Well formed formula Puns

Hello, I have trouble understanding why number 7 isnt a well formed formula. The one place predicate has another predicate inside it which has a constant and a constrained variable inside it. How is that not a well formed formula?

Thanks in advance!

Edit: forgot the image https://imgur.com/a/CYWRq0v

Translate "Everyone has a roommate who dislikes everyone."

Answer from the solution manual:

∀x∃y(R(x,y) ∧ ∀z¬L(y,z))

where R(x,y) = x has a roommate y and L(y,x) = y likes x.

My answer:

∀x∃y(R(x,y) ∧ ¬L(y,x)).

The only difference with mine is that I did not add the additional universal quantifier and bound variable ∀z. Do I need to add that? I thought that the first quantifier ∀x that is outside the parentheses should indicate that the "x" in "L(y,x)" is referring to everyone.

I was wondering whether or not a variable in a sentence of predicate logic needs to be attached to a predicate. My motivation for this is that I am trying to understand how to say something such as "I exist" or "I think therefore I am" in predicate logic. Since we have a quantifier for existence, it seems redundant or incorrect to try to have another predicate for existence. I am struggling to see how I would write something like "I think therefore I am" in PL. My initial attempt would be something like: ∃x(Tx -> x), but this appears to be incorrect to me. If I try to then express the earlier sentence "I am" in PL, it would then look something like: ∃xx; this, however, does not appear to me to be well formed. So I am not sure exactly how I am supposed to talk simply about something existing without ascribing a predicate.

If the stuff below doesn't make sense I also have it typed up here: http://mathb.in/27147

Well Formed Formula Questions:

Formula 1: An example of a formula that mixes propositions and predicates.

$\forall x (A \Rightarrow Bx)$ or something like $(A \Rightarrow Ba)$

I've seen formulas like this before, I was curious to know if mixing of propositions and predicates like this is normal or common? I know in at least one book that I've read this is acceptable but in other books it seems kind of vague.

Formula 2: A formula with a quantifier variable unrelated to the predicates variable.

$\forall x Ba$

I feel like there are several options for handling and interpreting a formula like this in plain english, in evaluating it as true or false, and in handling it in a derivation system. I was curious to know the most sensible and common ways to deal with it. It seems like quantifier rules would probably work fine if a formula like this is allowed.

Formula 3: A quantifier attached to a proposition rather than a predicate.

$\forall x B$

Formula 1 made me curious about this variation. I basically have the same questions for formula 2 as I do for this case, but I also wanted to know is B a predicate or a proposition here? How am I supposed to know the difference? It seems to me grammatically like a predicate is always a proposition but a proposition is not always a predicate.

How do things differ when you swap the universal quantifiers above for existential quantifiers (if it matters)?

Where can I read more about this stuff? Is this just linguistic/grammar issues?

Notation questions:

I've gotten some flak in the past for using the universal quantifier like $(x)Px$ and $(\forall x) Px$ instead of $\forall x Px$ as well as with the conditional using $A \supset B$ instead of $A \Rightarrow B$. I recognize that probably the more important issue is to be consistent with whichever notation I use but I wanted to get some clarification on what I should prefer or how to know which one I should pick?

Some Background Info:

The reason these questions came up for me is because I wrote a lisp program to spit out some first order practice arguments. The program works but I spent a good deal of time trying to decide if I should allow these cases as valid output.

This seems like an important semantic distinction to make.

I have been reading up recently on predicate logic and the concept of a "well-formed formula". I honestly have no idea how to wrap my head around it. As far as I know, a sentence is considered a well-formed formula right? Can someone well-versed in logic explain to me like I'm 5 about this concept if possible?

Why do we need the small supplementary "well-formed" when we say "formula" to indicate that the symbols are correctly arranged? When we say "sentence" regularly, we say just that (and assume no syntactical errors), so why do we not do that in PL?

No idea what I'm doing

Hi I am a fairly new user. I have set up an income/expense spreadsheet on Google Sheets that tracks my fixed month to month expenses, daily expenses and future bank balances needed on upcoming dates to pay upcoming monthly bills. I am looking for help in 2 areas. First how do I insert a new row for daily expenses that also pastes the formula, and is there a way to populate a certain cell with the expense using a drop down menu.

EDIT: thanks everyone for the helpful replies!

Not sure if this is the right steps to get to the answer.

Steps:

1. O has 8 electrons, but loses 2 of them
2. To balance out with Fe, Fe also needs 6 electrons.
3. Chemical formula is Fe2O3

Would I also use the same steps for this question? Predict the chemical formula for the ionic compound formed by Ca²⁺ and ClO₃⁻

Does it measure player activity in a certain place and center it around that area, or make sure it isn't centered around players that aren't moving so it won't favor them? I'm just curious.

i think europe is seeking authenticity. edit: i really don't think any language sounds ugly! i'm turkish, it's a phonetic mess but i love turkish songs as much as serbian or german.

Hi,

In my app I'm trying to retrieving json data from an API and displaying it as Text widgets. And I just found out some texts cannot be displayed such as the one I'm coming across like this one: î●sell♥potato 🙼⫷⫸⩎

Then the exception String is not well-formed utf-16 is throwed.

Is there anyway that I can ensure the decoded string from api is well-formed utf-16(removing/replacing those special emoji characters is fine) or is there anyway that I can turn it into well-formed utf-16?

Edit:

My code flow starts as getting json data from api:

dynamic res;
  try {
    var response = await http.get(Uri.parse(url));
    response.statusCode == HttpStatus.ok
        ? res = fromjson(response.body)
        : print('something wrong');
  } catch (exception) {}
  return res;

for json parsing, I used this website, and in my real code, it looks like this

So I used json.decode to to decode the json data, and the data is a list of maps(of type <Record>).

And the json url is here, where the error string is the map at index 54 of which the playerName key.

And it is that key value I'm trying to put inside a Text Widget, which causes the exception

The thought is, because ice formed under high pressure cannot optimize for hydrogen bonds (why it normally expands when frozen), that it would require less energy to undergo the phase transition