Nota Notae Est Nota Rei Ipsius and the Ontological Argument

(By popular demand, I repost the following old Powerblogs entry.)

"The mark of a mark is a mark of the thing itself." I found this piece of scholasticism in C. S. Peirce. (Justus Buchler, ed., Philosophical Writings of Peirce, p. 133) It is an example of what Peirce calls a   'leading principle.'

Let's say you have an enthymeme:

   Enoch was a man
   -----
   Enoch died.

Invalid as it stands, this argument can be made valid by adding a premise. (Any invalid argument can be made valid by adding a premise.) Add 'All men die' and the argument comes out valid. Peirce writes:

     The leading principle of this is nota notae est nota rei ipsius.
     Stating this as a premiss, we have the argument,

     Nota notae est nota rei ipsius
     Mortality is a mark of humanity, which is a mark of Enoch
     -----
     Mortality is a mark of Enoch.

But is it true that the mark of a mark is a mark of the thing itself? There is no doubt that mortality is a mark of humanity in the following sense: The concept humanity includes within its conceptual content the superordinate concept mortal, which implies that, necessarily, if anything is human, then it is mortal. But mortality is not a mark, but a property, of Enoch. I am alluding to Frege's distinction between a Merkmal and an Eigenschaft. Frege explains this distinction in various places, one being The Foundations of Arithmetic, sec. 53. But rather than quote Frege, I'll explain the distinction in my own way using a totally original example.

Consider the concept bachelor. This is a first-order or first-level concept in that the items that fall under it are not concepts but objects. The marks of a first-order concept are properties of the objects that fall under the concept. Now the marks of bachelor are unmarried, male, adult, and not a member of a religious order. These marks are themselves concepts, concepts one can extract from bachelor by analysis. Given that Tom falls under bachelor, he has these marks as properties. Thus unmarried, etc. are not marks of Tom, but properties of Tom, while unmarried, etc. are not properties of bachelor but marks of bachelor.

To appreciate the Merkmal (mark)-Eigenschaft (property) distinction, note that the relation between a concept and its marks is entirely different from the relation between a concept and its instances. A first-order concept includes its marks without instantiating them, while an object instantiates its properties without including them.

This is a very plausible line to take. It makes no sense to say of a concept that it is married or unmarried, so unmarried cannot be a property of the concept bachelor. Concepts don't get married or remain single. But it does make sense to say that a concept includes certain other concepts, its marks. On the other hand, it makes no sense to say of Tom that he includes certain concepts since he could do such a thing only if he were a concept, which he isn't. But it does make sense to say of Tom that he has such properties as being a bachelor, being unmarried, being an adult, etc.

Reverting to Peirce's example, mortality is a mark of humanity, but not a mark of Enoch. It is a property of Enoch. For this reason the scholastic formula is false. Nota notae NON est nota rei ipsius. The mark of a mark is not a mark of the thing itself but a property of the thing itself.

No doubt commenter Edward the Nominalist will want to wrangle with me over this slight to his scholastic lore, and I hope he does, since his objections will aid and abet our descent into the labyrinth of this fascinating cluster of problems. But for now, two quick applications.

One is to the ontological argument, or rather to the ontological argument aus lauter Begriffen as Kant describes it, the ontological argument "from mere concepts." So we start with the concept of God and analyze it. God is omniscient, etc. But 'surely' existence is also contained in the concept of God. For a God who did not exist would lack a perfection, a great-making property; such a God would not be id quo maius cogitari non posse. He would not be that than which no greater can be conceived. To conceive God, then, is to conceive an existing God, whence it follows that God exists! For if you are conceiving a nonexistent God, then you are not conceiving God.

Frege refutes this version of the OA -- not the only or best version I hasten to add -- in one sentence: Weil Existenz Eigenschaft des Begriffes ist, erreicht der ontologische Beweis von der Existenz Gottes sein Ziel nicht. (Grundlagen der Arithmetik, sec. 53)  "Because existence is a property of concepts, the ontological argument for the existence of God fails to attain its goal." What Frege is saying is that the OA "from mere concepts" rests on the mistake of thinking of existence as a mark of concepts as opposed to a property of concepts.  No concept for Frege is such that existence is included within it. Existence is rather a property of concepts, the property of having an instance.

The other application of my rejection of the scholastic formula above is to the logical question of the correct interpretation of singular propositions. The scholastics treat singulars as if they are generals as I explained fully in previous posts. But if Frege is right, this is a grave logical error since it rides roughshod over the mark/property distinction. To drag this all into the full light of day will take many more posts.

Comments

[Edward nominalist]

First point, it is high time that someone wheeled out the supposed distinction between the 'is' of predication and the 'is' of identity, so I have forestalled that with a comment here http://ocham.blogspot.com/2011/07/do-we-need-identity.html .

Second point, I cannot find the 'nota notae' formulation in my formidable collection of Latin logic, using the built-in searcher (the site now contains 1,820 pages). Widening the search to other scholastic sites shows up nothing, and only on widening it to Google do I find Baldwin's dictionary, where the principle is located in Aristotle, with the Latin formulation probably originating in the early modern period.

Aristotle's formulation, e.g. "'Of whatever the species is predicated, the genus is predicable" does not seem problematic, when regarded as a linguistic or logical principle. It was formulated by the scholastics as the dici de omni, which Ockham defines in Book III part 1 chapter 2 of the of the Summa Logicae. "Est autem dici de omni quando nihil est sumere sub subiecto, quin de eo dicatur praedicatum". I.e. to say that 'Every S is P' is to say that of whatever S is true, P is true. This is impeccable. It is the formulation "nota notae est nota rei ipsius" which is weak.

[Edward nominalist]

PS I have listed another objection to the 'singulars as universals' thesis here http://ocham.blogspot.com/2011/07/singulars-as-universals.html , courtesy of Colwyn Williamson. Let's nail them all to the door, please.

[Bill Vallicella]

As we learned from Bill Clinton, much indeed does depend on what the meaning of 'is' is. His contributions to the philosophy of logic are not to be denigrated. I'll look at your post.

>>to say that 'Every S is P' is to say that of whatever S is true, P is true. This is impeccable.<<

Well, sometimes. 'Every cat is an animal' is such that whatever 'cat' if true of, 'animal' is true of. But not so for 'Caissa is a cat.' (I am referring to a cat of mine who died about a year ago, not to the goddess of chess.) For it makes no sense to say that 'Caissa' is true of a particular cat.

Why not? Well, 'Caissa' is not a predicate but a name. If it were a predicate what properties or property would its use predicate of the thing to which it applies?

The Fregean point, which I've made before, is that *animal* is a mark of *cat* not a property of it while *cat* is a property, not a mark of Caissa. After all, Caissa is not a concept, but an individual.

To budge me from this Fregean view you would have to show that the relation between Caissa and *cat* is the same as the relation between *cat* and *animal.* But they are logically very different relations: Caissa falls under *cat,* but *cat* does not fall under *animal.* A cat is an animal, but the concept *cat* is not an animal.

[Bill Vallicella]

I wonder how much of this scholastic stuff is based on use-mention confusion. A scholastic will write, exasperatingly, 'Man is an animal.' But that's ambigous as between:

1. A man is an animal
and
2. The concept *man* is an animal.

The first is true, the second false.

[Edward nominalist]

>>To budge me from this Fregean view you would have to show that the relation between Caissa and *cat* is the same as the relation between *cat* and *animal.*

Straw man. What you want me to show is transparently wrong. What I am actually claiming is

(1) the relation between 'Caissa' and 'cat' is the same as the relation between 'cat' and 'animal'.

Note the quote marks which signify a linguistic entity.

(2) the relation between *Caissa* and *cat* is the same as the relation between *cat* and *animal.*

Note the asterisk I have added, which are intended to signify a conceptual entity. You are trying to claim a category mistake which I am not making.

You will now assert that there is no (singular) concept *Caissa*, but you have given absolutely zero argument for this anywhere (apologies if I have missed one).

[Edward nominalist]

>>For it makes no sense to say that 'Caissa' is true of a particular cat. Why not? Well, 'Caissa' is not a predicate but a name. If it were a predicate what properties or property would its use predicate of the thing to which it applies? <<

The property which its use predicates is the property of being Caissa. The concept it signifies is the concept of being Caissa.

[Bill Vallicella]

With all due respect my good man, you are digging your hole deeper. As I have argued ad nauseam, there are no properties of the form 'the property of being a,' where 'a' is a proper name, and where 'being a' means the same as 'being identical to a.' And the same goes for concepts. 'Caissa' does not express a concept.

So we are brought back to the business of haecceity-properties/individual concepts. I say there aren't any. You say there are. How can we resolve this?

[Bill Vallicella]

>>What I am actually claiming is

(1) the relation between 'Caissa' and 'cat' is the same as the relation between 'cat' and 'animal'.

Note the quote marks which signify a linguistic entity.

(2) the relation between *Caissa* and *cat* is the same as the relation between *cat* and *animal.*<<

Of course I deny both of these as well. 'Cat' is true of Caissa, but 'Caissa' is not true of Caissa. And there is no such concept as *Caissa.* Am I simply begging the question against you, and you against me?

Since you think I have provided no reason to discount individual/singular concepts, I shall have to write another post!

[Edward nominalist]

>>With all due respect my good man, you are digging your hole deeper.

No, the bottom of the hole has been my starting point, for many years. My whole and entire theory is a theory of singular concepts.

>>As I have argued ad nauseam, there are no properties of the form 'the property of being a,' where 'a' is a proper name, and where 'being a' means the same as 'being identical to a.'

Yes.

>>And the same goes for concepts.

No. As I also have argued ad nauseum.

>>I say there aren't any. You say there are. How can we resolve this?

To be clear, I say there aren't any haecceity properties but there are haecceity concepts and haecceity predicates. As I said about ten posts ago, it is self-evident that there are some predicates to which no property corresponds. E.g. 'the former', 'the latter'. So you can't use the assumption that there are no haecceity properties to prove that there are no haecceity predicates. And if there are such predicates, what they signify must be a haecceity concept, or a singular meaning or whatever term you like.

>>How can we resolve this?

By considering seriously the possibility that there may be such predicates. We have considered and not rejected as obviously false the idea that proper names can be syntatically and semantically predicates (obstacle #1 to those trained in MPC).

Now, once we have got in the right frame of mind, we need to consider the idea of haecceity predicates.

I have any number of posts on my blog that we could consider, I which I could rewrite. But consider my remarks here as a preliminary, the 'terms of engagement' as the management consultants like to say.

And now it is the evening and I am off with wife for dinner in modest style.

Very sad about Amy Winehouse, by the way. And the news of the Norwegian massacre equally tragic. How could a Christian do such a thing?

[Edward nominalist]

>>Since you think I have provided no reason to discount individual/singular concepts, I shall have to write another post!
<<

[Crossed posts] Agreed. And now it really is time for supper, and a beer, and reflection on the sadness of this world, and of human nature.

[Bill Vallicella]

I heard about Winehouse. It's about noon here. She will preempt tonight's oldies show.

Predicates don't prove properties, but they don't prove concepts either. But later on this.

Enjoy your London evening. The "sadness of this world" and its utter hopelessness and manifest unimprovability are reasons to take religion seriously. And Dawkins be damned. (Figuratively speaking.)