@EliahKagan it is a new thing to me, I am a case beginner, so I don't think I can say much interesting. I like it because I think it is less confusing than relying on syntax. I can try to find some good examples when I get back, but I think my favourite case so far is the instrumental case because it has the most impressive impact.

This one allows you to put some noun in a form such that we know it is being used for something. On a simple level, something like toothbrush-instrumental-case-ending teeth clean instead of a toothbrush is used for cleaning teeth. But what I find fun is that we can express causality by saying something like thing-instrumental-case-ending effect. It ends up very different to any structure in English that expresses the same relationship between ideas

@EliahKagan I think number bonds are addition facts. I mean that, given a sum like 2+3, she would have to count on her fingers or something like that. We do an awful lot of mental math here which would be slow for her. Anyway she had surely spent hundreds of hours with people trying to enable her to memorize those facts, to no avail. But by the time she reached GCSE the board her school used for math exams had stopped producing a non-calculator paper altogether so it wasn't such a big deal.

Given a small number of perceptible objects, could she subitize?

Yes

Could 2 + 3 be computed in such a way, by spatial thinking?

(I am not assuming the answer to this question for that student or in general.)

I would think so

But boolean algebra with Venn diagrams and set-theoretic notation was okay, because the notation was clearly documented?

I suppose so yes

She was good at foreign languages :)

One of the things she was able to do perfectly fine was learn and follow a procedure with few enough steps. As she got older, she could manage more steps. So really there was nothing she couldn't do as long as she had learned the procedure for dealing with it.

She had a very positive attitude to whatever she was trying to do. She would never get upset by things going wrong. That was something for me to learn from her

@Zanna We might--and in fact do--want to try and work with objects that we can use in place of predicates. That is an informal characterization. I don't exactly mean that for names "x" and "y" we would want to write "xy" with a meaning like "Fy". I mean we may hope there is some object x that gives us power comparable to what we would have with F in a second-order system.

There is more than one way to go with this, but a way that is very useful is to talk about what one might call "knowledge of which things y are such that Fy" or "the bundle of complete information about which things y are such that Fy."

I am tempted to call this a "property" but that is not intuitively right, because if my universe of discourse is macroscopic physical objects on my kitchen table, and it happens that all the cups are blue, and it happens that all the blue things are cups, it's still a hard sell to say that blueness and cupness are the same property.

To say that would be to claim that "everything is blue iff it's a cup" doesn't communicate any actual information other than logical truths (or, sometimes, to say that it's true merely by convention). That claim does not seem right.

I should have said from the outset that, for the time being, I am specifically talking about capturing something meaningful about unary predicates in objects.

The intuitive goal of these unary-predicate-like objects is to be able (in principle) to answer the question, "Do you hold for x?" for any x. Then to actually make that claim, a new binary predicate is introduced, something like H where Hyx means, "It holds for y that x." (I could have defined it to take the arguments in the other order; the reason I've done it this way will be apparent soon, or perhaps already is.)

Introducing the predicate I am calling "H" dispenses with the need for many other predicates, since many of the claims we'd otherwise need to introduce predicates to say can now be said with H. Furthermore, when an atomic sentence with the predicate H is written, both arguments are objects. Both are quantified over. To state it very squishily, we can automatically talk about things that are about other things.

One way to look at these unary-predicate-like objects is as opaque knowers of answers to yes-no questions about arbitrary objects (including unary-predicate-like objects). Then H is the predicate used to state what facts they know. That's the way I've just introduced H.

It seems like a useful thing

But another way to look at these unary-predicate-like objects is that they are collections, where Hyx expresses that "y is a member of x."

The collection x contains y when the answer it gives when asked "Do you hold for y?" is "Yes."

The collection x does not contain y when the answer it gives when asked "Do you hold for y?" is "No."

Not just any collection will do. It must be unordered, and it must not store any information besides which things are its members.

This is to say that these unary-predicate-like objects are sets and "H" means "∈".

Oooh

:)

:D

So a first-order set theory (i.e., set theory on top of first-order logic) lets us do most of what we what we would want a higher-order logic for.

(It may also be able to do some things that a higher-order logic doesn't give you automatically. But you could use a higher-order set theory--a higher-order logic with axioms of set theory for first-order objects--and you'd presumably get that too.)

There are two major limitations.

First, in my kitchen table example, the set of blue things is equal to the set of cups.

(This is assuming that the introduction of sets into my universe of discourse has not made the "blue iff a cup" fact no longer true. The most intuitive way to do this is just to say sets are neither blue nor cups.)

So you cannot use "the set of blue things" to mean "blueness", nor "the set of cups" to mean "cupness" ("cupliness"?), if you want to distinguish those properties.

This is extensionality.

Whereas if you want to distinguish blueness from cupness, you need a notion of them that is intensional rather than extensional.

That's "intensional" with an "s". Even if I have misspelled it previously. :)

@EliahKagan !!! Oh no

@EliahKagan Though I am not 100% confident its example is factually accurate (I'm not saying it's not, I'm not sure, I'll try to remember to ask a biologist), I think the explanation of extensionality and intensionality in the Stanford Encyclopedia of Philosophy article on intentionality (with a "t") is very good:

@EliahKagan I misspelled it previously because I didn't know that word so I mistook it for intention, but I realised the next time you mentioned it that it was a different thing

@Zanna I believe I misspelled it previously. I am not sure. That's what I was referring to though.

> Although the meaning of the word ‘intentionality’ in contemporary philosophy is related to the meanings of such words as ‘intension’ (or ‘intensionality’ with an s) and ‘intention,’ nonetheless it ought not to be confused with either of them. On the one hand, in contemporary English, ‘intensional’ and ‘intensionality’ mean ‘non-extensional’ and ‘non-extensionality,’ where both extensionality and intensionality are logical features of words and sentences.

> For example, ‘creature with a heart’ and ‘creature with a kidney’ have the same extension because they are true of the same individuals: all the creatures with a kidney are creatures with a heart. But the two expressions have different intensions because the word ‘heart’ does not have the same extension, let alone the same meaning, as the word ‘kidney.’

> On the other hand, intention and intending are specific states of mind that, unlike beliefs, judgments, hopes, desires or fears, play a distinctive role in the etiology of actions. By contrast, intentionality is a pervasive feature of many different mental states: beliefs, hopes, judgments, intentions, love and hatred all exhibit intentionality.

> In fact, Brentano held that intentionality is the hallmark of the mental: much of twentieth century philosophy of mind has been shaped by what, in this entry, will be referred to as “Brentano’s third thesis.”

That's all one paragraph, but I didn't know how to blockquote it in SE chat without splitting it.

@Zanna Does that bother you?

This is the same thing I expressed there.

I'm not saying that means it shouldn't bother you (or that it should), I just want to make sure that's clear.

I don't know if I wrote it out in full before or not, but here's a way to write the axiom of extensionality formally:

∀x ∀y ((∀z (z ∈ x ↔ z ∈ y)) → x = y)

Suppose on your table we don't know that the blue things are the same things as the cups. But we do know there are exactly 5 blue things, and that there are exactly 5 cups. Or perhaps we don't even know that, but we paired them off, so we know there's the same number of them, even if we're not sure that that number is.

Then the number of blue things is equal to the number of cups. This is so, even though what it means to be the number of blue things and what it means to be the number of cups are not the same meanings. One might say that numbers are extensional, too.

Similarly, when we're talking about object on my kitchen table (in the thought experiment about my kitchen table), the noun phrases "the set of blue things" and "the set of cups" don't have the same meaning, but they are names for the same thing.

* about objects on my kitchen table

@EliahKagan no but it is a good illustration of the limitation especially because from the point of view of this universe it is merely accidental that the set of blue things equals the set of cups in this other universe

@EliahKagan yes indeed