predication and genus

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wtf
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Re: predication and genus

Post by wtf » Sun Oct 22, 2017 2:02 am

Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
I know I'm being simple but how can you call 'all sets' a set then? Is that what you mean by a class, all the sets?
Oh I see, you're right. I misspoke myself. I should have said, The class of all sets is not a set. Equivalently, there is no set of all sets.

When I said, "The set of all sets is not a set" that was an error on my part.

Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
Ah! I think I see what you mean the 'set of all sets' is just false, that right. It's just that you have to talk about it as a 'set' to get the idea across?
I confusingly misspoke myself. The class or collection of all sets can not be a set.
Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
Also, is Vx(R(x)) not the same as the set of all red things?
Well that's a statement or a proposition that says, "Everything is red." It doesn't specify the class of red things, it says that everything is a red thing.

There is a class or collection of all red things, but it can't be a set unless you first specify some larger set that contains all the things we're talking about. The point is that you can't just take a predicate by itself to define a set. You have to start with a known set and then reduce it via a predicate. So if I have a set of widgets, I can form the set of all red widgets. You have to have an enclosing set. You first have a set of widgets, then you can apply the predicate to that set. But you can't apply a predicate to define a set without starting with some collection already known to be a set. Otherwise you run into Russell's paradox.

Let me know if that's clear or if I'm still not explaining this well. Was the Wiki page on Russell's paradox helpful?

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Arising_uk
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Re: predication and genus

Post by Arising_uk » Sun Oct 22, 2017 5:01 pm

wtf wrote:When I said, "The set of all sets is not a set" that was an error on my part.
Understood.

Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
Ah! I think I see what you mean the 'set of all sets' is just false, that right. It's just that you have to talk about it as a 'set' to get the idea across?
... Well that's a statement or a proposition that says, "Everything is red." It doesn't specify the class of red things, it says that everything is a red thing. ...
Fair point.
There is a class or collection of all red things, but it can't be a set unless you first specify some larger set that contains all the things we're talking about. ...
Are we talking about domains? If so could we get away with the domain being all the objects in the universe and then Vx(R(x) v ¬R(x)) which would 'give' you all the red objects?
...
Let me know if that's clear or if I'm still not explaining this well. ...
I think I get you.
Was the Wiki page on Russell's paradox helpful?
Sort of but it's a heavy read nowadays for this old man. :)
p.s.
Sorry, just realised that the PredL just says for "all X it is either red or it is not" but no way to say which is or isn't. Are sets different in that they say what is what? Or which is which if you prefer. :)
Last edited by Arising_uk on Mon Oct 23, 2017 1:22 pm, edited 2 times in total.

ficino
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Re: predication and genus

Post by ficino » Mon Oct 23, 2017 2:37 am

Averroes wrote:
Fri Oct 20, 2017 8:41 pm
ficino wrote:If there is a genus F, then is it the case that for all x, if x is F, x must be in the genus F?

If you say that there is a genus F, then you are saying that there are species which are defined with F as their genus. Of such species which belong in genus F, F is predicated of them. Every genus is predicated of the species which fall under it. This means that every genus is a predicate. But not every predicate is a genus. There are predicates which are accidents and properties as well. So you should not confuse these two notions, namely predicate and genus.


Thank you, Averroes.

Re: "not every predicate is a genus": yes, some are too narrow, so to speak. In Aristotle, color is a genus in the category of Quality. It is broken down into black and white (or the "pale" so beloved of translators) and ranges in between. But if I'm only saying, "Socrates is colored," then Socrates or the trope, Socrates' "coloredness," has the genus color predicated of him/it, no?

The point of my question was to wonder whether things have predicates that fall under no genus. I should think that the only candidates for predicates that are in no genus are the so-called transcendentals that cut across categories and genera and thus are not genera, e.g. being, one, true. Otherwise, if x is F, I should think that x is in the genus F, if F is a genus, or if F is a species or subspecies of some genus, x is in that. ???

I'm not applying this to God for obvious reasons. I mentioned First Mover, which of course in the tradition is identified with God, but one has to argue for that identification.

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Re: predication and genus

Post by Averroes » Thu Oct 26, 2017 11:06 pm

wtf wrote:
Sat Oct 21, 2017 4:05 am
Averroes wrote:
Fri Oct 20, 2017 8:41 pm
So, the following discussion must be construed in an Aristotelian logical context. In modern logic, concepts such as predicate is different from that in Aristotelian logic. In a modern context, a predicate is construed as a function, where as in ancient logic it is a term. There are other major differences as well such as the use of quantifiers and truth functional connectives in modern logic, all of which are absent in ancient logic.
Yes that makes sense. I'll have to spend more time with your post to understand what Aristotle means by a term.
In my previous posts you will not find that information.

There are interesting entries in the Standford Encyclopedia of Philosophy (SEP) and also in the internet Encyclopedia of Philosophy (IEP) on Aristotle's logic, which covers Aristotelian logic in an accessible language; both the entries on each of these encyclopedias are named "Aristotle's logic". On the Stanford entry, there is the following:

SEP wrote:4.1 Terms
Subjects and predicates of assertions are terms. A term (horos) can be either individual, e.g. Socrates, Plato or universal, e.g. human, horse, animal, white. Subjects may be either individual or universal, but predicates can only be universals: Socrates is human, Plato is not a horse, horses are animals, humans are not horses.

The word universal (katholou) appears to be an Aristotelian coinage. Literally, it means “of a whole”; its opposite is therefore “of a particular” (kath’ hekaston). Universal terms are those which can properly serve as predicates, while particular terms are those which cannot.

This distinction is not simply a matter of grammatical function. We can readily enough construct a sentence with “Socrates” as its grammatical predicate: “The person sitting down is Socrates”. Aristotle, however, does not consider this a genuine predication. He calls it instead a merely accidental or incidental (kata sumbebêkos) predication. Such sentences are, for him, dependent for their truth values on other genuine predications (in this case, “Socrates is sitting down”).

Consequently, predication for Aristotle is as much a matter of metaphysics as a matter of grammar. The reason that the term Socrates is an individual term and not a universal is that the entity which it designates is an individual, not a universal. What makes white and human universal terms is that they designate universals.
Link: https://plato.stanford.edu/entries/aristotle-logic/#Ter
Yet on another entry on Stanford labelled "Anciet logic", we have the following further specifications:
Aristotle restricts and regiments the types of categorical sentence that may feature in a syllogism. The admissible truth-bearers are now defined as each containing two different terms (horoi) conjoined by the copula, of which one (the predicate term) is said of the other (the subject term) either affirmatively or negatively. Aristotle never comes clear on the question whether terms are things (e.g., non-empty classes) or linguistic expressions for these things. Only universal and particular sentences are discussed.
Link: https://plato.stanford.edu/entries/logi ... #NonModSyl
You might be accustomed to the concept of predicate in modern logic. But it is different in Aristotelian logic. As we have seen above, a predicate is a term in Aristotelian logic, while in modern Frege-Russel logic, it is a function. Here, we are talking about the mathematical notion of a function; which, as you must already know, can be defined as something which takes input(s) as argument(s) and returns a value. In the case of the predicate in modern logic, it takes name(s) of object(s) as arguments and returns the value of either "TRUE" or "FALSE". Let us take a relevant example, and compare the modern and the Aristotelian approach to symbolization.

Statement: God, the Almighty is the Creator of the heavens and the earth and anything in between.

1. In Aristotle logic, this will be called a singular statement, because God, the Almighty is a Unique Being. There are four types of statements Aristotle recognised, and these are singular, universal (e.g. All bodies are divisible), particular (e.g. Some bodies are divisible), and indefinite (e.g. A body is divisible).
So, let G stand for "God, the Almighty", and C stand for "the Creator of the heavens and the earth and anything in between". The symbolism in Aristotelian logic would be: G is C. Here, 'G' is the subject term and C is the predicate term.

2. In moder logic, predicates are construed as functions, which takes name(s) as input(s)/argument(s) and returns a value.

Let Cxy represent: 'x created y' (two place predicate)
Let Bxyz represent: 'x is between y and z' (three place predicate)

'Ref(w)' means the reference of name 'w'.
Let Ref(a)= God, the Almighty; Ref(e)=the earth; Ref(h)=the heavens

Paraphrase: God created the heavens and the earth, and for all x, if x is between the earth and the heavens, then God created x.

Modern logic symbolization of the statement: (Cae & Cae) & ∀x(Bxeh⊃Cax)

You can do this comparison for the other types of statements Aristotle recongnised. And to further understanding, you can also investigate how Aristotle's symbolism is related to his deductive theory of the syllogism, for this is the ultimate purpose of symbolization. These are covered on each of the entries in SEP and IEP.

Averroes
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Re: predication and genus

Post by Averroes » Thu Oct 26, 2017 11:12 pm

wtf wrote:
Averroes wrote:But as for God, the Almighty, such things cannot be said. For God, the Almighty is Absolute Oneness, and there is and can be no species to which He belongs. Since there is no-one like Him, the Almighty, therefore He does not belong in/under any species. As He does not belong to any species, there can also be no genus in which He would belong.
Yes that makes perfect sense. A collection that can't be contained in any other collection. In math that's a proper class. Like the set of all sets.
This is not what I meant by the paragraph you quoted. It has to be remembered that the context of the discussion I am having with ficino is Aristotelian logic and philosophy. And as such, collections such as species and genus in Aristotle philosophy have determinate meanings. In Aristotelian philosophy, a species would be a collection of individuals, for example the species man is the collection of all individual man and woman; the species horse the collection of all individual horses etc... In a species there would be individuals resembling each other in a substantial respect. With the reiteration of this proviso, my replies have to be construed accordingly.

God, the Almighty is not a collection at all. A collection is a manifold or at least implies one. But God, the Almighty is Absolutely One and there is none like Him.
What I meant specifically in the paragraph you quoted, is that: God, the Almighty does not belong to any collection that might be defined as a species or genus because He is Absolutely One and there is none like Him.

When a species is defined, the definition gives the essence of that species. According to Stanford Encyclopedia of Philosophy: "In general, however, it is not individuals but rather species (eidos: the word is one of those Plato uses for “Form”) that have essences. A species is defined by giving its genus (genos) and its differentia (diaphora): the genus is the kind under which the species falls, and the differentia tells what characterizes the species within that genus. As an example, human might be defined as animal (the genus) having the capacity to reason (the differentia)."
_______________________
And therefore there can be no predicate that describes a class larger than God.
Here again,that statement is not what I am saying and it was addressed above. But nevertheless, I reiterate that God, the Almighty is not a collection or a class. God, the Almighty is Absolutely One and there is none like Him.
_______________________
I agree that whatever God is,(...)
God, the Almighty is the Creator of the heavens and the earth and anything in between. God is the All-Wise, the All-Knowing, the Most Compassionate, the Eternal; all beautiful names belong to God, the Almighty. Everyone knows God, the Almighty. This knowledge is, so to speak hardwired in every human being and every other created being. There is none like God, the Almighty. God, the Almighty is One.
(...)God must have the property of not being a member of any collection.
Respectfully, I understand what you mean here, but I will make an important distinction. A collection is too vague here. Please, allow me to explain myself by giving some examples.

For example if by collection one understands a species, which can be defined as the collection of spatio-temporal beings who have similar characteristics such as the species man, horses etc..., then I agree that God, the Almighty does not belong to any such collections or any such kind of collections, because there is none like Him.

But if by collection one understand the collection of the names of beings (unqualified) who possess wisdom and knowledge, then in my perspective, the name of God, the Almighty belongs to such a collection. So in the collection of the name of all beings who have wisdom, there will be the names of all those who have wisdom. And certainly since God, the Almighty is All-Wise, hence His Name must be in that set. However, the Wisdom of God is not limited but our (human) wisdom is limited. The Wisdom of God encompasses all our collective wisdom and goes beyond that, i.e. God's Wisdom is unlimited and infinite.

Here it is important to consider that it is not God, the Almighty Who is being put in a set but only His Name which is being included within a set/collection. When I write the name "Aristotle" on paper, the piece of paper does not itself contain Aristotle (the man) but only contain the name "Aristotle" which refers to Aristotle(the man). So my definition of a collection here is the aggregation of linguistic terms and not the beings to whom those linguistic terms refer to.

Averroes
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Re: predication and genus

Post by Averroes » Thu Oct 26, 2017 11:16 pm

ficino wrote:
Mon Oct 23, 2017 2:37 am

Re: "not every predicate is a genus": yes, some are too narrow, so to speak. In Aristotle, color is a genus in the category of Quality. It is broken down into black and white (or the "pale" so beloved of translators) and ranges in between. But if I'm only saying, "Socrates is colored," then Socrates or the trope, Socrates' "coloredness," has the genus color predicated of him/it, no?
The four labels, namely definition, genus, property and accident, are relationships that a predicate may have with respect to the subject in a statement. In Topics Book 1 chapters 4-9, Aristotle explains that in quite some details. In me saying that "not every predicate is a genus", it must be understood in the context of a statement and with respect to the subject of that proposition. It is not about being too narrow, but it is about context.

Let us take the example you gave: Socrates is colored. In the later example, 'color' is predicated of 'Socrates', so 'color' is a predicate here. Now with respect to 'Socrates', 'color' is a property and not a genus. So here you cannot say "the genus color", but you must say "the property color".
But in the statement: white is a color, the predicate 'color' is the genus of 'white' because 'white' is a kind or species of color. Whereas for the example of Socrates, 'Socrates' was not a species of color, but color was an accidental property of Socrates; meaning that Socrates could have been non-colored.

So you have to be careful. The statement: "Socrates or the trope, Socrates' "coloredness," has the genus color predicated of him/it, no?" is wrong, because 'color' is not the genus of 'Socrates' in "Socrates' coloredness", but it is an accident of Socrates.
You might ask, what would help you in identifying which of the four predicables apply in a given statement. Well, Aristotle provided the test for that.
Aristotle wrote:For every predicate of a subject must of necessity be either convertible with its subject or not: and if it is convertible, it would be its definition or property, for if it signifies the essence, it is the definition; if not, it is a property-for this was what a property is, viz. what is predicated convertibly, but does not signify the essence. If, on the other hand, it is not predicated convertibly of the thing, it either is or is not one of the terms contained in the definition of the subject; and if it is one of those terms, then it will be the genus or differentia, inasmuch as the definition consists of genus and differentia; whereas if it is not one of those terms, clearly it would be an accident, for accident was said to be what belongs to a subject without being either its definition or its genus or property.
[Topics Book 1 chapter 8]
What this means I will explain, if God wills, with some examples.

Suppose we are given a statement, S is P, where P is the predicate term and S is the subject term. Remember we are given "S is P" (pay attention to the order, S followed by P). So Aristotle says that if the statement is convertible, i.e. if "P is S" can be said and has the same meaning as "S is P", then P is either the essence or the property of S. Otherwise, P is either the genus, differentia or accident of S.

The examples, according to Aristotle:

1. "Man is a rational animal" has the same meaning as "A rational animal is a man", because "rational animal" is the definition of man. So, because the subject and predicate are related through definition, they are convertible.

2. Another example, "man is an animal who speaks"; converting we get: "an animal who speaks is man". As we can see both the previous pairs are convertible, but here the predicate "animal who speaks" is convened to be a property of man and not his essence.

3. Yet another example, "man is an animal", here if we convert we get "An animal is man", and we observe that the two previous pairs of statements have not the same meaning. So here they are not convertible. What this means is that 'animal' is not the 'essence' or 'property' of man. It can either be its genus or differentia or accident. And in this case, we know that Aristotle take animal to be the genus of man. You can do the same with "man is rational".

4. Last example, "Socrates is (a) sleeping (thing)". If we convert we get, "a sleeping thing is Socrates". But here we can see that 'a sleeping thing' need not be 'Socrates', and it can even be false when Socrates wakes up. So, the statements are not convertible, so "sleeping" is either a genus, differential or accident of Socrates. We know that 'sleeping' is a temporary attribute of Socrates when he is indeed sleeping, so 'sleeping' is an accident of Socrates.

By employing this method you can test how a predicate relates to a subject in a statement with respect to one of the predicables, i.e. definition, property, genus, differentia and accident.

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Re: predication and genus

Post by Averroes » Thu Oct 26, 2017 11:34 pm

ficino wrote:
Mon Oct 23, 2017 2:37 am
The point of my question was to wonder whether things have predicates that fall under no genus. I should think that the only candidates for predicates that are in no genus are the so-called transcendentals that cut across categories and genera and thus are not genera, e.g. being, one, true.
There is in fact an interesting argument Aristotle gives concerning Being and Unity, and this closely resembles the paradox of the set of all sets.
Aristotle wrote:For if universals are always more truly first principles, clearly the answer will be "the highest genera," since these are predicated of everything. Then there will be as many first principles of things [20] as there are primary genera, and so both Unity and Being will be first principles and substances, since they are in the highest degree predicated of all things. But it is impossible for either Unity or Being to be one genus of existing things. For there must be differentiae of each genus, and each differentia must be one; but it is impossible either for the species of the genus to be predicated of the specific differentiae, or for the genus to be predicated without its species. Hence if Unity or Being is a genus, there will be no differentia Being or Unity. But if they are not genera, neither will they be first principles, assuming that it is the genera that are first principles.
[Metaphysics Book 3 998b]
Aquinas has interpreted this passage thus:
Aquinas wrote:433. That being and unity cannot be genera he proves by this argument: since a difference added to a genus constitutes a species, a species cannot be predicated of a difference without a genus, or a genus without a species. That it is impossible to predicate a species of a difference is clear for two reasons. First, because a difference applies to more things than a species, as Porphyry says; ‘ and second, because, since a difference is given in the definition of a species, a species can be predicated essentially of a difference only if a difference is understood to be the subject of a species, as number is the subject of evenness in whose definition it is given. This, however, is not the case; but a difference is rather a formal principle of a species. Therefore a species cannot be predicated of a difference except, perhaps, in an incidental way. Similarly too neither can a genus, taken in itself, be predicated of a difference by essential predication. For a genus is not given in the definition of a difference, because a difference does not share in a genus, as is stated in Book IV of The Topics; nor again is a difference given in the definition of a genus. Therefore a genus is not predicated essentially of a difference in any way. Yet it is predicated of that which “has a difference,” i.e., of a species, which actually contains a difference. Hence he says that a species is not predicated of the proper differences of a genus, nor is a genus independently of its species, because a genus is predicated of its differences inasmuch as they inhere in a species. But no difference can be conceived of which unity and being are not predicated, because any difference of any genus is a one and a being, otherwise it could not constitute any one species of being. It is impossible, then, that unity and being should be genera.
Reference: http://dhspriory.org/thomas/Metaphysics3.htm#8

And a commentary of modern logician named Joseph Bochenski in his book Ancient Formal logic thus:
Bochenski wrote:Second, Aristotle teaches that "being" and "one" are not genera, i.e. that there is no all-embracing class. The proof runs as follows:
(1) for all A: if A is a genus, there is a B which is its difference;
(2) for all A and B: if B is the difference of A, then A is not the genus of B.
Suppose now that there is an all-embracing genus V; then, for all A, V would be a genus of A[by definition]; but, as V is a genus, it must have some differences, asy B [by (1)]; now V cannot be a genus of B [by(2)]; consequently V is not the all-embracing genus and we get a contradiction.
From the point of view of recent logic, that proof probably offends the rules the theory of types or syntactical categories. Nevertheless, the doctrines of categories is syntactically important: it is an attempt to classify not only objects but also types of objects, and it includes an explicit rejection of the all-comprehending class.
The same result was reached again in 1908, after Aristotle's doctrine had been forgotten.
[Ancient Formal logic pages 34-35]
The reference to 1908 is probably a reference to Zermelo's axiomatisation of set theory. All this means that there can be no all-encompassing class/set.

wtf
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Re: predication and genus

Post by wtf » Fri Oct 27, 2017 10:11 pm

Averroes wrote:
Thu Oct 26, 2017 11:34 pm
The reference to 1908 is probably a reference to Zermelo's axiomatisation of set theory. All this means that there can be no all-encompassing class/set.
I don't agree with this interpretation.

Nothing in set theory says that there can't be an all-encompassing collection, or class. It just can't be a set. For example the collection of all sets is perfectly well defined. It just doesn't happen to be a set.

This idea is a basic part of modern math in the form of category theory. There is a category of sets. It doesn't happen to be a set, but that does not stop us from writing down its properties and proving theorems about it and using it to explore other mathematical structures.

There are many subtleties in the interplay between category theory and set theory. Practitioners are careful about them. But mathematicians consider all-encompassing collections every day. It's a standard part of modern math.
Last edited by wtf on Fri Oct 27, 2017 10:43 pm, edited 1 time in total.

Averroes
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Re: predication and genus

Post by Averroes » Fri Oct 27, 2017 10:41 pm

wtf wrote:
Fri Oct 27, 2017 10:11 pm
Averroes wrote:
Thu Oct 26, 2017 11:34 pm
The reference to 1908 is probably a reference to Zermelo's axiomatisation of set theory. All this means that there can be no all-encompassing class/set.
I don't agree with this interpretation.

Nothing in set theory says that there can't be an all-encompassing collection, or class. It just can't be a set. For example the collection of all sets is perfectly well defined. It just doesn't happen to be a set.
Of course you can disagree. But again, I have to remind you that the conversation I am having on this thread is about ancient logic. And the modern logician's opinion that I was commenting was himself discussing ancient logic in his book I quoted. So all through, the subject matter has been mostly ancient logic. Now, how does this matter? Well, before the distinction between classes, sets, collections and the likes in modern mathematics and logic; all these terms had the same meaning in ancient logic and mathematics. And here again, I am not making this distinction because the subject is ancient logic. And clearly, the paradox of an all-encompassing class/set had been found and explitely rejected by Aristotle, more than 2000 years before the 19-20 century mathematicians rediscovered this. And personally, I find Aristotle's argument to be beautiful.

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Re: predication and genus

Post by wtf » Fri Oct 27, 2017 10:45 pm

Averroes wrote:
Fri Oct 27, 2017 10:41 pm
Of course you can disagree. But again, I have to remind you that the conversation I am having on this thread is about ancient logic. And the modern logician's opinion that I was commenting was himself discussing ancient logic in his book I quoted. So all through, the subject matter has been mostly ancient logic. Now, how does this matter? Well, before the distinction between classes, sets, collections and the likes in modern mathematics and logic; all these terms had the same meaning in ancient logic and mathematics. And here again, I am not making this distinction because the subject is ancient logic. And clearly, the paradox of an all-encompassing class/set had been found and explitely rejected by Aristotle, more than 2000 years before the 19-20 century mathematicians rediscovered this. And personally, I find Aristotle's argument to be beautiful.
Of course I agree that the main discussion is about ancient logic, about which I'm admittedly ignorant.

But you made your claim in the context of Zermelo's axiomitization of set theory. My comments are valid in that context. Once you invoked 1908 you left the domain of classical logic. Surely that's fair to point out. If I misunderstood your reference to Zermelo or 1908 I'll accept clarification. When you said, "The reference to 1908 is probably a reference to Zermelo's axiomatisation of set theory. All this means that there can be no all-encompassing class/set," perhaps I misunderstood the referent of "this." I took it as referring to Zermelo, not Aristotle.

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Re: predication and genus

Post by Averroes » Fri Oct 27, 2017 11:18 pm

wtf wrote:
Fri Oct 27, 2017 10:45 pm
But you made your claim in the context of Zermelo's axiomitization of set theory. My comments are valid in that context. Once you invoked 1908 you left the domain of classical logic. Surely that's fair to point out. If I misunderstood your reference to Zermelo or 1908 I'll accept clarification. When you said, "The reference to 1908 is probably a reference to Zermelo's axiomatisation of set theory. All this means that there can be no all-encompassing class/set," perhaps I misunderstood the referent of "this." I took it as referring to Zermelo, not Aristotle.
The reference to 1908 is not mine, but that of a modern logician Joseph Bochenski in his book Ancient logic. The reference to Zermelo is my tentative interpretation, I said "probably", for this is what easily comes to mind when this year and subject is mentioned, and in the context of the statement it makes sense to me. But again, if you impose modern distinctions on the ancients, it will be unfair and anachronistic. The claim was not made in the context of Zermelo but Zermelo was taken as an example of a recognition by the moderns of the inherent contradiction of the set of all sets and an attempt to find a solution to it. The parallel is that Aristotle found this more than 2000 years before and he too tried to wrestle his way out with the means that he had! That is as far as the reference goes, and it is already something tremendous, at least for me, to testify with my own eyes and mind the achievements of Aristotle. With all due respect, going into the modern distinctions between sets and classes and the likes, is irrelevant and misses the point here.

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Re: predication and genus

Post by wtf » Sat Oct 28, 2017 1:20 am

@Arising_uk,

I want to mention something that might put what follows into context.

In grade school they tell us that a set is a "collection of objects." That leads us into error, as shown by Russell's paradox.

Rather, a set is a very particular technical gadget in set theory. In fact if you ask a set theorist what is a set, they'll tell you that nobody knows. A set is whatever satisfies the axioms of set theory. [I don't know if every set theorist would say that. Some would, I'm sure. I'm making a point about sets, not about set theorists!]

When I say something's a set, I only mean that it satisfies the axioms of set theory. I'm not adding any mental context like "collection," or "concept defined by a predicate," or anything like that. In other words sometimes it's helpful to put on our formalist hat.
Arising_uk wrote:
Sun Oct 22, 2017 5:01 pm
Are we talking about domains? If so could we get away with the domain being all the objects in the universe and then Vx(R(x) v ¬R(x)) which would 'give' you all the red objects?
I am concerned that you are trying to find some way to characterize the collection of all red things. It makes me think we're not entirely on the same page.

It's simple to define the collection of all red things. It's "x such that x is red." That is a predicate-based definition of the collection, or the concept, of red things.

It might or might not be a set. It depends on whether your universe is a set. To know if something's a set, you ask if it can be a member of some class. The collection of all red things is a member of the universe. So if the universe is a set, so is the collection of red things. If not, then you haven't got an enclosing set so your collection isn't a set.

Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
Sort of but it's a heavy read nowadays for this old man. :)
Yeah the page is a little heavy going.

Do you understand Russell's paradox? It's worth going through.

Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
p.s.
Sorry, just realised that the PredL just says for "all X it is either red or it is not" but no way to say which is or isn't.
Agreed. We have the collection of red things. The question isn't how to express it as a predicate or formula. The question is whether the resulting collection is a set.

"Set" is a loaded term. It's a technical term. It's nothing like the sets they teach you about in grade school. That's the takeaway here. That would be my tl;dr.
Arising_uk wrote:
Sun Oct 22, 2017 12:26 am
Are sets different in that they say what is what? Or which is which if you prefer. :)
Well that is a very interesting question. Sets are a LOT different than things that say "what is what."

I hope you don't mind if I talk about this a little. It goes beyond the current discussion but it gets at the nature of mathematical sets.

There are many sets that have no definition at all. Their membership is essentially random.

As one way to see this, consider the set of natural numbers 1, 2, 3, 4 [with or without 0, I don't care]. This is a countably infinite set.

In set theory if we have a set we may form its powerset, which is the set of all subsets of the original set. And we know from Cantor that the powerset of the natural numbers is an uncountably infinite set.

However if we look at all the predicates there could possibly be, there are only countably many of them. Syntactically a predicate is a finite-length string of symbols and there are only countably many finite strings of symbols as long as the alphabet is at most countable.

There "aren't enough" predicates to be able to characterize every set with a predicate. In set theory we believe in the existence of sets whose contents could never be described by a finite-length description, like "all the red things." Most sets of natural numbers are random. There is nothing you can say about them other than to list their elements. And any such list must be infinite, and not finitely describable in any way.

Philosophically, if you believe that in order for a thing to exist there must be an algorithm that cranks it out, then the full powerset of the natural numbers does not exist.

In some philosophies of math such as various flavors of constructivism and intuitionism, the full powerset of the natural numbers doesn't exist.

In standard set theory, we just allow it all. If an axiom says a set exists, then that set exists, even if we have no way to describe its elements.

Summing it all up, predicates are both too strong and too weak to define sets.

They're too strong in the sense that a predicate like "x is a set" defines a collection that's too big to be a set.

They're too weak in the sense that most sets aren't defined by any predicate at all.

wtf
Posts: 615
Joined: Tue Sep 08, 2015 11:36 pm

Re: predication and genus

Post by wtf » Sat Oct 28, 2017 1:55 am

Averroes wrote:
Fri Oct 27, 2017 11:18 pm
With all due respect, going into the modern distinctions between sets and classes and the likes, is irrelevant and misses the point here.
I fully agree. I'm hitting the nail with the only hammer I have. I wanted to mention that since I just made a long post along the same admittedly irrelevant lines.

I'll take some more time to digest the rest of your very interesting post.

Averroes
Posts: 182
Joined: Thu Jul 20, 2017 8:48 pm

Re: predication and genus

Post by Averroes » Sat Oct 28, 2017 6:22 am

wtf wrote:
Sat Oct 28, 2017 1:55 am
Averroes wrote:
Fri Oct 27, 2017 11:18 pm
With all due respect, going into the modern distinctions between sets and classes and the likes, is irrelevant and misses the point here.
I fully agree. I'm hitting the nail with the only hammer I have. I wanted to mention that since I just made a long post along the same admittedly irrelevant lines.

I'll take some more time to digest the rest of your very interesting post.
Well, concerning my posts, all the nails were already hammered in. I am a neat carpenter, at least in my perception! Of course, it goes without saying that discussing set theory in the proper context, without imposing it on the ancients, is not at all irrelevant and it is even encouraged if it is to educate others who show an interest in understanding it. So please do not let the proviso in my post impede you to share your thoughts.

Belinda
Posts: 1932
Joined: Fri Aug 26, 2016 10:13 am

Re: predication and genus

Post by Belinda » Sat Oct 28, 2017 7:02 pm

There is an infinity of what we , or any individual, might possibly predicate of any given phenomenon.

The set of unmanifested + manifested predicates is therefore infinite, but not the set of manifested predicates which, given that time is finite, is finite.
If God is that which transcends time and also is immanent in time then God is the set of infinite and unmanifested sets, which includes the set of manifested sets.
Am I right?

I agree with ficino:
If there is a genus F, then is it the case that for all x, if x is F, x must be in the genus F? (In asking this question I leave out the problem of analogical predication of names of God.)

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