predication and genus

What is the basis for reason? And mathematics?

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ficino
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predication and genus

Post by ficino » Thu Oct 12, 2017 3:24 pm

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.)

Thank you, f

wtf
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Re: predication and genus

Post by wtf » Fri Oct 13, 2017 1:50 am

What is a genus in this context? There's a technical term genus in math, but it's quite far removed from anything your question might possibly be about.

If you mean some sort of logical or conceptual category, your question suffers from vagueness. For example if by genus you mean set, then you run into Russell's paradox. But it by genus you mean proper class, then what you say is true. If x is F then x is in F.

I totally fail to understand what God has to do with this.

ficino
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Re: predication and genus

Post by ficino » Fri Oct 13, 2017 3:29 am

Many thanks for answering.

I am thinking of genus as in Aristotle's Categories and throughout his works. In Ari, a genus is a "kind," which is divided into species by various differentiae. I.e. "There are kinds in the sense in which plane is the kind of plane figures and solid of solids; for each of the figures is in the one case a plane of such and such a kind, and in the other a solid of such and such a kind; and this is what underlies the differentiae. Again, in formulae their first constituent element, which is included in the essence, is the kind, whose differentiae the qualities are said to be." Metaphysics V.28, 1024a36-b6. So, e.g. "biped," of which "man" is a species. Or "color," in which white/pale is a species.

I'm wondering whether, if we say there is a genus of movers, the First Unmoved Mover must be in the genus of movers if it is to be a mover. Aquinas does not want God to be in any genus. But he also argues to God's existence by identifying God with the First Mover. And in Aquinas' argument from motion, doesn't the First Mover need to be in a genus of movers for the argument to go through? It doesn't seem to me that one can say that the First Mover is supreme in an ordered series of movers AND deny that the First Mover is in a genus of movers. So the First Mover seems not to be identical with Aquinas' God.

I'm sorry, I don't know enough to know the difference between "set" and "proper class"! And I am mystified by the Wikipedia link, but thank you for linking it. You can see that my background is not in math or modern logic.

wtf
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Re: predication and genus

Post by wtf » Fri Oct 13, 2017 3:52 am

ficino wrote:
Fri Oct 13, 2017 3:29 am

I'm wondering whether, if we say there is a genus of movers, the First Unmoved Mover must be in the genus of movers if it is to be a mover. Aquinas does not want God to be in any genus. But he also argues to God's existence by identifying God with the First Mover. And in Aquinas' argument from motion, doesn't the First Mover need to be in a genus of movers for the argument to go through? It doesn't seem to me that one can say that the First Mover is supreme in an ordered series of movers AND deny that the First Mover is in a genus of movers. So the First Mover seems not to be identical with Aquinas' God.
This seems out of place on the philosophy of math subforum. It has nothing to do with math. I noticed that you usually post in the religion section. Perhaps that might be the right place for your question. Questions about God and first movers are not any part of math.

But since we're here, why must there be a first mover? The standard mathematical counterexample is the negative integers ..., -4, -3, -2, -1. In this model, each number has a predecessor (or "mover" if you like) yet there is no first mover. This model comes up all the time in discussions about William Lane Craig's cosmological argument. You could if you like think of each number as an "event" that is "caused by" the event immediately to its left. Every event has a cause, yet there is no first cause.

That's the thing about math. Given any metaphysical or physical theory, math can whip up a model. Math itself is agnostic about what's true. Math is a toolkit for building models so that we can talk about them with precision. But it doesn't tell you whose model is the right one to describe the state of our world.

ficino
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Re: predication and genus

Post by ficino » Fri Oct 13, 2017 3:28 pm

Thank you. I've only posted 6 times on here, or thereabouts. But I'll try moving this to the religion subforum. I posted here because I thought the question of predication and genus had to do with logic.

Averroes
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Re: predication and genus

Post by Averroes » Fri Oct 13, 2017 6:06 pm

Thank you. I've only posted 6 times on here, or thereabouts. But I'll try moving this to the religion subforum. I posted here because I thought the question of predication and genus had to do with logic.
It is alright ficino, it is not only a math subforum, but also a logic subforum. And the OP has a place here. Give me some time I will reply to you if God wills. Do not worry.

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Harbal
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Re: predication and genus

Post by Harbal » Fri Oct 13, 2017 6:11 pm

Averroes wrote:
Fri Oct 13, 2017 6:06 pm
I will reply to you if God wills.
What possible objection could God have?

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attofishpi
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Re: predication and genus

Post by attofishpi » Sun Oct 15, 2017 11:07 pm

Harbal wrote:
Fri Oct 13, 2017 6:11 pm
Averroes wrote:
Fri Oct 13, 2017 6:06 pm
I will reply to you if God wills.
What possible objection could God have?
No, it's willy not wonte.

wtf
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Re: predication and genus

Post by wtf » Mon Oct 16, 2017 2:02 am

ficino wrote:
Fri Oct 13, 2017 3:28 pm
Thank you. I've only posted 6 times on here, or thereabouts. But I'll try moving this to the religion subforum. I posted here because I thought the question of predication and genus had to do with logic.
I did some Googling and evidently this is something to do with ancient logic. I don't mean to discourage anyone from talking about classical logic. I don't know anything abut it and can be of no help. What I read does not seem to have anything to do with math as I understand it, but maybe it did back in the day. The examples are triangles being part of the genus polygon, So the examples are mathematical even if I don't recognize it as the kind of math I know. So maybe there's some philosophical aspect that's beyond my limited knowledge.

If anyone wants to explain more about this I'm certainly interested in learning. I still don't see the connection to God.

ps -- What is "analogical predication of names of God?" I would like to know what that means so that I can learn something. I do remember a story by Arthur C. Clarke called The Nine Billion Names of God.

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Re: predication and genus

Post by Averroes » 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?
Genus, differentia, species and the likes are found in Aristotelian logic. Aristotelian logic has now been superseded by First Order Logic. 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.

A genus is a larger class containing under it smaller classes. In the statement ‘Man is a mammal’ mammal is a genus with respect to man.

A species is a smaller class contained under a larger one. In the statement: ’Aristotle is a man’, man is species with respect to Aristotle.

When one talks of genus, one is also talking of species and individuals. An Individuals belongs in/under a species and a species belongs in/under a genus. So if one says that x belongs in/under the genus F, then either x is a species or x is an individual. If x is an individual, then it must also belong in a species, and the species will belong in a genus.

For an individual man such as for example Aristotle, it can be said that he belonged to the species man, and as man is a species which belong to the genus mammal, Aristotle also belonged in the genus mammal.

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.

The genus and the differentia together make up the definition of a species. The definition of something expresses the essence of that something. Now according to Aristotle thoughts, only species can be defined. From Stanford Encyclopedia of Philosophy in the entry on Aristotle’s logic, we have this:
Since a definition defines an essence, only what has an essence can be defined. What has an essence, then? That is one of the central questions of Aristotle’s metaphysics; once again, we must leave the details to another article. 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).
Reference link: https://plato.stanford.edu/entries/aristotle-logic/#Def

Now from what Aristotle says in the categories, it is clear that the notion of a genus and a species applies to more than one thing:
A substance-that which is called a substance most strictly, primarily, and most of all--is that which is neither said of a subject nor in a subject, e.g. the individual man or the individual horse. The species in which the things primarily called substances are, are called secondary substances, as also are the genera of these species. For example, the individual man belongs in a species, man, and animal is a genus of the species; so these-both man and animal-are called secondary substances.
(…)
Every substance seems to signify a certain 'this'. As regards the primary substances, it is indisputably true that each of them signifies a certain 'this'; for the thing revealed is individual and numerically one. But as regards the secondary substances, though it appears from the form of the name-when one speaks of man or animal—that a secondary substance likewise signifies a certain 'this', this is not really true; rather, it signifies a certain qualification, for the subject is not, as the primary substance is, one, but man and animal are said of many things. However, it does not signify simply a certain qualification, as white does. White signifies nothing but a qualification, whereas the species and the genus mark off the qualification of substance-they signify substance of a certain qualification. (One draws a wider boundary with the genus than with the species, for in speaking of animal one takes in more than in speaking of man.) [Categories chapter 5]
So, the genera and species are only said of many things.

But since God, the Almighty, is Absolute Oneness, and there is none like Him, therefore there is and can be no species to which He belongs. But that now does not mean that nothing can be predicated of God, the Almighty. But what can be said is this: that which is predicated of God, the Almighty cannot be said to be a genus or a differentiae or species.

Genus, species and differentia belong to the class of things known as the predicables in ancient logic. The other predicables are: property and accident.
ficino wrote:…if we say there is a genus of movers, the First Unmoved Mover must be in the genus of movers if it is to be a mover.
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.


Anyway, that is why, I would, as well, not include God, the Almighty in any genus or species.

Averroes
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Re: predication and genus

Post by Averroes » Fri Oct 20, 2017 8:42 pm

Harbal wrote:What possible objection could God have?
This is a good question, but I cannot reply to such a question in this thread as it is off-topic. If you are still interested in the answer, please follow the following link to my forum, I have answered the question there: http://philosophyforum.aba.ae/viewtopic.php?f=8&t=6

wtf
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Re: predication and genus

Post by wtf » Sat Oct 21, 2017 4:05 am

Averroes wrote:
Fri Oct 20, 2017 8:41 pm
Aristotelian logic has now been superseded by First Order Logic.
Oh that is awesome. Something I can relate to. Your post made this entire thread finally make sense to me. Thanks much!
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.
Averroes wrote:
Fri Oct 20, 2017 8:41 pm
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 reminds me of Cantor, the founder of set theory. He had religious beliefs that are forgotten today but that were personally important to him. He believed that beyond his endless hierarchy of infinities was Absolute Infinity, which he identified with God. Mathematicians still study Cantor's infinities, but nobody thinks God is the supremum of that process.

I agree that whatever God is, God must have the property of not being a member of any collection. And therefore there can be no predicate that describes a class larger than God.

Is that about right?

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Arising_uk
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Re: predication and genus

Post by Arising_uk » Sat Oct 21, 2017 9:34 pm

wtf wrote:...
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. ...
What is a class then? For myself the 'set of all sets' sounds like it must be a set?

wtf
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Re: predication and genus

Post by wtf » Sat Oct 21, 2017 9:45 pm

Arising_uk wrote:
Sat Oct 21, 2017 9:34 pm
wtf wrote:...
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. ...
What is a class then? For myself the 'set of all sets' sounds like it must be a set?
Can't be. If the set of all sets is a set, then we can reduce it via a predicate to form the set of all sets that are not members of themselves. But that set both is and isn't be a member of itself, contradiction. So the set of all sets is not a set. And more generally, we may not form a set from a predicate. Rather, we can only form a set by applying a predicate to an existing set.

So for example if R(x) is a unary predicate meaning "x is red," then we may NOT form the set of all red things. In grade school, a set is just a collection. But in formal math, a set is much more restrictive. Sets can only be formed in carefully specified ways. Unrestricted formation via a predicate is not allowed.

Lot of nice history behind all this. Frege proposed that sets were defined by predicates, and Russell found the hole in that idea.

https://en.wikipedia.org/wiki/Russell%27s_paradox

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Arising_uk
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Re: predication and genus

Post by Arising_uk » Sun Oct 22, 2017 12:26 am

wtf wrote:Can't be. If the set of all sets is a set, then we can reduce it via a predicate to form the set of all sets that are not members of themselves. But that set both is and isn't be a member of itself, contradiction. So the set of all sets is not a set. And more generally, we may not form a set from a predicate. Rather, we can only form a set by applying a predicate to an existing set.

So for example if R(x) is a unary predicate meaning "x is red," then we may NOT form the set of all red things. In grade school, a set is just a collection. But in formal math, a set is much more restrictive. Sets can only be formed in carefully specified ways. Unrestricted formation via a predicate is not allowed. ...
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?

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?

Also, is Vx(R(x)) not the same as the set of all red things?

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