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Plato, Self-Predication, the Verb to Be and the Arity of Relations

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ChanceIsChange
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Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ChanceIsChange » December 16th, 2018, 8:43 am

Hello,

In Plato's dialogues, it is sometimes stated that that every Form is itself. For example, the Form of Justice is said to be just. I am wondering whether some characters in Plato's dialogues commit linguistic errors regarding this self-predication of Forms. Specifically, I stumbled over the following two.

a) Imho, not every Form is itself in the sense above. As an example, take the Form of Evenness. This is the property of being divisible by 2. Obviously, only numbers can have this property. But Evenness itself is a Form and not a number. Therefore, it can't possibly be even, i.e. divisible by 2. But of course, Evenness is obviously itself in that it is identical itself.
As far as I can see, there are (at least) four distinct meanings of the word "is":

1. Existence; as in "The Sun is", which means the same as "The Sun exists."

2. The (has-as-its-essence)-relation, which hold between something that is and its essence; perhaps as in "Evenness is to be divisible by 2", which means the same as "To be divisible by 2 is the essence of Evenness."

3. Characterization, the (has-as-a-property)-relation; as in "The Sun is bright", which means the same as "The Sun has brightness as a property."
3.5. The (belongs-to)-relation; as in "The Sun is a star", , which means the same as "The Sun belongs to the class of all stars." Since classes are extensions of properties, the (belongs-to)-relation is essentially a special variant of the (has-as-a-property)-relation. The sentence "The Sun belongs to the class of all stars" means the same as "The Sun has starness."

4. The identity relation; as in "The Sun is the central star of the Solar System", which means the same as "The Sun is identical to the central star of the Solar System."

Do Plato's characters mix up these last two meanings of the word "is" when they say that every Form is identical to itself? Obviously, every Form is identical to itself, but, as is shown by the Form of Evenness, not every Form characterizes itself. By realizing the latter, we can also solve the problem of the third man.


b) Another problem with self-predication is demonstrated by the Form of Equality. Plato's characters have stated that Equality is equal. If they mean that in the sense of (a.3), then wouldn't they commit another fallacy? Here is why I think they would. Equality is a binary relation, not a unary one. Therefore, a sentence like "Equality is equal", if understood in the sense (a.3), is not just false, like the sentence "Evenness is even", but even meaningless, since the binary relation of equality cannot be predicated of a single entity, including itself. Saying "Equality is equal" is like saying "the number 5 is less than" and then stopping.
In my opinion, the fact that some Forms, such as Equality, are binary relations and not unary ones proves that not every Form characterizes itself.


My preliminary conclusion is that self-predication of Forms should not be understood as self-characterization. Is that conclusion correct? If yes, did Plato really let some of his characters make the elementary mistakes of confusing the different arities of different relations and the different meanings of the word "is", and why? If no, did Plato himself make such basic mistakes?

Regards,

ChanceIsChange

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by Fooloso4 » December 16th, 2018, 2:03 pm

@ChanceIsChange

I think you are on the right track in rejecting self-predication.

ChanceIsChange:
As an example, take the Form of Evenness. This is the property of being divisible by 2.
A and B are not even because each can be divided by two. They are even because one is not greater than the other. A and C may be uneven but each can be divided into two with each one (each half) of A and of C being even.
Do Plato's characters mix up these last two meanings of the word "is" when they say that every Form is identical to itself?
Each form is one, singular, unique. There is some ambiguity to the notion of identity There is a sense in which identity entails difference, a comparison of one thing to another. A and B are identical if in some sense there is no difference, but unless B is another name for A they are not one but two. Since each may be composed of parts and the part is not identical to the whole there is another problem of identify. A man is not the same as his finger. The finger is not the essence of a man. An essence is literally “what it is to be”. The question of what it is to be a man requires making a distinction between what is and what is not a man.

Since the forms are singular there is no part of the form that is not what it is. The Form Even cannot itself be even because to be even means to relate one thing to another. The Even cannot be divided into parts, for to be one is to be without parts, indivisible. In Greek mathematics the ‘one’ is the unit of measure. The one is that which is counted. Two is two ones. If two is divided there is no longer two, but if one is divided we have two ones. Irrational numbers arise when there is no common unit of measure, where there is no one. Note that in the example above when A is divided a unit of measure is preserved. It is not the same unit of measure but there is still a unit of measure, a one.

The Forms are in this sense units of measure, the ‘one’ by which we can say (provided the Form is known) that this act is just and that one is not. If we know the Form or one ‘man’ we can identify this as a man but not that.

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ktz » December 16th, 2018, 5:51 pm

I'm not sure that I'm familiar enough with the source material to be a useful contributor here, but maybe you can humor me as I wonder if there's some relevant issue with the use-mention distinction in the argument you are making.

Correct me if I'm wrong, but I thought Plato's Theory of Forms was basically asserting that the physical world imperfectly models the most ideal version of something in some ultimate reality where there exists the true forms of things. In the realm of forms, there is an ideal form of a chair, more perfect than all physical chairs, and an ideal form of a table, etc. So with my perhaps incorrect understanding, to say that "The Form of Evenness is not even" is to criticize the mention of the form of evenness in the use-mention distinction. The use of the Form of Evenness, being an ideal from the realm of forms, is the even-est thing possible which all other even things are modeled after, so it would meet the criteria of being even -- all even numbers would be shadows of this ideal form of evenness which is perhaps described mathematically by the number n such that n/2 is element of the integers. The Form of Equality would be incomprehensible in the physical world, but we see shadows in the physical realm when we describe things as being equal or not.

I have my issues and complaints with this theory on a more general basis, but given my limited understanding, I would disagree that any basic mistakes are being made by Plato here.
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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ChanceIsChange » December 17th, 2018, 5:55 am

Fooloso4 wrote:A and B are not even because each can be divided by two. They are even because one is not greater than the other. A and C may be uneven but each can be divided into two with each one (each half) of A and of C being even.
Aren't you talking about Equality here rather than Evenness? I meant Evenness in the number-theoretic sense, that is as the property (unary relation) of being a number whose half is a whole number. Didn't Plato mean the same thing when using (the Greek equivalent of) the word "Evenness", for example when stating that the number 4 is even?
Fooloso4 wrote:There is some ambiguity to the notion of identity There is a sense in which identity entails difference, a comparison of one thing to another. A and B are identical if in some sense there is no difference, but unless B is another name for A they are not one but two.
I think that our notion of identity in this context is quite unambiguous and precise. Identity is the binary relation that relates every entity to itself and nothing else. So, yes, entities A and B are identical to each other if and only if "A" and "B" are names for one and the same thing.
Fooloso4 wrote:Since the forms are singular there is no part of the form that is not what it is. The Form Even cannot itself be even because to be even means to relate one thing to another.
This is similar to my argument about Equality below, but I think (please correct me if I have misunderstood something) that you're mixing up Evenness with Equality.
ktz wrote:I'm not sure that I'm familiar enough with the source material
If I've misunderstood some things about Plato, I would also be happy for a correction.
ktz wrote:So with my perhaps incorrect understanding, to say that "The Form of Evenness is not even" is to criticize the mention of the form of evenness in the use-mention distinction.
You mean that I'm saying that the word "Evenness" is not even, right? Actually, that's not what I'm saying. What I'm saying is that the Form of Evenness, which is what the word "Evenness" refers to, is not even, i.e. does not have Evenness. If I wanted to say that the word "Evenness" is not even, I would say ""Evenness" is not even" instead of "Evenness is not even", that is, I would use the expression ""Evenness"" to refer to the word "Evenness" and not use the word "Evenness" to refer to the Form of Evenness. If you still think that there's an issue with use and mention, please tell me where.
ktz wrote:I thought Plato's Theory of Forms was basically asserting that the physical world imperfectly models the most ideal version of something in some ultimate reality where there exists the true forms of things.
I also understand the Theory of Forms (ToF) in that way, and I have my reservations about it. There are (at least) two ways the Forms can be understood: as properties or as prototypes. I have no problem with the former, but I have qualms about the latter. Here is why.
ktz wrote:The use of the Form of Evenness, being an ideal from the realm of forms, is the even-est thing possible which all other even things are modeled after, so it would meet the criteria of being even -- all even numbers would be shadows of this ideal form of evenness which is perhaps described mathematically by the number n such that n/2 is element of the integers.
The only candidates for even entities are numbers, since only they can be divided by the number 2. A chair, for example, can't be divided by 2 in the first place. Therefore, the purportedly most even thing, the Form of Evenness understood as a prototype, must also be a number. Now, every number is either divisible bu 3, or it isn't. Also, there are even numbers divisible by 3 and others that aren't. The prototypical Evenness, however, can probably only have properties common to all even numbers. In any case, it can neither be divisible by 3 nor not divisible by 3 since both of these are particular properties unrelated to evenness. That, however, contradicts the fact that Evenness is number and therefore either divisible by 3 or not. Therefore, I believe that Evenness is not itself even. Rather, I think that it is a Form understood as a property and neither a number nor a prototype.

According to my opinion, another example that puts self-predication of all Forms in an even greater predicament is given by the Form of Equality. Equality is a binary (2-ary) relation. Therefore, it seems impossible for Equality to be equal, since that would require Equality to be a unary relation, that is a property, and not a binary one. For me, the only way out of this problem appears to lie in understanding binary relations as properties of ordered pairs. In that case, for Equality to be equal, it would have to be the ideal, prototypical ordered pair of entities equal to each other. But wouldn't that contradict the simplicity, i.e. incomposite-ness and oneness, of the Form of Equality?

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ktz » December 17th, 2018, 12:28 pm

ChancelsChange wrote:
ktz wrote:So with my perhaps incorrect understanding, to say that "The Form of Evenness is not even" is to criticize the mention of the form of evenness in the use-mention distinction.
You mean that I'm saying that the word "Evenness" is not even, right? Actually, that's not what I'm saying. What I'm saying is that the Form of Evenness, which is what the word "Evenness" refers to, is not even, i.e. does not have Evenness. If I wanted to say that the word "Evenness" is not even, I would say ""Evenness" is not even" instead of "Evenness is not even", that is, I would use the expression ""Evenness"" to refer to the word "Evenness" and not use the word "Evenness" to refer to the Form of Evenness. If you still think that there's an issue with use and mention, please tell me where.
I'm not saying that you're referring to the word Evenness, but you are conflating the your current concept of "Evenness" with the Form of Evenness, and trying to apply your logical intuition when Plato's dialogues would indicate it has a transcendent character beyond the imagining and intuitive reasoning of us lowly physical world humans. The Form of Evenness, as I understand Plato's dialogues are trying to say, is the most transcendently Even thing possible. This form transcends our physical world your intuitions about Evenness and so the Form of Evenness can be even just by Plato's definition, even if it contradicts our intuitive reasoning about what is or isn't even and we have no idea why, and lack any intuitive ability to understand what some transcendent Form of Evenness could even look like.


I also understand the Theory of Forms (ToF) in that way, and I have my reservations about it. There are (at least) two ways the Forms can be understood: as properties or as prototypes. I have no problem with the former, but I have qualms about the latter. Here is why.



The only candidates for even entities are numbers, since only they can be divided by the number 2. A chair, for example, can't be divided by 2 in the first place. Therefore, the purportedly most even thing, the Form of Evenness understood as a prototype, must also be a number. Now, every number is either divisible bu 3, or it isn't. Also, there are even numbers divisible by 3 and others that aren't. The prototypical Evenness, however, can probably only have properties common to all even numbers. In any case, it can neither be divisible by 3 nor not divisible by 3 since both of these are particular properties unrelated to evenness. That, however, contradicts the fact that Evenness is number and therefore either divisible by 3 or not. Therefore, I believe that Evenness is not itself even. Rather, I think that it is a Form understood as a property and neither a number nor a prototype.

According to my opinion, another example that puts self-predication of all Forms in an even greater predicament is given by the Form of Equality. Equality is a binary (2-ary) relation. Therefore, it seems impossible for Equality to be equal, since that would require Equality to be a unary relation, that is a property, and not a binary one. For me, the only way out of this problem appears to lie in understanding binary relations as properties of ordered pairs. In that case, for Equality to be equal, it would have to be the ideal, prototypical ordered pair of entities equal to each other. But wouldn't that contradict the simplicity, i.e. incomposite-ness and oneness, of the Form of Equality?
I think complaining about the pragmatic uselessness of Plato's prototypical Forms is reasonable, and is part of why Plato's Theory of Forms is probably a bit obsolete of an idea, but allow me to play the devil's advocate in this case. Your intuition about numbers are based on your past experience with human constructions and subject to human fallibility in understanding. For example, your past usages of "Equality" have limited its arity to 2 usages. But in Polish notation, for example, there is no such restriction on the arity of the equality operator:

(= (2 (1+1) (3-1) (4/2) ) ) is an expression that resolves to true.

Basically what I am saying is that your reasoning in this case is subject to your fallible human intuition about the equality operator, the principle of bivalence, and booleans. Just like how you could write a programming language that accepts (= 1) as a reflexive expression that resolves to true, The Realm of Forms appears to lack the limitations of human language and mathematics like restrictions based on Zermelo-Fraenkel set theory upon we have built human conceptions of truth in mathematics.


Similarly, I might contest your proposition that the application of even and odd is necessarily restricted to numbers. Evenness is a concept with a specific character in biodiversity as well, referring to species evenness in ecological systems, and even and odd have characteristic applications in literary analysis, with evenness being a property of God and Oddness being a property of the devil. Somehow, Plato's theory of forms would subsume all these applications of evenness somehow, or the Form of Evenness would reveal the flawed nature of mankind's ideas about Evenness when our applications do not resemble the Form of Evenness.

So although I might not believe it exists, conceptually I think one implication for you to consider is that there may be many unknown unknowns that can hinder our fallible human reasoning about perfection without our awareness. Your issue with prototypical nature perhaps lacks the considerations of what is the true understanding of the ideal form of these abstractions -- perhaps through no fault of your own, but the fallibility of mankind's understanding as a whole. Consider, for example, the possibility that our physical world brains just aren't big enough to appreciate the true character of these Forms?
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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by Fooloso4 » December 17th, 2018, 12:52 pm

ChanceIsChange:
I meant Evenness in the number-theoretic sense, that is as the property (unary relation) of being a number whose half is a whole number.
It is important to see that Greek number theory is not the same as modern theories. There is no zero. Two is the first number. Arithmos means to count. There must be a unit that is counted, some one. An even number means it can be divided into two parts without one left over. For example, if five apples are grouped into two there will always be one apple that cannot be in one of the groups if both are to be equal, that is, if there is to be an even number in each.
Didn't Plato mean the same thing when using (the Greek equivalent of) the word "Evenness", for example when stating that the number 4 is even?
The number 4 is even because it can be divided into two equal parts, each with the same number of ‘ones’.
I think that our notion of identity in this context is quite unambiguous and precise. Identity is the binary relation that relates every entity to itself and nothing else. So, yes, entities A and B are identical to each other if and only if "A" and "B" are names for one and the same thing.
But Forms only have one name that name the one thing that it alone is.
This is similar to my argument about Equality below, but I think (please correct me if I have misunderstood something) that you're mixing up Evenness with Equality.
The are not the same but related, although we do say, divide the pie evenly, which is to say into equal parts. The key is the Greek understanding of number. A number (which is always a number of units) is even if it can be divided into two equal parts. A and B are equal if the number of units of measure of each are even, without one having more than the other.

What is more important is the connection between ‘one’ and ‘Form’. Just as the one allows us to say how many, the Form allows us to say what something is. For example, if I point to a bowl of fruit and ask how many, the question cannot be answered or will be answered differently depending on how many of what is to be counted - pieces of fruit, apples instead of oranges. If we are to determine whether an action is just, we need to know the measure, the one, the eidos or idea or form or kind or look (all terms that translate eidos) that allows us to identify the act as just or not and to give an account.

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by JaxAg » December 17th, 2018, 1:29 pm

Is Russell's paradox related to this problem? Consider the set of all sets that are members of themselves. Is the set of sets that include themselves a member of the set of 'sets that include themselves' or not?

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by JaxAg » December 18th, 2018, 6:49 am

Ooops! Just reread the above post and realised I missed out the words 'do not'. (As mistakes go, this is a biggy). To clarify by an example: The set of all cars is not a car, so it's not a member of itself. The set of all mathematical objects is itself a mathematical object, so it is a member of itself. So we can identify two new sets (maybe supersets?): the set M of all sets that are members of themselves, and the set N of all sets that are not members of themselves. If N is not a member of itself, then it is a member of itself. And vice versa. I was wondering whether this paradox is in someway related to the problem of the self-predication of Forms?

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by Fooloso4 » December 18th, 2018, 11:12 am

The Forms are not sets.

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by JaxAg » December 18th, 2018, 11:46 am

Fooloso4 wrote:
December 18th, 2018, 11:12 am
The Forms are not sets.
Yes. That is true. I was wondering if there might be a useful analogy in there? eg, might theree be some Forms which do characterise themselves, and some which do not, and if so, would this lead to paradoxical results?

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ChanceIsChange » December 18th, 2018, 1:45 pm

JaxAg wrote:I was wondering whether this paradox is in someway related to the problem of the self-predication of Forms?
Fooloso4 wrote:The Forms are not sets.
Indeed, sets are merely the extensions of Forms. Still, Russel's Paradox can easily be modified so as to apply to Forms: simply consider the Form of Not Being a Self-Characterizing Form. This Form creates a paradox for the ToF pertaining to self-predication. However, this paradox would persist even if we gave up self-predication as a general property of Forms.

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by Consul » December 18th, 2018, 1:46 pm

Fooloso4 wrote:
December 18th, 2018, 11:12 am
The Forms are not sets.
They are universalssubstantial ones (sortal or particularizing universals: kinds, sorts, types, genera, species) or non-substantial ones (non-sortal or characterizing universals: properties/qualities or relations). By the way, self-predication of sortal universals makes no sense. Consider e.g. "Doghood is a dog" (as opposed to e.g. "Beauty is beautiful").
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by JaxAg » December 18th, 2018, 2:25 pm

ChanceIsChange wrote:
December 18th, 2018, 1:45 pm


Indeed, sets are merely the extensions of Forms. Still, Russel's Paradox can easily be modified so as to apply to Forms: simply consider the Form of Not Being a Self-Characterizing Form. This Form creates a paradox for the ToF pertaining to self-predication. However, this paradox would persist even if we gave up self-predication as a general property of Forms.
. I can't think of any paradoxes that don't involve putting a negative in 'by hand'. Perhaps there's a clue there?

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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by ChanceIsChange » December 21st, 2018, 3:33 pm

Sorry for the late and lengthy answer.
ktz wrote:I'm not saying […]us lowly physical world humans.
I admit that my concept of evenness is probably an imperfect image of the Form of Evenness. I also admit that logical intuition might sometimes be flawed. In addition, I am aware of the transcendence of the Forms. However, I don't see a problem with applying logical intuition per se to the Forms. That is because:

1. imho, our logical intuition is, as its name suggests, not physical, but logical. Some things derived from logical intuition, such as the Law on Non-Contradiction (LNC), can hardly be reasonably doubted (although I'm not 100% sure). I think that our logical intuition is not derived (at least not completely) from physical phenomena, but rather that it is a (maybe sometimes distorted) dianoetic image (at least in part) of the perfect noetic knowledge held by the part of our souls which always remains in the realm of the Forms and there beholds perfect being. One reason I think that logical intuition is at least partly independent from physical experience is this: I am quite certain, except for a fundamental doubt of mine about everything, that some logical propositions, like LNC, are true, even though physical experience seems to suggest otherwise. For example, I see a tree standing outside the window. That tree appears to be green and not green, seemingly contradicting LNC. Still, I’m almost 100% sure that LNC holds true. Later, I realize that the sentences “The tree is green” and “The tree is not green” are actually ambiguous. The first sentence means that the tree is partly green (its leaves are green), whereas the second sentence means that the tree is not wholly green (its trunk is not green, but brown). These two true propositions are not at all each other’s negations or otherwise mutually exclusive and thus pose no threat to LNC. So, my logical intuition was right and did not succumb to physical intuition.
2. The transcendence of the Forms has to be qualified. As far as I know, all Forms except for the Form of Oneness-Goodness-Beauty are beings (German: Seiende), that is, they have being (German: Sein). This means that they are not absolutely transcendent, for they don’t transcend being (German: das Sein). Now, I expect that in the realm of being, the laws of logic rule supreme. That's, well, logical. Therefore, all Forms except for the Form of the Good ought to obey the laws of logic. For example, they can’t defy LNC. Our logical intuition, if cleansed from opinions about the physical world that masquerade as logical intuition, should therefore be applicable to the Forms except for the Form of Goodness. In particular, the Form of Evenness should obey the laws of logic.
ktz wrote:The Form of Evenness, as I understand Plato's dialogues are trying to say, is the most transcendently Even thing possible.
That would mean that the Form of Evenness has the most evenness among all entities. Then, there would still be that of which the Form of Evenness has the most, namely evenness. What would that be? Probably itself, right? If not, then evenness would be still higher than the Form of Evenness, since without evenness, the Form of Evenness couldn't have the most evenness among all entities. Either way, Forms as properties seem to be "higher" than Forms as prototypes.
ktz wrote:This form transcends […] could even look like.
As I have said above (if what I’ve said is wrong, please tell me), just because something transcends the physical world, it need not be unamenable to our logical intuition. For example, we have no problem accepting that the Form of Evenness doesn’t have a spatial or temporal extension, physical shape, spatial or temporal location or parts. Also, we must not forget that Plato himself was also one of those “lowly physical humans”, unless you actually believe the story of him being Apollo’s son ;). In that case, I’ll tell to you that, although it’s something only very few people know about, I’m actually descended from both Thor and Zeus through some of my ancestors and thus have the authority to rightly claim that everything, even the Form of Goodness beyond being and everything, ultimately stems from a completely incomprehensible Turtle. But be warned: if you’re not a descendent of a god, merely trying to think about that Turtle will drive you mad ;). Joking apart, Plato cannot simply define a mere being (Seiendes) such as Evenness as having certain properties that contradict logic. Of course, you’re right in saying that our logical intuition could sometimes be false, and that therefore, Plato’s concept maybe doesn’t violate logic, but rather only our notion of logic.
Please don’t get me wrong – when it comes to highly transcendent “things” like the One, their transcendence can’t be radical enough for my taste. The topic “Why do people make statements about God?” in the philosophy of religion forum demonstrates that. Here, however, we’re talking about normal Forms, which are mere beings (Seiende).
ktz wrote:Your intuition about numbers are based on your past experience with human constructions and subject to human fallibility in understanding.
I accept that dianoetic thought can be fallible but dispute that it is only based on experience. For example, we have a dianoetic concept of the number 5. You might say that we have abstracted it from the group of five sheep over there, the pack of five wolves hunting the sheep and the team of those five players on TV. But how could those concrete groups be the source of our concept of the number 5 rather than, say, our concept of group-of-mammals? My answer is that they aren’t, and that the Form of Fiveness, noetically apprehended by the highest part of our soul, is the pre-image of our dianoteic concept of the number 5.
ktz wrote:For example, your past usages of "Equality" have limited its arity to 2 usages. But in Polish notation, for example, there is no such restriction on the arity of the equality operator:

(= (2 (1+1) (3-1) (4/2) ) ) is an expression that resolves to true.
You’re making a good point here, and I’ll come back to it shortly. But, I ask you, isn’t the equality operator =2 with arbitrary arity (incl. 0) derived from the binary equality operator =1 thus:

For all tuples T : (=2 T) :⇔ for all components x, y of T: x =1 y ?
ktz wrote:Basically what I am saying […]truth in mathematics.
Now I’m coming back to your point. I think that Plato’s Forms are indeed no ordinary universals like properties or relations. Rather, they appear to be the unmanifested gist or “soul” behind ordinary universals. For example, I’ve always felt that there is the property of being beautiful, but also something else, Beauty itself, not yet specifically manifested as a property. Another example would be the Form of Largeness, which can manifest as the property of being large or as the (greater-than)-relation. Yet another example would be the above-mathematical Form of Equality, manifesting as the mathematical entities =1 and =2. I personally think that the Forms are these “souls” which are apprehended noetically, whereas the specific manifestations of the Forms are the still abstract, but already limited mathematicals including numbers and properties. These are the objects of dianoia. So, I think you are right in realizing the limitations of mathematics.
Concerning the laws of logic you have mentioned, I don’t believe in the Law of Bivalence (e.g. because of the Sea-Battle-Argument) and therefore do not accept Boolean logic, but I do believe in the Law of the Excluded Middle.
ktz wrote:Similarly, I might contest your proposition […] Form of Evenness.
Exactly: Evenness manifests in many different definite and limited ways. However, concerning the association of Evenness with God and Oddness with the Devil, I might add that the Pythagoreans associated the odd with the good and the even with the bad.
ktz wrote:So although I might not believe[…]the true character of these Forms?
I agree with you that our dianoetic understanding is still a long way from infallible, complete noetic understanding of the Forms. Still, prototypicality leads to problems in my opinion. For example, the Form of Materiality should be material as the prototypical material entity, but immaterial as a Form. Also, prototypicality leads the problem raised by the Third Man Argument. I might add Cosul’s objection that Doghood is not a dog, at least not in the same way that Beauty is beautiful. Even if all Forms are prototypical in some sense, there must be two different self-predication relations, one that applies to all Forms, and another one which only applies to some Forms, e.g. Beauty.
Also, you are right to doubt as to whether our brains are sufficient to understand the Forms. Just bear in mind that Plato also had a human brain, and that at least that part of our (and all living things’) souls that resides in the Above-Heavenly Realm is not restricted by our brain anatomy.

Talking of Plato’s human brain, I think that Fooloso4 has made some good points, e.g.
Fooloso4 wrote:It is important to see that Greek number theory is not the same as modern theories. There is no zero. […] an even number in each.
,
Fooloso4 wrote:The number 4 is even because it can be divided into two equal parts, each with the same number of ‘ones’.
and
Fooloso4 wrote:The key is the Greek understanding of number. A number (which is always a number of units) is even if it can be divided into two equal parts.
To me, this suggests that the Greek mind sometimes thought in a rather concrete way. Wouldn’t I be justified to suppose that sometimes, the ones who aren’t thinking abstractly enough aren’t we, but rather Plato and/or his characters? Here are some backings for my claim that at least Plato’s characters sometimes think too concretely or are imprecise:

1. “At Phaedo 102b ff., Socrates points out that Simmias is taller than Socrates (and hence tall), but that Simmias is also shorter than Phaedo (and hence short). Thus, Simmias is both tall and short.” (https://plato.stanford.edu/entries/plat ... eTheFor128) Sorry, and please correct me if I’m wrong, but to me, this looks like sophistry. Isn’t Socrates treating the binary (is-larger-than)-relation as if it were unary (a property)?
2. In the Parmenides, there is an apparent problem with Forms being in more than one entity and place at once. Imho, there is no problem if one realizes that partaking is only a physical metaphor for an abstract relation, which simply holds between sensibles and Forms without the Forms being literally present in different entities in the spatio-temporal world.
3. Plato or his characters seem to believe that there are concrete entities. I have my doubts that there are any concrete entities at all, for which I’ll probably open another topic.
4. I have read about there being "two 2's" when adding 2 to 2 to get 4, and that therefore, the number 2 is not the same as the ideal number 2 because there are multiple numbers 2, but only one ideal number 2. Here, the apparently physically inspired thought seems to be that we are literally adding 2 to 2 to get 4, rather than just metaphorically expressing that the function + sends the ordered pair (2, 2) to 4, i.e. that the binary right-unique relation + holds between (2, 2) and 4.
Fooloso4 wrote:But Forms only have one name that name the one thing that it alone is.
But the Form of Justice, for example, has many names, including "Justice", "Gerechtigkeit", "عدل" and "Δίκη", doesn't it?
Fooloso4 wrote:The are not the same but related, although we do say, divide the pie evenly, which is to say into equal parts.
That is correct imho.
Fooloso4 wrote:What is more […]and to give an account.
I agree. How exactly does this relate to the problem of self-predication?
JaxAg wrote:might theree be some Forms which do characterise themselves, and some which do not?
That it is so was my original opinion, which I have modified somewhat, see above.
JaxAg wrote:might theree be some Forms which do characterise themselves, and some which do not, and if so, would this lead to paradoxical results?
Yes, the Form of Non-Self-Characterization still leads to Russel’s Paradox for Forms.

Needless to say, but if I have misunderstood something, please correct me.

Fooloso4
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Re: Plato, Self-Predication, the Verb to Be and the Arity of Relations

Post by Fooloso4 » December 21st, 2018, 5:08 pm

ChanceIsChange:
Sorry, and please correct me if I’m wrong, but to me, this looks like sophistry. Isn’t Socrates treating the binary (is-larger-than)-relation as if it were unary (a property)?
One question that occurs in the dialogue Sophist and elsewhere is the relationship between the philosopher and the sophist. Aristophanes seemed to regard him as a sophist. The difference seems to be a matter of intention. Socrates often used sophistic or rhetorical arguments. It is always important to see the arguments in context. The Phaedo is about the fate of the soul. Socrates’ friends are more troubled by his death than he is. He must charm away their childish fears (77d-e). Several of the arguments are based on the Forms:
As I recall it, when the above had been accepted, and it was agreed that each of the Forms existed, and that other things acquired their
name by having a share in them, he followed this up by asking: If you say these things are so, when you then say that Simmias is taller than Socrates but shorter than Phaedo, do you not mean that there is in Simmias both tallness and shortness? - I do. (102b)



Answer me then, he said, what is it that, present in a body, makes it living?—A soul.
And is that always so?—Of course.
Whatever the soul occupies, it always brings life to it?—It does.
Is there, or is there not, an opposite to life?—There is.
What is it?—Death.
So the Soul will never admit the opposite of that which it brings along, as we agree from what has been said?
Most certainly, said Cebes.
Well, and what do we call that which does not admit the form of the
even?—The uneven.
What do we call that which will not admit the just and that which will not admit the musical?
The unmusical, and the other the unjust.
Very well, what do we call that which does not admit death?
The deathless, he said.

Now the soul does not admit death?—No.
So the soul is deathless?—It is. (105d-e)
Each of the arguments for the immortality of the soul fail, and Plato shows the careful reader enough to see how and why they do fail. Hence the problem of misology (89d).

But Forms only have one name that name the one thing that it alone is.
But the Form of Justice, for example, has many names, including "Justice", "Gerechtigkeit", "عدل" and "Δίκη", doesn't it?
To be more precise I should have said in Greek it only has one name, and in English etc. it only has one name. That is, it is not also ‘beauty’ or the ‘Good’ or anything else.

What is more […]and to give an account.
I agree. How exactly does this relate to the problem of self-predication?
It doesn’t. It points the relationship between arithmos, epistemology, and ontology. It is in each case the "one" that allows us to say something true, to count and to give an account. It also points toward the problem of the one and the many. Each Form is one, but the Forms are many. Each unit is one but there are many units. The many are One.

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