Predicate logic, pls help

[Naoko]

Hi,
I have difficulty symbolizing sentences from English to first order logic (FOL) language with quantifiers and predicates. Could someone please help me symbolize these sentences with some expanations?


The black cat is happy .



Lara ate exactly two apples.


Every city is either smaller than London or polluted.

London is not a polluted city.

Since everyone admires someone, there is definitely someone who also admires John.

 

[pka]

I have difficulty symbolizing sentences from English to first order logic (FOL) language with quantifiers and predicates. Could someone please help me symbolize these sentences with some expanations?
The black cat is happy .
Lara ate exactly two apples.
Every city is either smaller than London or polluted.
London is not a polluted city.
Since everyone admires someone, there is definitely someone who also admires John.

Naoko, we have no access to your notes/textbook. So we do not know what symbols you use.
How would translate \(\bf\forall\forall\exists\exists~?\)

 

[Naoko]

Here`s the example how do we symbolise. As im studyng humanities, its difficult for me, but logic course is mandatory.
∀ - general quantifier, ∃- identifying quatinfier.


Here's an example for the first one:
Our domain of discourse is the set of all people.
Let Janet be represented by jj.
Let T(x,y)T(x,y) be the symbol for xx trusts yy.
We have a couple of statements like "Everyone" and "anyone", so there's probably going to be some ∀∀'s in here. First, let's try to encode "Everyone trusts anyone". This could be done with

∀x∀yT(x,y).∀x∀yT(x,y).

Now we should add in the "except Janet" part. The end result will be something like

∀x(x≠j→∀yT(x,y)).∀x(x≠j→∀yT(x,y)).


And this is where I hate these kinds of questions. It may be perfectly valid to stop there, but someone who reads further into the sentence would notice that "except Janet" seems to imply that there's somebody Janet doesn't trust. So you may want to add that in:

∀x(x≠j→∀yT(x,y))∧∃x(¬T(j,x)).

 

[pka]

Here`s the example how do we symbolise. As im studyng humanities, its difficult for me, but logic course is mandatory.
∀ - general quantifier, ∃- identifying quatinfier.

Since everyone admires someone, there is definitely someone who also admires John.
Let's use \(\mathscr{A}(x,y)\) to mean that \(x\text{ admires }y\)
\((\forall x)(\exists y) \left[\mathscr{A}(x,y) \Rightarrow (\exists z)[\mathscr{A}(z,\text{ John}]\right]\)

BTW \(\forall\forall\exists\exists\) is translated "for every upsidedown A there is a backwards E".

 

[HallsofIvy]

If "the black cat is happy" then there exist a cat that is black and happy.

Let "C(x)" represent the statement "x is a cat", let "B(x)" represent the statement "x is black" and let "H(x)" represent the statement "x is happy". The \(\displaystyle \exists x| C(x)\wedge B(x)\wedge H(x)\).

 

[thePhilosophyWizard]

Hi I have a question about translating into predicate logic.

• There is none like you, Lord; you are great, and your name is mighty in power. (L, G, M)

I need help translating this English sentence to a predicate logic system, thanks