Island of castaway thoughts

1:00 PM

An axiom of extensionality relates the "∈" and "=" predicates--which one has already because this is for when one's underlying logic gives "="--by saying:

for all s, for all t: ((all x: x ∈ s ↔ x ∈ t) → x = t)

We could write the major break as "if and only if" rather than just "only if" like this but we don't have to:

for all s, for all t: ((all x: x ∈ s ↔ x ∈ t) ↔ x = t)

Zanna

I feel like I have learned a lot!

Eliah Kagan

More typos! :(

@EliahKagan Should be:

for all s, for all t: ((all x: x ∈ s ↔ x ∈ t) → s = t)

@EliahKagan Should be:

for all s, for all t: ((all x: x ∈ s ↔ x ∈ t) ↔ s = t)

(I accidentally wrote x on the right instead of s.)

The reason we don't have to is that when our underlying logic is equipped with identity, its axioms of identity let us conclude from x = t that anything true for x is true for t. It is true that:

all x: x ∈ s ↔ x ∈ s

So when s = t, we can infer, by substitution into that formula, that:

all x: x ∈ s ↔ x ∈ t

@Zanna Cool! I hope the wrong stuff I later corrected but could not edit has not made things too frustrating.

So you can see that having one's logic equipped with "=" (and associated axioms) is very powerful and useful but that in a set theory we may do without it if we like.

So, I had said:

38 mins ago, by Eliah Kagan

So, what I am about to say should be taken with a vast asterisk.

Zanna

@EliahKagan I guess I didn't bother to read those symbols properly since I knew what they were symbolising anyway. Wow I'm so lazy

anyway I wasn't confused

Eliah Kagan

The vast asterisk applies specifically to my claim that, when we use an underlying logic that doesn't give us "=", this way of defining it by saying equal thngs have all the same members, taken together with the axiom that says equal things are members of all the same things, is sufficient to ensure that "=" works the way it ordinarily does in logic and mathematics.

Specifically, that our system is incapable of distinguishing between things x and y in cases where x = y, i.e., in its language we cannot say anything about x whose truth value could be different if x were replaced by y.

I said that everything we can say in our system comes down to, or can be expressed as, atomic sentences of the form a ∈ b, connected together.

There are two important comments on this, the second of which I'd characterize as a caveat.

Zanna

> The limits of my language means the limits of my world

Eliah Kagan

1:13 PM

The first, though, is that I've talked about predicates (in our system without identity, just "∈") and how one makes an atomic sentence by combining an n-ary predicate with n terms, and I've talked about truth-functional sentential connectives and how one makes a compound sentence by attaching n sentences (atomic or otherwise) together with an n-ary sentential connective... but many of the sentences I've written have more than that.

Specifically, they have quantifiers.

So far I've used only universal quantification. I've talked about "for all x", "for all y".

There's also existential quantification: "there exists some x"

These are also part of the formal language.

the formal way to write "for all x" is: ∀x

i.e., with an upturned A

and the formal way to write "there exists an x" (i.e. "for some x") is: ∃x

i.e., with an upturned E

I have been told that the ∀ is not merely upside-down but also backwards, and that the ∃ is not merely backwards but also upside-down. :)

I've been using quantification all along, and it's not a problem for my claim that when "∈" is our only predicate, anything we can say about anything is in terms of "∈".

But I wanted to mention it; plus, it is quite important for other purposes.

Also, I should say, we also have variables of quantification (often just called variables), like x.

Zanna

the upside down A looks like a container of something, which is helpful

Eliah Kagan

Nice--I've never noticed that! :)

Zanna

:)

Eliah Kagan

The actual caveat in terms of what is expressible, though, is that the signatures of formal systems are not limited to predicates (e.g., = anytime we have identity, ∈ in set theory, Away and Found above) and constants (e.g., smith and towel above). Some formal systems also have function symbols.

For example, before I expressed, "Smith found the towel." But suppose I wanted to express "Smith found his umbrella."

The word "his" is no problem here; just plug "Smith" in for it. That is, this readily reduces to the problem of expressing, "Smith found Smith's umbrella."

There are a few approaches.

One approach is to have an Umbrella predicate, where Umbrella(x, y) expresses "x is an umbrella of y".

In the a terser dialect, one might write Uxy.

Zanna

1:29 PM

that seems like it could get long

Eliah Kagan

Then ∃x (Umbrella(x, smith) ∧ Found(smith, x)) expresses that Smith found (at least) one of his umbrellas.

Zanna

hahaha

Eliah Kagan

If one has = then one can express that Smith found his unique umbrella.

There are several reasonable ways to say this; one is:

∃x (Umbrella(x, smith) ∧ (∀y (Umbrella(y, smith) → y = x)) ∧ Found(smith, x))

There are two major alternatives to this.

And it does seem like an alternative would be nice.

Zanna

it does :S

Eliah Kagan

For example, one of the uses of formal logic is mathematics. You don't want to have to write AddsTo(x, y, z) instead of x + y = z. (There are other reasons it might not be--though could be--translated quite that way even if there were no alternative, but that's the idea.) You don't want to have to do that because you want to be able to build even more complex sentences and still have them make sense.

To bring it back to what we were talking about before... you can do the same thing to both sides of an equation, right?

Zanna

1:41 PM

yesss

Eliah Kagan

But so far our logic does not have any elegant way to express doing something to a term.

Zanna

:(

Eliah Kagan

So one approach is to use function symbols.

Whereas a sentential connective combines with sentences to form a sentence, and a predicate combines with terms to form a sentence, a function symbol combines with terms to form a term.

For example:

Found(smith, umbrella(smith))

In this dialect, function symbols like umbrella are lower-case to make it easier to distinguish them from predicates like Found.

Here, umbrella is a unary function symbol.

Function symbols are also called "functions" and various other things. I am avoiding calling them functions because they do not actually capture what we mean by functions in mathematics, for a reason I hope to get to.

You can probably see the appeal of Found(smith, umbrella(smith)).

It's short. It's syntactically more similar to natural language and to formal languages we enjoy using (like whatever language we're using when we do math, and also like many programming languages, though not like sed or bash so... :).

It's doesn't use any quantifiers.

It doesn't use any explicit quantifiers.

So, what does Found(smith, umbrella(smith)) mean if Smith has no umbrella, or if Smith has multiple umbrellas?

There are various approaches; to my knowledge, two are mainstream.

The first approach is a cop-out. It says, don't have your system have function symbols for anything where this can happen.

Zanna

maybe it means he found himself a new umbrella while shopping

Eliah Kagan

:)

The issue is that umbrella(smith) means "the thing that is an umbrella of Smith".

It has to refer to some specific thing so that a claim can be made about it.

In natural language, "Smith's umbrella" is ambiguous, and that ambiguity can be handy. But the goal of expressing it symbolically is to get rid of that ambiguity.

The second approach is to interpret Found(smith, umbrella(smith)) as saying "Smith has an unique umbrella and Smith found it."

That is, as saying that.

So if Smith has no umbrella or multiple umbrellas, Found(smith, umbrella(smith)) comes out false.

This is intuitively appealing: it's false, after all, that the present king of France is in my living room. So too would it be false that Smith found his nonexistent umbrella (or his nonexistent unique umbrella).

However, one might be tempted to make a more sweeping claim about how this works, like, "Sentences about nonexistent things are false."

The problem is, Found(smith, umbrella(smith)) is false, which makes ¬Found(smith, umbrella(smith)) true.

This is actually a very deep problem, and the major alternative to function symbols has it too. But one part of the appeal of function symbols, in my view, is the illusion that they are simpler than they are.

One can altogether avoid this problem--or, a cynic might say, replace it with other problems. One approach is to use free logic, which lets atomic sentences whose arguments fail to refer to anything be true. I have not studied free logic. Another approach is to use a logic with more than two truth values. Multivalued logics aren't necessary for the things people often assume they're necessary for, but I do think they have some value.

There are also, um, wrong "approaches," such as pretending that some of one's sentences do not really have a truth value, but then still relying on parts of standard logic that are incompatible with that, such as the law of the excluded middle and proof techniques that rely on it (like proof by contradiction).

So, the major alternative to function symbols is definite descriptions.

I like this much better. It is very powerful. Anything you can do with function symbols, you can do with predicates and definite descriptions. It also does not conceal its existence and uniqueness claims.

So, without function symbols or definite descriptions, we had:

39 mins ago, by Eliah Kagan

∃x (Umbrella(x, smith) ∧ (∀y (Umbrella(y, smith) → y = x)) ∧ Found(smith, x))

With definite descriptions:

Found(smith, ɿx (Umbrella(x, smith)))

"Smith found the thing that is an umbrella of Smith."

The symbol "ɿ" should be a turned lower-case Greek iota (a turned "ι") in the same way that the symbols "∀" and "∃" are a turned "A" and a turned "E", respectively, but Unicode doesn't quite have anything for that, so I used the closest I could find. It would be excellent if SE chat had MathJax enabled so LaTeX could be used (though writing the upturned iota is a bit challenging even with LaTeX -- at least it's challenging for me).

With either function symbols or a notation for definite descriptions, the notation for functions in mathematics like sin and + can be introduced as alternate forms involving particular function symbols or definite descriptions. I say "alternate forms involving" rather than "alternate forms of" because the ways of doing the translation that avoid diminishing the power such notations are expected to have may be somewhat surprising.

This is what I alluded to when I mentioned that function symbols don't really capture what is going on with mathematical functions. I still hope to get to this.

What either function symbols or definite descriptions buys us, though, is that notation like cos x or a + b, can be defined as an alternate form of something, and the thing it is an alternate form of is the same no matter what bigger thing it is part of,

There, by "thing" I mean term or formula/sentence.

In contrast, imagine expressing "addition is associative" with just a AddsTo predicate (used like: AddsTo(x, y, z) to express what we'd prefer to say by x + y = z).

You can do it, of course. It's not even all that bad.

(Here's something much like it.)

But what you get doesn't really seem like it clearly conveys the meaning of what you are trying to say.

Zanna

2:54 PM

@EliahKagan that's no fun

Eliah Kagan

I've might be okay for some simple systems.

@EliahKagan I could've and should've written the slightly syntactically simpler:

Found(smith, ɿx Umbrella(x, smith))

(I had some extra parentheses.)

So, just like a term with a function symbol like umbrella(smith) can fail, as when Smith has no unique umbrella... a term with a definite description like Found(smith, ɿx Umbrella(x, smith)) can also fail, and does so under the same circumstances.

@EliahKagan * It might be okay for some simple systems.

However, the definite description is quite explicit about its relationship to quantification.

The syntax resembles that of quantification and is sometimes said to be a form of quantification.

Zanna

@EliahKagan the one unique thing, his umbrella

Eliah Kagan

Definite descriptions are also highly flexible; if you have them, you could introduce function symbols as specialized syntax for some of them.

Having function symbols in the signature of one's system, so they are primitive, is something people sometimes want to do, and it is something that is traditionally done in some systems. For example, the typical formalization of Peano arithmetic has a function symbol S that means "the successor of" (i.e., "the next number after") and a constant, 0, for zero.

You don't actually need function symbols or constants, though.

I'm not speaking specifically in terms of Peano arithmetic here (my goal is not to give an alternative formalization of it), but suppose my universe of discourse is the nonnegative integers. Suppose my system's signature has one predicate, P, for "immediately precedes". (I could do it with a predicate for "immediately follows" instead if I wanted.) So P is primitive.

Suppose further that my system has constants (nothing primitive like 0 or smith) and no function symbols (nothing like "the successor of" or "the predecessor of").

Then I can say 0 is an alternate syntax for ɿy ¬∃x Pxy, and if I want to express "the successor of x" as s(x) then I can say terms that look like that (with any term for x, substituted appropriately) are an alternate syntax for ɿy Pxy.

Sometimes such definitions are expressed this way, though I don't know if the := notation is considered formal in the strictest sense:

0 := ɿy ¬∃x Pxy
s(x) := ɿy Pxy

So... if my thinking were less muddled, I wouldn't have presented these topics in quite this order, or I would at least have warned you that I was going to. Most of what I've been talking about is formal logic rather than set theory itself!

But the reason I introduced the notion of function symbols now, rather than somewhat later, is this... imagine if you had a system with one or more function symbols but built on an underlying logic that does not provide identity (i.e., that does not give you = and associated axioms of identity).

Do you see what might be considered objectionable about this?

Zanna

3:19 PM

@EliahKagan 0 is another way of saying the thing such that there is no thing that precedes that thing, and s(x) is another way of saying the thing that x precedes ?

Eliah Kagan

Yes.

Of course, those definitions are only reasonable if one's universe of discourse doesn't have things like (finite) negative numbers.

Zanna

sure, you mentioned that when you were describing the universe being discussed :)

Eliah Kagan

Yes.

Zanna

@EliahKagan I don't think I have exactly got what function symbols are

it might be relevant to say that I haven't ever got comfortable with the idea of functions in math

we didn't do those either

Eliah Kagan

That's relevant, but function symbols don't really capture what is going on with functions in math.

In the sense that names are references, function symbols are relative references.

My example of a function symbol--umbrella for "the umbrella of," was perhaps confusing because the real-world idea that inspires it doesn't entail uniqueness. The problem is that all the common examples that people tend to assume entail uniqueness do not. A common example is to have f [or father] mean "the father of", so that when s [or smith] denotes Smith, fs [or father(smith)] means "Smith's dad" and ffx (or father(father(smith))) means "smith's paternal grandpa."

There are certainly examples from mathematics, though. I emphasizes that functions in mathematics -- the kind you were lamenting not having studied -- are not really captured by function symbols in logic. But you can use function symbols for math. The "the successor of" function symbol in the typical formalization of Peano arithmetic is an example of this.

Function symbols have something in common with functions in computer programming. You know at least one programming language that has functions: Bash. But Bash is so weird, there isn't really a connection, I don't think.

I am not sure if more examples will help.

Zanna

3:40 PM

it's ok

Eliah Kagan

But I would like to provide them.

Zanna

:)

Eliah Kagan

Suppose we have a function symbol for "the favorite comedian of."

Then we can translate "Bob's favorite comedian is Sue" as:

favorite_comedian(bob) = sue

The specific symbols favorite_comedian, bob, and sue are up to us.

Zanna

ok

Eliah Kagan

"Smith found Bob's favorite comedian's unique umbrella":

Found(smith, umbrella(favorite_comedian(bob)))

Zanna

3:44 PM

the umbrella seemed problematic to me because there would end up being too many symbols

Eliah Kagan

Can you elaborate?

Specifically, what do you mean by "too many symbols"?

Zanna

well, I might not be able to say anything sensible

Eliah Kagan

I'm not sure I know what you mean.

Do you mean that "____'s only umbrella" is rarely something you'd want to express?

Zanna

what I thought was that there are many different things we might want to talk about, and that there are lots of different relationships they can have to each other. I was thinking about a feature of natural languages that I like, case

Eliah Kagan

@Zanna To clarify terminology, by "symbol" I mean things like the function symbol umbrella, constants like smith, sue, and bob, and predicates like Found.

Zanna

3:50 PM

whereby the relationship of a noun to something else is expressed by changing the noun itself

Eliah Kagan

My understanding is that, in some languages, including English, there are fewer case distinctions, and a rigid word order serves some of the purposes case once served.

Zanna

that's my understanding too

Eliah Kagan

An analogy to commands in a shell occurs to me. I think the distinction between options (i.e., flags), operands attached to options, and non-option arguments is related to case.

I mean as far as the commands themselves are concerned. The shell does not respect these distinctions.

Zanna

so thinking about cases, I though that the umbrella might not only belong to smith. It might be going somewhere or coming from somewhere or be used for something. These are things that cases express. But the umbrella symbol can only have one term and always relates to it in the same way

Eliah Kagan

Oh, I think that's actually no problem here.

If umbrella is a binary function symbol (i.e., a function symbol of arity 2), it's true that it's ill-formed (i.e., fails to satisfy the formal syntactic constraints for what a term is) to write umbrella() or umbrella(smith, sue).

But you can still express those other things.

So, first, you can talk about both umbrella(smith) and umbrella(sue). That's actually the purpose of a function symbol. You get a different compound term by binding different terms to it as arguments.

Also, even when smith and sue are different people, there's no prohibition on umbrella(smith) = umbrella(sue).

Zanna

3:59 PM

they share an umbrella

Eliah Kagan

The things case expresses are expressible with predicates (and, if you want, function symbols) of however high arity you need. So, for example, you cae have a ternary predicate F that represents some verb, where in Fxyz the first argument x has the role of the subject, the second argument y has the role of a direct object, and the third argument z has the role of an indirect object.

As in the case of Found, it matters quite a bit for this F what order you put the arguments in.

"x gave y to z"

Or you could construct everything you're interested in as a set and then you only need "∈"! :D

I should say, though, that there are things systems like this don't express.

In particular, the kind of logic I am talking about does not have any non-truth-function sentential connectives.

I cannot express "because" with a sentential connective.

You could have a system that has that (and I believe there is work on that), but it would be a different kind of logic.

Zanna

@EliahKagan that expression looks joyful

Eliah Kagan

:)

The kinds of systems I'm talking about also don't have modals. You can't say, "Smith should find his umbrella," except opaquely with a "should find" predicate or the like.

Similarly, there's no symbol that you can attach to a sentence to assert that it is necessarily so.

That's the sort of thing modal logic is used for.

Zanna

hmm :)

so I suppose I should go back to where this interesting sidetrack began

Eliah Kagan

Somewhat similarly, if the B predicate means "is a bird" and the F predicate means "fllies", then Bx → Fx expresses "if x is a bird then x flies," but it does not express "if x were a bird then x would fly," and the kinds of systems I'm talking about don't have a means to do that.

Zanna

4:10 PM

@EliahKagan no...

@EliahKagan it takes a lot of effort to make things out of logic

Eliah Kagan

@Zanna With Unicode in SE chat that is definitely so. :)

In seriousness, though, I'm not sure I know what you mean.

Zanna

@EliahKagan lol

Eliah Kagan

@Zanna Assuming an expected interpretation of a godmother function symbol, godmother(sue) means "Sue's only godmother".

What are we to make of that if our underlying logic, which provides us with the ability to use function symbols, does not supply any notion of identity?

For simplicity, consider how one would express that there is only one thing.

More precisely: exactly one thing.

This would be so if our universe of discourse has only one thing.

But it can be meaningfully expressed even if it is false.

This expresses that there is at most one thing, and some underlying logics insist there's at least one thing (i.e., they don't allow an empty universe of discourse), in which case it is sufficient:

∀x ∀y (x = y)

Otherwise (or in this case too):

∃x ∀y (x = y)

If we have predicate G that means "is good" then we can say ~~only~~ exactly one thing is good by writing:

∃x (Gx ∧ ∀y (Gy → y = x))

Btw, there's a notation for that, usually defined in terms of it:

∃!x Gx

Zanna

@EliahKagan I am not getting that one

Eliah Kagan

There exists some x such that:

x is good, and

for every y, if y is good, then y is the same thing as x.

So it's saying something is good, and all the things that are good are that thing. So exactly one thing is good.

Does that make sense?

Zanna

4:31 PM

yes

@EliahKagan what is that "!"?

Eliah Kagan

It's an exclamation point.

:)

Zanna

hahaha

Eliah Kagan

∃!x ... means "there exists a unique x such that ..."

If our system has constants (i.e., primitive names) like smith, it already carries a notion of uniqueness, since (in a standard logic) smith is taken to name exactly one thing. That's what lets you use it. The goal is to speak unambiguously, so many useful yet vexing properties of natural language are deliberately avoided in formal logic.

Having function symbols takes this a big step further.

Zanna

@EliahKagan that is good, in that case

Eliah Kagan

Yes.

I believe some people to use a logic that doesn't supply identity but that does have function symbols.

I don't think anything prohibits it.

Zanna

4:37 PM

we had to define = by saying that if two sets are equal they have all the same things in them, and they are in all the same things

Eliah Kagan

@Zanna If one's underlying logic does not supply "=" and one does set theory, then yes, one does define "=" to mean "has all the same element as." (That equal sets are elements of all the same sets is not part of the definition of "=" though; instead, it is introduced as an axiom.)

But if one's underlying logic does supply "=" then one doesn't do that.

Zanna

@EliahKagan yes

Eliah Kagan

I think there are good conceptual reasons to prefer to use an underlying logic with a primitive "=" and axioms of identity. However, there's nothing wrong with using one that does not as the basis for set theory. But there is typically just one primitive, then: "∈"

And "∈" is a predicate. x ∈ y expresses "x is a member of y".

Suppose you threw in a function symbol f. Let's take that to mean "the favorite set of." Suppose for simplicity that every set has some set that is its favorite set. (Our universe of discourse is sets, so I could have said "object" or "thing" instead of "set.")

Zanna

it's alright for y to have only one member...

Eliah Kagan

@Zanna Is that a question, or...?

Zanna

4:43 PM

thinking aloud

Eliah Kagan

x ∈ y does not express that y has only one member. It merely expresses that it has x as a member. There are sets with only one member though. They are called singleton sets, or singletons.

Zanna

@EliahKagan I realise that it does not mean that

Eliah Kagan

∃!x (x ∈ y) would express that y has exactly one member, though.

@Zanna Ah, okay. Sorry.

Zanna

I was trying to think about this

Eliah Kagan

Oh. Well, you can express that there's only one thing in set theory. But it is not true in any set theory.

Zanna

4:45 PM

@EliahKagan no apology needed :) I am just being slow

@EliahKagan that notation is useful...

Eliah Kagan

@EliahKagan If you've built your set theory on an underlying logic that supplies identity, just express it by saying there's something everything is equal to. Otherwise, express it by saying everything has all the same members and is a member of all the same sets as everything else. However, one would not in practice ever consider something a set theory if it permitted there to be one thing. That's way too small an ontology to be useful for the purposes set theories are aimed at. :)

@Zanna Yes. If you don't have it, you can write: ∃x (x ∈ y ∧ ∀w (w ∈ y → w = x))

I should say, in set theory one develops more notation to make things convenient. Usually in practice one would express that a y has exactly one member by writing: |y| = 1

Zanna

hahaha

Eliah Kagan

@EliahKagan I meant if it only permitted there to be one thing.

So, suppose fx means "x's favorite thing" (so, for example, and imagining these are in our system's universe of discourse, perhaps the set of integers's favorite thing is the set of telegrams).

Zanna

:)

Eliah Kagan

Suppose our set theory has only one predicate, "∈", but has a function symbol, "f". Suppose further that our set theory is built on an underlying logic that does not supply a notion of identity. Then we define "=" to mean "has the same members as" and we introduce an axiom that says equal things are members of the same things. Usually this is sufficient to make "=" work in the expected way, but because of "f", it's not sufficient.

Zanna

4:59 PM

because it gives us some more information that isn't about only being in a set?

Eliah Kagan

Right.

You can always prove from that axiom, given two sets that have the same members (i.e., that are equal) that they are members of the same sets, and thus you cannot distinguish them by "∈" alone. But you cannot prove those equal sets have the same favorite set, i.e., it does not follow from x = y that fx = fy. Because though x = y by definition means x and y have the same members, perhaps some member of fx is not a member of fy.

Zanna

I can see that

Eliah Kagan

So, that's the caveat I was saying applies to what I said about how (to state it a bit differently), when "∈" is the only predicate, you can only express "∈"-claims.

It was sort of silly for me to go on this vast tangent for that.

But function symbols and definite descriptions were part of what I had eventually wanted to present.

Do they make sense?

Like, do function symbols make sense now?

Zanna

I think so

Eliah Kagan

How about definite descriptions?

Zanna

5:13 PM

@EliahKagan that makes sense I think, or at least is not surprising

@EliahKagan I seem to have some idea about that yes

Eliah Kagan

Cool. Would you be interested to try out a small example?

Zanna

yes definitely!

Eliah Kagan

Excellent--what example would you like to try?

Zanna

oh you want me to make an example?

Eliah Kagan

You don't have to.

But yes.

Like, what sort of relationship between things do you want to use for it?

Here's an example of an example... :)

...in the US, every state has a state bird.

Zanna

5:21 PM

:D

@EliahKagan oooh how nice

Eliah Kagan

At least I think so.

Zanna

was that the whole example example?

Eliah Kagan

In a sense, yes, but I had not intended to stop there. :)

Zanna

oh haha I didn't want you to wait for me while I waited for you

Eliah Kagan

@Zanna Yes, deadlocks are best avoided.

Zanna

5:25 PM

haha :)

Eliah Kagan

So one could introduce a binary predicate B where Bxy means "x has y as a state bird". One could then express, using just B and =, various claims about state birds. Some of those claims might be more convenient to express using a unary function symbol f where fx means "the unique state bird of x", or with definite descriptions.

Also, if you like the state bird example, you could use that. But I urge you to come up with something like B and b (I suggest not calling them B and b, to avoid confusing in this conversation) even if you prefer to talk about state birds instead, because there is value in making a language.

(These are very simple languages, so making one is not impressive, but it is valuable.)

@EliahKagan To be clear, if you prefer to talk about state birds instead of whatever you come up with, you should do that! I'm recommending you come up with an example even if that is not then the example you proceed to try out.

Zanna

maybe it could be possible to talk about which spices are included in particular dishes

Eliah Kagan

5:40 PM

I observe that your phrasing is very tentative. :)

Yes, that can be expressed.

How would you like to express it?

Zanna

I'm being distracted by a game show my mum is watching and I'm not able to move myself away from it because this is the only bearably warm room in the house

in your example the state bird is a unique thing

but there are quite a lot of spices

Eliah Kagan

The example doesn't require that the state bird be unique, though.

And I am not totally sure every US state has just one state bird.

Also, there are some things in my universe of discourse that presumably don't have state birds.

Do you see why I say this?

@Zanna Is it at least a good game show?

Zanna

@EliahKagan I don't think so

Eliah Kagan

@Zanna New York's state bird is the bluebird. To express this with my B predicate, I must say something like Bxy with New York plugged in for x and the bluebird plugged in for y. So New York and the bluebird must be in my system's universe of discourse.

Zanna

@EliahKagan I wouldn't choose to watch this game show if the alternative was sitting in the room with nothing to do except think, but it's not so bad that it would be annoying if I were not trying to do something else while it's on

@EliahKagan so birds don't have state birds

Eliah Kagan

5:50 PM

It intuitively seems they would not.

Zanna

hahahaha

I thought of that

Eliah Kagan

On the other hand, what basis do we have to insist that no bird is the same thing as a US state?

Like, what if the bluebird, which is the state bird of New York, is also the same thing as Delaware?

Even supposing my system is supposed to accord with reality, what fact would you point to, to cast doubt on this remarkable conjecture?

Suppose I have a specific living being that is a bluebird. For clarity, I'll name it: call this particular bluebird Cassidy.

Zanna

ok XD

Eliah Kagan

One might insist that Cassidy is not Delaware because, even though Delaware is a very abstract idea, we nonetheless can speak of where it is, and we can find that it's not in all the same places Cassidy is in.

Zanna

although Cassidy might be in it

Eliah Kagan

5:56 PM

Yes.

I might try to rebut this by insisting my system is not powerful enough to express claims about where things are. But most people would say this means that, whiile my system may have some interpretations in which Cassidy is Delaware, those interpretations are really assigning different, unusual meanings to the system's primitives.

But the bluebird is extremely abstract. It's not obvious that it is in any particular place or what it means to say it is.

We might say that if the bluebird and Delaware are the same thing, then this is another case where primitives of my system (like "is the state bird of") are being given unusual meanings.

But mightn't we just as well say that if the bluebird and Delaware are not the same thing, then this is another case where primitives of my system (like "is the state bird of") are being given unusual meanings?

Because then one is using the system in a way that allows them to be distinguished.

Relatedly, is it false to say that "3 meters = 9 kilograms"?

Zanna

@EliahKagan yes?

Eliah Kagan

I have no objection to that answer, but it is not obvious to me that it is so.

Zanna

we might have three metres of very heavy string

Eliah Kagan

I agree that having 3 meters of string that mass 9 kilograms does not make 3 meters = 9 kilograms.

Zanna

hahahaha

Eliah Kagan

6:06 PM

Is x = y always false when x is a length and y is a mass?

Zanna

how do we find out whether 3 metres = 9 kilograms?

@EliahKagan yes?

Eliah Kagan

@Zanna I don't think there is a way to figure that out, because I don't think it's physically meaningful. But you seem to think there is a way to find out, because you're expressing the view that it is false that 3 meters = 9 kilograms.

@Zanna How about when x is a length and y is a time?

Zanna

@EliahKagan then again we can't compare them

@EliahKagan I don't think there is a way to find out but I think things being the same as each other is very rare and special

Eliah Kagan

@Zanna I hope I am not becoming tiresome here, asking multiple questions of the same form... but do you have the same view of mass and energy?

Zanna

for mass and energy we have got a conversion factor

Eliah Kagan

6:15 PM

That also works with length and time.

Before special relativity, I don't think it was known how conceptually deep it was that people often measured length in units of time anyway (e.g., "My office is only ten minutes away from home.").

Zanna

oh, ok, yes...

Eliah Kagan

Suppose we permit units of mass to be used to represent energies and so forth, so we say a particular energy is equal to the corresponding mass times c^2.

Then c = 1.

Zanna

@EliahKagan no no, only I hope I am not being tiresome by not understanding

Eliah Kagan

I don't agree you're being tiresome. Also, I'm not sure I agree that you're not understanding.

Zanna

@EliahKagan ok...

Eliah Kagan

6:22 PM

I'm not saying it's a profound fact of the world or that c = 1 (though some people do seem to think so). I don't think that expresses a claim about the world that can be readily checked. But it's a reasonable convention to adopt under some circumstances.

The thing that's neat about dimensional analysis is that it is useful for expressing deep insights that are experimentally verifiable, but also it is a choice.

Zanna

@EliahKagan it seems like we could make some uses of it

I've just realised that we are going to be late for dance class

so I am running off

I will be back tomorrow

Eliah Kagan

@Zanna People sometimes go much farther.

Zanna

thank you for your time and awesome help

Eliah Kagan

@Zanna Cool--ttyl!

Zanna

^_^

Eliah Kagan

6:24 PM

@Zanna I feel like I promised set theory and talked about logic instead. :(

But I think there are some advantages of having some first-order logic explicitly for set theory.

(As described thus far, I have not committed to first-order logic, and higher order logics could be used, but I figured I'd mention first-order logic in case you felt like searching for something. The systems in which set theories are developed--and also the most popular formal logical systems in general--are first-order systems.)

I want to ask you about what kind of dance the class is about... but I also want to not keep you from getting to the dance class... so I will try to remember to ask you about that in the future.

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