rationalskepticism.org

Dolorosa wrote:I was wondering if someone could help me interpret read the formulas in points 3(a),(b) and 4(a),(b) - https://faculty.washington.edu/smcohen/320/heracli.htm

3.(a) ∃x ∃F (Fx & Fbarx)
"There exists x such that there exists F such that Fx is true and Fbarx is true"
3.(b) ∀x ∀F (Fx & Fbarx)
"For all x and For all F Fx is true and Fbarx is true"
4.(a) ∀x ∃F (Fx & Fbarx)
"For all x there exists F such that Fx is true and Fbarx is true"
4.(b) ∀F ∃x (Fx & Fbarx)
"For all F there exists x such that Fx is true and Fbarx is true"

The text is ambiguous about how Fbar relates to F, it's intended to denote the "opposite" of F, which may or may not relate to the general logical concept of negation.

A bit clearer is that he intends the variable x to denote "things" and the variable F to denote "properties", so he's running through the four permutations of the quantifiers on X and F. So the suggestions become:
3.(a) Heraclitus believed that there is at least one object that has at least one property such that both the property and its opposite are true of that object.
3.(b) Heraclitus believed that for every object all its properties and their opposites hold true.
4.(a) Heraclitus believed for every object there is at least one property such that both the property and its opposite are true.
4.(b) Heraclitus believed that for every property there is at least one object such that the property and its opposite both hold true.

In terms of learning to read this kind of symbolic logic you'd need to find a course or primer on "first order logic". The plain text answer is more straightforward, but a bit of practice at what the implications of logic are is probably needed to really understand what's going on.

However, just to read, it's not that complicated. You go from left to right, adhering to the following conventions:
https://en.wikipedia.org/wiki/List_of_logic_symbols

• ∃ reads "there exists". This is a quantifier, which means there is at least one object in your domain which satisfies the following formulas.
• ∀ reads "for all". This is a quantifier, which means that every object in your domain (which can be empty - i.e. having zero things in it, in some logics) satisfies the following formulas.
• Lower case letters like x are just read "x", these typically denote members of the domain.
• Upper case letters like F are just read "F", these typically denote formulas, which philosophers (although not logicians) will often think of as properties. Note: Thinking of them as properties has no formal implications.
• ¬ reads "not". It's logical negation and is also sometimes written as ~. So ¬F would mean "not F", which simply reverses the truth value (in classical bivalent logic - bivalent means two truth values, "true" and "false", it can mean more subtle things in some more advanced logics that we should not worry about here) of F, i.e. it's false if F is true and true if F is false.

He also used an overline that I can't quickly replicate here, which I'd just read as "F bar", so I've reformatted it as Fbar here to make it naturally readable in that same way.