handle these challenges yourself, working from the example I provided -- not asking me to take up my time to do them.

However, if you cannot figure out how to use Aristotelian principles to solve these challenges, let me know, and as a consolation for the F you will receive, I'll give you one of the solutions, so that you can try again on the other and maybe bring your overall grade up to a C. ;-)

REB

You still don't get it. You cannot prove what I asked you to proved with the 19 valid categorical syllogisms. Aristotle's syllogistic logic does not handle relations. It only handle 1-ary predicates. That is why Frege had to invent a new kind of logic to do mathematics with.

Ba'al Chatzaf

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sorry Im late to this thread, havent read any of it, and have nothing to contribute... but, I thought Aristotle has low credibility becasue he was an "arm-chair philosopher" as in- no experimentation to confirm his thought experiments....

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sorry Im late to this thread, havent read any of it, and have nothing to contribute... but, I thought Aristotle has low credibility becasue he was an "arm-chair philosopher" as in- no experimentation to confirm his thought experiments....

What I am talking about is purely technical. The 19 valid categorical syllogisms do NOT handle binary relations well and n-ary relations for n > 2 not at all.

Ba'al Chatzaf

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  • 1 month later...

Try this. A lamp post is left of all the animals. A tiger is an animal, therefore the lamp post is left of the tiger.

The Sun is bigger than all animals. a tiger is an animal therefore the Sun is bigger than the tiger.

Too easy!

Relations can be handled by "immediate inference by converse relation", which was recognized by Aristotle. Thus, in every relational expression it's possible to find the converse, e.g.:

A is the mother of B => B is the child of A

A is larger than B => B is not larger than A

A loves B => B is loved by A

etc...

So the above arguments are easily dealt with.

L = lamp post

A = animals

T = Tiger

1. Some L is left of All A

2. All T is A

3. All A is right of Some L (converse of 1.)

4. All T is right of Some L (2, 3 Dictum de Omni)

5. Some L is left of All T (converse of 4.)

The second argument is much the same. The machinery of the predicate calculus is not necessary to handle multiply quantified relational propositions.

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