Expressions logically equivalent to "such that"

Question

The question arises from the rewording of some theorems, by example continuity of a function $f$, we have that

$$f\text{ is continuous at }c\iff (\forall\epsilon>0,\exists\delta>0:|x-c|<\delta\implies|f(x)-f(c)|<\epsilon)\tag{1}$$

then I think I can reword this as

$$f\text{ is continuous at }c\iff ((|x-c|<\delta\implies|f(x)-f(c)|<\epsilon)\implies (\forall\epsilon>0,\exists\delta>0))$$

The last statement seem strange to read but the last implication seems logically equivalent to $(1)$. Anyway this is just an experiment.

My question is, there is some expression logically equivalent to "such that" that can be written with basic logical operators as $\lor$, $\land$, $\lnot$, $\iff$ and $\implies$? Thank you in advance.

Answer 1

No, you can't do that; that's not how quantifiers (or conditionals) work.

"$\forall \epsilon>0$" just means, "For all $\epsilon>0$." It doesn't assert anything - e.g. it doesn't make sense to say ""$\forall \epsilon>0$" is false". Similarly with "$\exists \delta>0$."

So an expression of the form "...$\implies (\forall \epsilon>0,\exists \delta>0)$" is meaningless: "$\implies$" connects two sentences.

The issues go further than this: quantifiers bind variables. The expression "$\forall x(x>5)$" is a sentence; the expression "$x>5$" is not, and the expression "$(x>5)\implies (\forall x)$" is really not.

Answer 2

Abstractions. The closest thing to "such that" is a lambda operator or an abstraction. You can actually define a quantifier using an abstraction operator and property of attributes being universal or non-empty. See details in Fitch's "Symbolic Logic: An Introduction". Lambda operator can be defined using combinators. See details in Fitch's "Elements of Combinatory Logic"

Examples.

1. Abstraction or lambda operator. Suppose you have statement "sky is blue and transparent" you can also can paraphrase this like "sky has properties of being blue and transparent", you can paraphrase this property like "(x\x is blue and transparent)". Now you can say that "sky is blue and transparent" is logically equivalent to "(x\x is blue and transparent)""sky". Here used convention that we write property first and argument second. So we can write "(x\ x is red)""ball", this is logically equivalent to "ball is red".

2. Universal properties. Lets consider statement "(x\x=x)", that is property of being equal to itself. This property has property of being universal. We can write this like "A(x\x=x)".

3. Existential properties. Lets consider property of being equal to five "(x\x=5)". This property has property of being non-empty, that is of having some objects that has this property. We can write this like "E(x\x=5)".

4. Quantifiers in terms of abstraction and universal or existential attributes."A(x\x=x)" is equivalent to $\forall x x=x$. "E(x\x=5)" is equivalent to $\exists x x=5$. More generally "A(x\P(x))" is equivalent with $\forall x P(x)$. "E(x\Q(x))" is equivalent to $\exists x Q(x)$.

5. Abstraction or lambda operator in terms of combinators. We have analyzed quantifiers by dividing them into universal or non-empty attributes and abstraction operator. We can also analyze abstraction operator too. It can be shown that every (x\ $\phi(x)$) where $\phi(x)$ is some logical expression can be expressed using this expression and operators called combinators. Basically, suppose you have a function of several variables. Using combinators you can obtain same function but only variable places are changed or some arguments are missing or doubbled. There are several combinators: I, C, K, T, W.

I P(x,y,z) is equivalent P(x,y,z).

C P(x,y,z) is equivalent to P(x,z,y).

K P(x,y,z) is equivalent to P(x,z).

T P(x,y,z) is equivalent to P(y,x,z).

W P(x,y,z) is equivalent to P(x,y,y,z).

By applying this operators to formula you can compute abstraction of it.

6. Be cautious combinatory logic is incompatible with all the rules of classical logic.

Descriptions. Another thing that is very similar to "such that" is definite description operator $℩$. I found following information in Anderson's and Johnstone's book "Natural Deduction".

Examples.

1. Imagine you have a attribute of variable (we call it predicate) $\psi(x)$. It may be $x^2=0$. Expression $℩x(\psi(x))$ denotes "the x that has property $\psi(x)$". So $℩x(x^2=0)$ is equal to 0.
2. $℩$ operator can be expressed in terms of predicate logic. Expression $n=℩\alpha(\psi(\alpha))$ is equivalent to $\exists \alpha [\forall \beta [\psi(\beta)\equiv(\alpha=\beta)]\land(\alpha=n)]$.
3. Expression $\phi[℩\alpha(\psi(\alpha))]$ is equivalent to $\exists \alpha [\forall \beta [\psi(\beta)\equiv(\alpha=\beta)]\land(\phi(\alpha)]$.
4. Lets us assume we have some function $f(x)$. We want to express statement "the x such that f(x) is continuous". We may want to have this expression to substitute it then in some other formula. This expression can be written $℩x(\lim_{x \to c} f(x)=f(c))$
5. Be cautious. For descriptions identity relation may not be reflexive, symmetric or transitive.